Analytic Approximate Solutions to the Boundary Layer Flow Equation over a Stretching Wall with Partial Slip at the Boundary

PubMed Central

Ene, Remus-Daniel; Marinca, Vasile; Marinca, Bogdan

2016-01-01

Analytic approximate solutions using Optimal Homotopy Perturbation Method (OHPM) are given for steady boundary layer flow over a nonlinearly stretching wall in presence of partial slip at the boundary. The governing equations are reduced to nonlinear ordinary differential equation by means of similarity transformations. Some examples are considered and the effects of different parameters are shown. OHPM is a very efficient procedure, ensuring a very rapid convergence of the solutions after only two iterations. PMID:27031232

Analytical solution for the diffusion of a capacitor discharge generated magnetic field pulse in a conductor

NASA Astrophysics Data System (ADS)

Grants, Ilmārs; Bojarevičs, Andris; Gerbeth, Gunter

2016-06-01

Powerful forces arise when a pulse of a magnetic field in the order of a few tesla diffuses into a conductor. Such pulses are used in electromagnetic forming, impact welding of dissimilar materials and grain refinement of solidifying alloys. Strong magnetic field pulses are generated by the discharge current of a capacitor bank. We consider analytically the penetration of such pulse into a conducting half-space. Besides the exact solution we obtain two simple self-similar approximate solutions for two sequential stages of the initial transient. Furthermore, a general solution is provided for the external field given as a power series of time. Each term of this solution represents a self-similar function for which we obtain an explicit expression. The validity range of various approximate analytical solutions is evaluated by comparison to the exact solution.

Analytical solutions for coupling fractional partial differential equations with Dirichlet boundary conditions

NASA Astrophysics Data System (ADS)

Ding, Xiao-Li; Nieto, Juan J.

2017-11-01

In this paper, we consider the analytical solutions of coupling fractional partial differential equations (FPDEs) with Dirichlet boundary conditions on a finite domain. Firstly, the method of successive approximations is used to obtain the analytical solutions of coupling multi-term time fractional ordinary differential equations. Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the coupling FPDEs to the coupling multi-term time fractional ordinary differential equations. By applying the obtained analytical solutions to the resulting multi-term time fractional ordinary differential equations, the desired analytical solutions of the coupling FPDEs are given. Our results are applied to derive the analytical solutions of some special cases to demonstrate their applicability.

Solution of the advection-dispersion equation: Continuous load of finite duration

USGS Publications Warehouse

Runkel, R.L.

1996-01-01

Field studies of solute fate and transport in streams and rivers often involve an. experimental release of solutes at an upstream boundary for a finite period of time. A review of several standard references on surface-water-quality modeling indicates that the analytical solution to the constant-parameter advection-dispersion equation for this type of boundary condition has been generally overlooked. Here an exact analytical solution that considers a continuous load of unite duration is compared to an approximate analytical solution presented elsewhere. Results indicate that the exact analytical solution should be used for verification of numerical solutions and other solute-transport problems wherein a high level of accuracy is required. ?? ASCE.

Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion

NASA Astrophysics Data System (ADS)

Zhou, Quanlin; Oldenburg, Curtis M.; Rutqvist, Jonny; Birkholzer, Jens T.

2017-11-01

There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1-D, 2-D, and 3-D blocks. These infinite-series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, td. In this paper, approximate solutions were developed by combining the error-function-series solutions for early times and the exponential-series solutions for late times and by using time partitioning at the switchover time, td0. The combined solutions contain either the leading term of both series for normal-accuracy approximations (with less than 0.003 relative error) or the first two terms for high-accuracy approximations (with less than 10-7 relative error) for 1-D isotropic (spheres, cylinders, slabs) and 2-D/3-D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1-D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first-order heat/mass flux for 2-D/3-D rectangular blocks were derived for normal-accuracy approximations. These flux equations contain the early-time solution with a three-term polynomial in √td and the late-time solution with the limited-term exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of low-permeability blocks of varying shapes and sizes.

Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhou, Quanlin; Oldenburg, Curtis M.; Rutqvist, Jonny

There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1D, 2D, and 3D blocks. These infinite-series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, t d0. In this paper, approximate solutions were developed by combining the error-function-series solutions for early times and the exponential-series solutions for late times and by using time partitioning at the switchover time, t d0. The combined solutions contain either the leading term of both series for normal-accuracy approximations (with less than 0.003 relative error) or the firstmore » two terms for high-accuracy approximations (with less than 10-7 relative error) for 1D isotropic (spheres, cylinders, slabs) and 2D/3D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first-order heat/mass flux for 2D/3D rectangular blocks were derived for normal-accuracy approximations. These flux equations contain the early-time solution with a three-term polynomial in √td and the late-time solution with the limited-term exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of low-permeability blocks of varying shapes and sizes.« less

Revisiting the Fundamental Analytical Solutions of Heat and Mass Transfer: The Kernel of Multirate and Multidimensional Diffusion

DOE PAGES

Zhou, Quanlin; Oldenburg, Curtis M.; Rutqvist, Jonny; ...

2017-10-24

There are two types of analytical solutions of temperature/concentration in and heat/mass transfer through boundaries of regularly shaped 1D, 2D, and 3D blocks. These infinite-series solutions with either error functions or exponentials exhibit highly irregular but complementary convergence at different dimensionless times, t d0. In this paper, approximate solutions were developed by combining the error-function-series solutions for early times and the exponential-series solutions for late times and by using time partitioning at the switchover time, t d0. The combined solutions contain either the leading term of both series for normal-accuracy approximations (with less than 0.003 relative error) or the firstmore » two terms for high-accuracy approximations (with less than 10-7 relative error) for 1D isotropic (spheres, cylinders, slabs) and 2D/3D rectangular blocks (squares, cubes, rectangles, and rectangular parallelepipeds). This rapid and uniform convergence for rectangular blocks was achieved by employing the same time partitioning with individual dimensionless times for different directions and the product of their combined 1D slab solutions. The switchover dimensionless time was determined to minimize the maximum approximation errors. Furthermore, the analytical solutions of first-order heat/mass flux for 2D/3D rectangular blocks were derived for normal-accuracy approximations. These flux equations contain the early-time solution with a three-term polynomial in √td and the late-time solution with the limited-term exponentials for rectangular blocks. The heat/mass flux equations and the combined temperature/concentration solutions form the ultimate kernel for fast simulations of multirate and multidimensional heat/mass transfer in porous/fractured media with millions of low-permeability blocks of varying shapes and sizes.« less

An Analytical Solution for Transient Thermal Response of an Insulated Structure

NASA Technical Reports Server (NTRS)

Blosser, Max L.

2012-01-01

An analytical solution was derived for the transient response of an insulated aerospace vehicle structure subjected to a simplified heat pulse. This simplified problem approximates the thermal response of a thermal protection system of an atmospheric entry vehicle. The exact analytical solution is solely a function of two non-dimensional parameters. A simpler function of these two parameters was developed to approximate the maximum structural temperature over a wide range of parameter values. Techniques were developed to choose constant, effective properties to represent the relevant temperature and pressure-dependent properties for the insulator and structure. A technique was also developed to map a time-varying surface temperature history to an equivalent square heat pulse. Using these techniques, the maximum structural temperature rise was calculated using the analytical solutions and shown to typically agree with finite element simulations within 10 to 20 percent over the relevant range of parameters studied.

Error analysis of analytic solutions for self-excited near-symmetric rigid bodies - A numerical study

NASA Technical Reports Server (NTRS)

Kia, T.; Longuski, J. M.

1984-01-01

Analytic error bounds are presented for the solutions of approximate models for self-excited near-symmetric rigid bodies. The error bounds are developed for analytic solutions to Euler's equations of motion. The results are applied to obtain a simplified analytic solution for Eulerian rates and angles. The results of a sample application of the range and error bound expressions for the case of the Galileo spacecraft experiencing transverse torques demonstrate the use of the bounds in analyses of rigid body spin change maneuvers.

On the first crossing distributions in fractional Brownian motion and the mass function of dark matter haloes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hiotelis, Nicos; Popolo, Antonino Del, E-mail: adelpopolo@oact.inaf.it, E-mail: hiotelis@ipta.demokritos.gr

We construct an integral equation for the first crossing distributions for fractional Brownian motion in the case of a constant barrier and we present an exact analytical solution. Additionally we present first crossing distributions derived by simulating paths from fractional Brownian motion. We compare the results of the analytical solutions with both those of simulations and those of some approximated solutions which have been used in the literature. Finally, we present multiplicity functions for dark matter structures resulting from our analytical approach and we compare with those resulting from N-body simulations. We show that the results of analytical solutions aremore » in good agreement with those of path simulations but differ significantly from those derived from approximated solutions. Additionally, multiplicity functions derived from fractional Brownian motion are poor fits of the those which result from N-body simulations. We also present comparisons with other models which are exist in the literature and we discuss different ways of improving the agreement between analytical results and N-body simulations.« less

Approximate solutions for diffusive fracture-matrix transfer: Application to storage of dissolved CO 2 in fractured rocks

DOE PAGES

Zhou, Quanlin; Oldenburg, Curtis M.; Spangler, Lee H.; ...

2017-01-05

Analytical solutions with infinite exponential series are available to calculate the rate of diffusive transfer between low-permeability blocks and high-permeability zones in the subsurface. Truncation of these series is often employed by neglecting the early-time regime. Here in this paper, we present unified-form approximate solutions in which the early-time and the late-time solutions are continuous at a switchover time. The early-time solutions are based on three-term polynomial functions in terms of square root of dimensionless time, with the first coefficient dependent only on the dimensionless area-to-volume ratio. The last two coefficients are either determined analytically for isotropic blocks (e.g., spheresmore » and slabs) or obtained by fitting the exact solutions, and they solely depend on the aspect ratios for rectangular columns and parallelepipeds. For the late-time solutions, only the leading exponential term is needed for isotropic blocks, while a few additional exponential terms are needed for highly anisotropic rectangular blocks. The optimal switchover time is between 0.157 and 0.229, with highest relative approximation error less than 0.2%. The solutions are used to demonstrate the storage of dissolved CO 2 in fractured reservoirs with low-permeability matrix blocks of single and multiple shapes and sizes. These approximate solutions are building blocks for development of analytical and numerical tools for hydraulic, solute, and thermal diffusion processes in low-permeability matrix blocks.« less

An analytical model for solute transport in an infiltration tracer test in soil with a shallow groundwater table

NASA Astrophysics Data System (ADS)

Liang, Ching-Ping; Hsu, Shao-Yiu; Chen, Jui-Sheng

2016-09-01

It is recommended that an in-situ infiltration tracer test is considered for simultaneously determining the longitudinal and transverse dispersion coefficients in soil. Analytical solutions have been derived for two-dimensional advective-dispersive transport in a radial geometry in the literature which can be used for interpreting the result of such a tracer test. However, these solutions were developed for a transport domain with an unbounded-radial extent and an infinite thickness of vadose zone which might not be realistically manifested in the actual solute transport during a field infiltration tracer test. Especially, the assumption of infinite thickness of vadose zone should be invalid for infiltration tracer tests conducted in soil with a shallow groundwater table. This paper describes an analytical model for interpreting the results of an infiltration tracer test based on improving the transport domain with a bounded-radial extent and a finite thickness of vadose zone. The analytical model is obtained with the successive application of appropriate integral transforms and their corresponding inverse transforms. A comparison of the newly derived analytical solution against the previous analytical solutions in which two distinct sets of radial extent and thickness of vadose zone are considered is conducted to determine the influence of the radial and exit boundary conditions on the solute transport. The results shows that both the radial and exit boundary conditions substantially affect the trailing segment of the breakthrough curves for a soil medium with large dispersion coefficients. Previous solutions derived for a transport domain with an unbounded-radial and an infinite thickness of vadose zone boundary conditions give lower concentration predictions compared with the proposed solution at late times. Moreover, the differences between two solutions are amplified when the observation positions are near the groundwater table. In addition, we compare our solution against the approximate solutions that derived from the previous analytical solution and has been suggested to serve as fast tools for simultaneously estimating the longitudinal and transverse dispersion coefficients. The results indicate that the approximate solutions offer predictions that are markedly distinct from our solution for the entire range of dispersion coefficient values. Thus, it is not appropriate to use the approximate solution for interpreting the results of an infiltration tracer test.

Analytic approximations to the modon dispersion relation. [in oceanography

NASA Technical Reports Server (NTRS)

Boyd, J. P.

1981-01-01

Three explicit analytic approximations are given to the modon dispersion relation developed by Flierl et al. (1980) to describe Gulf Stream rings and related phenomena in the oceans and atmosphere. The solutions are in the form of k(q), and are developed in the form of a power series in q for small q, an inverse power series in 1/q for large q, and a two-point Pade approximant. The low order Pade approximant is shown to yield a solution for the dispersion relation with a maximum relative error for the lowest branch of the function equal to one in 700 in the q interval zero to infinity.

Nonlinear Schroedinger Approximations for Partial Differential Equations with Quadratic and Quasilinear Terms

NASA Astrophysics Data System (ADS)

Cummings, Patrick

We consider the approximation of solutions of two complicated, physical systems via the nonlinear Schrodinger equation (NLS). In particular, we discuss the evolution of wave packets and long waves in two physical models. Due to the complicated nature of the equations governing many physical systems and the in-depth knowledge we have for solutions of the nonlinear Schrodinger equation, it is advantageous to use approximation results of this kind to model these physical systems. The approximations are simple enough that we can use them to understand the qualitative and quantitative behavior of the solutions, and by justifying them we can show that the behavior of the approximation captures the behavior of solutions to the original equation, at least for long, but finite time. We first consider a model of the water wave equations which can be approximated by wave packets using the NLS equation. We discuss a new proof that both simplifies and strengthens previous justification results of Schneider and Wayne. Rather than using analytic norms, as was done by Schneider and Wayne, we construct a modified energy functional so that the approximation holds for the full interval of existence of the approximate NLS solution as opposed to a subinterval (as is seen in the analytic case). Furthermore, the proof avoids problems associated with inverting the normal form transform by working with a modified energy functional motivated by Craig and Hunter et al. We then consider the Klein-Gordon-Zakharov system and prove a long wave approximation result. In this case there is a non-trivial resonance that cannot be eliminated via a normal form transform. By combining the normal form transform for small Fourier modes and using analytic norms elsewhere, we can get a justification result on the order 1 over epsilon squared time scale.

Soliton polarization rotation in fiber lasers

NASA Astrophysics Data System (ADS)

Afanasjev, V. V.

1995-02-01

I have found the approximate analytical solution in explicit form for a vector soliton with an arbitrary component ratio. My solution describes the dependence of soliton intensity on polarization angle and also nonlinear polarization rotation. The analytical results agree well with the numerical simulations.

Exact analytic solution for the spin-up maneuver of an axially symmetric spacecraft

NASA Astrophysics Data System (ADS)

Ventura, Jacopo; Romano, Marcello

2014-11-01

The problem of spinning-up an axially symmetric spacecraft subjected to an external torque constant in magnitude and parallel to the symmetry axis is considered. The existing exact analytic solution for an axially symmetric body is applied for the first time to this problem. The proposed solution is valid for any initial conditions of attitude and angular velocity and for any length of time and rotation amplitude. Furthermore, the proposed solution can be numerically evaluated up to any desired level of accuracy. Numerical experiments and comparison with an existing approximated solution and with the integration of the equations of motion are reported in the paper. Finally, a new approximated solution obtained from the exact one is introduced in this paper.

Calculation of Thermal Conductivity Coefficients of Electrons in Magnetized Dense Matter

NASA Astrophysics Data System (ADS)

Bisnovatyi-Kogan, G. S.; Glushikhina, M. V.

2018-04-01

The solution of Boltzmann equation for plasma in magnetic field with arbitrarily degenerate electrons and nondegenerate nuclei is obtained by Chapman-Enskog method. Functions generalizing Sonine polynomials are used for obtaining an approximate solution. Fully ionized plasma is considered. The tensor of the heat conductivity coefficients in nonquantized magnetic field is calculated. For nondegenerate and strongly degenerate plasma the asymptotic analytic formulas are obtained and compared with results of previous authors. The Lorentz approximation with neglecting of electron-electron encounters is asymptotically exact for strongly degenerate plasma. For the first time, analytical expressions for the heat conductivity tensor for nondegenerate electrons in the presence of a magnetic field are obtained in the three-polynomial approximation with account of electron-electron collisions. Account of the third polynomial improved substantially the precision of results. In the two-polynomial approximation, the obtained solution coincides with the published results. For strongly degenerate electrons, an asymptotically exact analytical solution for the heat conductivity tensor in the presence of a magnetic field is obtained for the first time. This solution has a considerably more complicated dependence on the magnetic field than those in previous publications and gives a several times smaller relative value of the thermal conductivity across the magnetic field at ωτ * 0.8.

Approximate Solution to the Angular Speeds of a Nearly-Symmetric Mass-Varying Cylindrical Body

NASA Astrophysics Data System (ADS)

Nanjangud, Angadh; Eke, Fidelis

2017-06-01

This paper examines the rotational motion of a nearly axisymmetric rocket type system with uniform burn of its propellant. The asymmetry comes from a slight difference in the transverse principal moments of inertia of the system, which then results in a set of nonlinear equations of motion even when no external torque is applied to the system. It is often difficult, or even impossible, to generate analytic solutions for such equations; closed form solutions are even more difficult to obtain. In this paper, a perturbation-based approach is employed to linearize the equations of motion and generate analytic solutions. The solutions for the variables of transverse motion are analytic and a closed-form solution to the spin rate is suggested. The solutions are presented in a compact form that permits rapid computation. The approximate solutions are then applied to the torque-free motion of a typical solid rocket system and the results are found to agree with those obtained from the numerical solution of the full non-linear equations of motion of the mass varying system.

A semi-analytical study of positive corona discharge in wire-plane electrode configuration

NASA Astrophysics Data System (ADS)

Yanallah, K.; Pontiga, F.; Chen, J. H.

2013-08-01

Wire-to-plane positive corona discharge in air has been studied using an analytical model of two species (electrons and positive ions). The spatial distributions of electric field and charged species are obtained by integrating Gauss's law and the continuity equations of species along the Laplacian field lines. The experimental values of corona current intensity and applied voltage, together with Warburg's law, have been used to formulate the boundary condition for the electron density on the corona wire. To test the accuracy of the model, the approximate electric field distribution has been compared with the exact numerical solution obtained from a finite element analysis. A parametrical study of wire-to-plane corona discharge has then been undertaken using the approximate semi-analytical solutions. Thus, the spatial distributions of electric field and charged particles have been computed for different values of the gas pressure, wire radius and electrode separation. Also, the two dimensional distribution of ozone density has been obtained using a simplified plasma chemistry model. The approximate semi-analytical solutions can be evaluated in a negligible computational time, yet provide precise estimates of corona discharge variables.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhou, Quanlin; Oldenburg, Curtis M.; Spangler, Lee H.

Analytical solutions with infinite exponential series are available to calculate the rate of diffusive transfer between low-permeability blocks and high-permeability zones in the subsurface. Truncation of these series is often employed by neglecting the early-time regime. Here in this paper, we present unified-form approximate solutions in which the early-time and the late-time solutions are continuous at a switchover time. The early-time solutions are based on three-term polynomial functions in terms of square root of dimensionless time, with the first coefficient dependent only on the dimensionless area-to-volume ratio. The last two coefficients are either determined analytically for isotropic blocks (e.g., spheresmore » and slabs) or obtained by fitting the exact solutions, and they solely depend on the aspect ratios for rectangular columns and parallelepipeds. For the late-time solutions, only the leading exponential term is needed for isotropic blocks, while a few additional exponential terms are needed for highly anisotropic rectangular blocks. The optimal switchover time is between 0.157 and 0.229, with highest relative approximation error less than 0.2%. The solutions are used to demonstrate the storage of dissolved CO 2 in fractured reservoirs with low-permeability matrix blocks of single and multiple shapes and sizes. These approximate solutions are building blocks for development of analytical and numerical tools for hydraulic, solute, and thermal diffusion processes in low-permeability matrix blocks.« less

Analytical approximations to the dynamics of an array of coupled DC SQUIDs

NASA Astrophysics Data System (ADS)

Berggren, Susan; Palacios, Antonio

2014-04-01

Coupled dynamical systems that operate near the onset of a bifurcation can lead, under certain conditions, to strong signal amplification effects. Over the past years we have studied this generic feature on a wide range of systems, including: magnetic and electric fields sensors, gyroscopic devices, and arrays of loops of superconducting quantum interference devices, also known as SQUIDs. In this work, we consider an array of SQUID loops connected in series as a case study to derive asymptotic analytical approximations to the exact solutions through perturbation analysis. Two approaches are considered. First, a straightforward expansion in which the non-linear parameter related to the inductance of the DC SQUID is treated as the small perturbation parameter. Second, a more accurate procedure that considers the SQUID phase dynamics as non-uniform motion on a circle. This second procedure is readily extended to the series array and it could serve as a mathematical framework to find approximate solutions to related complex systems with high-dimensionality. To the best of our knowledge, an approximate analytical solutions to an array of SQUIDs has not been reported yet in the literature.

Nuclear magnetic resonance signal dynamics of liquids in the presence of distant dipolar fields, revisited

PubMed Central

Barros, Wilson; Gochberg, Daniel F.; Gore, John C.

2009-01-01

The description of the nuclear magnetic resonance magnetization dynamics in the presence of long-range dipolar interactions, which is based upon approximate solutions of Bloch–Torrey equations including the effect of a distant dipolar field, has been revisited. New experiments show that approximate analytic solutions have a broader regime of validity as well as dependencies on pulse-sequence parameters that seem to have been overlooked. In order to explain these experimental results, we developed a new method consisting of calculating the magnetization via an iterative formalism where both diffusion and distant dipolar field contributions are treated as integral operators incorporated into the Bloch–Torrey equations. The solution can be organized as a perturbative series, whereby access to higher order terms allows one to set better boundaries on validity regimes for analytic first-order approximations. Finally, the method legitimizes the use of simple analytic first-order approximations under less demanding experimental conditions, it predicts new pulse-sequence parameter dependencies for the range of validity, and clarifies weak points in previous calculations. PMID:19425789

Analytical Derivation and Experimental Evaluation of Short-Bearing Approximation for Full Journal Bearing

NASA Technical Reports Server (NTRS)

Dubois, George B; Ocvirk, Fred W

1953-01-01

An approximate analytical solution including the effect of end leakage from the oil film of short plain bearings is presented because of the importance of endwise flow in sleeve bearings of the short lengths commonly used. The analytical approximation is supported by experimental data, resulting in charts which facilitate analysis of short plain bearings. The analytical approximation includes the endwise flow and that part of the circumferential flow which is related to surface velocity and film thickness but neglects the effect of film pressure on the circumferential flow. In practical use, this approximation applies best to bearings having a length-diameter ratio up to 1, and the effects of elastic deflection, inlet oil pressure, and changes of clearance with temperature minimize the relative importance of the neglected term. The analytical approximation was found to be an extension of a little-known pressure-distribution function originally proposed by Michell and Cardullo.

Solutions of the Dirac Equation with the Shifted DENG-FAN Potential Including Yukawa-Like Tensor Interaction

NASA Astrophysics Data System (ADS)

Yahya, W. A.; Falaye, B. J.; Oluwadare, O. J.; Oyewumi, K. J.

2013-08-01

By using the Nikiforov-Uvarov method, we give the approximate analytical solutions of the Dirac equation with the shifted Deng-Fan potential including the Yukawa-like tensor interaction under the spin and pseudospin symmetry conditions. After using an improved approximation scheme, we solved the resulting schr\\"{o}dinger-like equation analytically. Numerical results of the energy eigenvalues are also obtained, as expected, the tensor interaction removes degeneracies between spin and pseudospin doublets.

Approximate analytical solution for induction heating of solid cylinders

DOE PAGES

Jankowski, Todd Andrew; Pawley, Norma Helen; Gonzales, Lindsey Michal; ...

2015-10-20

An approximate solution to the mathematical model for induction heating of a solid cylinder in a cylindrical induction coil is presented here. The coupled multiphysics model includes equations describing the electromagnetic field in the heated object, a heat transfer simulation to determine temperature of the heated object, and an AC circuit simulation of the induction heating power supply. A multiple-scale perturbation method is used to solve the multiphysics model. The approximate analytical solution yields simple closed-form expressions for the electromagnetic field and heat generation rate in the solid cylinder, for the equivalent impedance of the associated tank circuit, and formore » the frequency response of a variable frequency power supply driving the tank circuit. The solution developed here is validated by comparing predicted power supply frequency to both experimental measurements and calculated values from finite element analysis for heating of graphite cylinders in an induction furnace. The simple expressions from the analytical solution clearly show the functional dependence of the power supply frequency on the material properties of the load and the geometrical characteristics of the furnace installation. In conclusion, the expressions developed here provide physical insight into observations made during load signature analysis of induction heating.« less

Interaction and charge transfer between dielectric spheres: Exact and approximate analytical solutions.

PubMed

Lindén, Fredrik; Cederquist, Henrik; Zettergren, Henning

2016-11-21

We present exact analytical solutions for charge transfer reactions between two arbitrarily charged hard dielectric spheres. These solutions, and the corresponding exact ones for sphere-sphere interaction energies, include sums that describe polarization effects to infinite orders in the inverse of the distance between the sphere centers. In addition, we show that these exact solutions may be approximated by much simpler analytical expressions that are useful for many practical applications. This is exemplified through calculations of Langevin type cross sections for forming a compound system of two colliding spheres and through calculations of electron transfer cross sections. We find that it is important to account for dielectric properties and finite sphere sizes in such calculations, which for example may be useful for describing the evolution, growth, and dynamics of nanometer sized dielectric objects such as molecular clusters or dust grains in different environments including astrophysical ones.

Analytic solution and pulse area theorem for three-level atoms

NASA Astrophysics Data System (ADS)

Shchedrin, Gavriil; O'Brien, Chris; Rostovtsev, Yuri; Scully, Marlan O.

2015-12-01

We report an analytic solution for a three-level atom driven by arbitrary time-dependent electromagnetic pulses. In particular, we consider far-detuned driving pulses and show an excellent match between our analytic result and the numerical simulations. We use our solution to derive a pulse area theorem for three-level V and Λ systems without making the rotating wave approximation. Formulated as an energy conservation law, this pulse area theorem can be used to understand pulse propagation through three-level media.

Approximate Solution of Time-Fractional Advection-Dispersion Equation via Fractional Variational Iteration Method

PubMed Central

İbiş, Birol

2014-01-01

This paper aims to obtain the approximate solution of time-fractional advection-dispersion equation (FADE) involving Jumarie's modification of Riemann-Liouville derivative by the fractional variational iteration method (FVIM). FVIM provides an analytical approximate solution in the form of a convergent series. Some examples are given and the results indicate that the FVIM is of high accuracy, more efficient, and more convenient for solving time FADEs. PMID:24578662

Full analytical solution of the bloch equation when using a hyperbolic-secant driving function.

PubMed

Zhang, Jinjin; Garwood, Michael; Park, Jang-Yeon

2017-04-01

The frequency-swept pulse known as the hyperbolic-secant (HS) pulse is popular in NMR for achieving adiabatic spin inversion. The HS pulse has also shown utility for achieving excitation and refocusing in gradient-echo and spin-echo sequences, including new ultrashort echo-time imaging (e.g., Sweep Imaging with Fourier Transform, SWIFT) and B 1 mapping techniques. To facilitate the analysis of these techniques, the complete theoretical solution of the Bloch equation, as driven by the HS pulse, was derived for an arbitrary state of initial magnetization. The solution of the Bloch-Riccati equation for transverse and longitudinal magnetization for an arbitrary initial state was derived analytically in terms of HS pulse parameters. The analytical solution was compared with the solutions using both the Runge-Kutta method and the small-tip approximation. The analytical solution was demonstrated on different initial states at different frequency offsets with/without a combination of HS pulses. Evolution of the transverse magnetization was influenced significantly by the choice of HS pulse parameters. The deviation of the magnitude of the transverse magnetization, as obtained by comparing the small-tip approximation to the analytical solution, was < 5% for flip angles < 30 °, but > 10% for the flip angles > 40 °. The derived analytical solution provides insights into the influence of HS pulse parameters on the magnetization evolution. Magn Reson Med 77:1630-1638, 2017. © 2016 International Society for Magnetic Resonance in Medicine. © 2016 International Society for Magnetic Resonance in Medicine.

From analytical solutions of solute transport equations to multidimensional time-domain random walk (TDRW) algorithms

NASA Astrophysics Data System (ADS)

Bodin, Jacques

2015-03-01

In this study, new multi-dimensional time-domain random walk (TDRW) algorithms are derived from approximate one-dimensional (1-D), two-dimensional (2-D), and three-dimensional (3-D) analytical solutions of the advection-dispersion equation and from exact 1-D, 2-D, and 3-D analytical solutions of the pure-diffusion equation. These algorithms enable the calculation of both the time required for a particle to travel a specified distance in a homogeneous medium and the mass recovery at the observation point, which may be incomplete due to 2-D or 3-D transverse dispersion or diffusion. The method is extended to heterogeneous media, represented as a piecewise collection of homogeneous media. The particle motion is then decomposed along a series of intermediate checkpoints located on the medium interface boundaries. The accuracy of the multi-dimensional TDRW method is verified against (i) exact analytical solutions of solute transport in homogeneous media and (ii) finite-difference simulations in a synthetic 2-D heterogeneous medium of simple geometry. The results demonstrate that the method is ideally suited to purely diffusive transport and to advection-dispersion transport problems dominated by advection. Conversely, the method is not recommended for highly dispersive transport problems because the accuracy of the advection-dispersion TDRW algorithms degrades rapidly for a low Péclet number, consistent with the accuracy limit of the approximate analytical solutions. The proposed approach provides a unified methodology for deriving multi-dimensional time-domain particle equations and may be applicable to other mathematical transport models, provided that appropriate analytical solutions are available.

Double power series method for approximating cosmological perturbations

NASA Astrophysics Data System (ADS)

Wren, Andrew J.; Malik, Karim A.

2017-04-01

We introduce a double power series method for finding approximate analytical solutions for systems of differential equations commonly found in cosmological perturbation theory. The method was set out, in a noncosmological context, by Feshchenko, Shkil' and Nikolenko (FSN) in 1966, and is applicable to cases where perturbations are on subhorizon scales. The FSN method is essentially an extension of the well known Wentzel-Kramers-Brillouin (WKB) method for finding approximate analytical solutions for ordinary differential equations. The FSN method we use is applicable well beyond perturbation theory to solve systems of ordinary differential equations, linear in the derivatives, that also depend on a small parameter, which here we take to be related to the inverse wave-number. We use the FSN method to find new approximate oscillating solutions in linear order cosmological perturbation theory for a flat radiation-matter universe. Together with this model's well-known growing and decaying Mészáros solutions, these oscillating modes provide a complete set of subhorizon approximations for the metric potential, radiation and matter perturbations. Comparison with numerical solutions of the perturbation equations shows that our approximations can be made accurate to within a typical error of 1%, or better. We also set out a heuristic method for error estimation. A Mathematica notebook which implements the double power series method is made available online.

Exact solutions to the fermion propagator Schwinger-Dyson equation in Minkowski space with on-shell renormalization for quenched QED

DOE PAGES

Jia, Shaoyang; Pennington, M. R.

2017-08-01

With the introduction of a spectral representation, the Schwinger-Dyson equation (SDE) for the fermion propagator is formulated in Minkowski space in QED. After imposing the on-shell renormalization conditions, analytic solutions for the fermion propagator spectral functions are obtained in four dimensions with a renormalizable version of the Gauge Technique anzatz for the fermion-photon vertex in the quenched approximation in the Landau gauge. Despite the limitations of this model, having an explicit solution provides a guiding example of the fermion propagator with the correct analytic structure. The Padé approximation for the spectral functions is also investigated.

Exact solutions to the fermion propagator Schwinger-Dyson equation in Minkowski space with on-shell renormalization for quenched QED

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jia, Shaoyang; Pennington, M. R.

With the introduction of a spectral representation, the Schwinger-Dyson equation (SDE) for the fermion propagator is formulated in Minkowski space in QED. After imposing the on-shell renormalization conditions, analytic solutions for the fermion propagator spectral functions are obtained in four dimensions with a renormalizable version of the Gauge Technique anzatz for the fermion-photon vertex in the quenched approximation in the Landau gauge. Despite the limitations of this model, having an explicit solution provides a guiding example of the fermion propagator with the correct analytic structure. The Padé approximation for the spectral functions is also investigated.

Simplified multiple scattering model for radiative transfer in turbid water

NASA Technical Reports Server (NTRS)

Ghovanlou, A. H.; Gupta, G. N.

1978-01-01

Quantitative analytical procedures for relating selected water quality parameters to the characteristics of the backscattered signals, measured by remote sensors, require the solution of the radiative transport equation in turbid media. Presented is an approximate closed form solution of this equation and based on this solution, the remote sensing of sediments is discussed. The results are compared with other standard closed form solutions such as quasi-single scattering approximations.

Optimal homotopy asymptotic method for flow and heat transfer of a viscoelastic fluid in an axisymmetric channel with a porous wall.

PubMed

Mabood, Fazle; Khan, Waqar A; Ismail, Ahmad Izani Md

2013-01-01

In this article, an approximate analytical solution of flow and heat transfer for a viscoelastic fluid in an axisymmetric channel with porous wall is presented. The solution is obtained through the use of a powerful method known as Optimal Homotopy Asymptotic Method (OHAM). We obtained the approximate analytical solution for dimensionless velocity and temperature for various parameters. The influence and effect of different parameters on dimensionless velocity, temperature, friction factor, and rate of heat transfer are presented graphically. We also compared our solution with those obtained by other methods and it is found that OHAM solution is better than the other methods considered. This shows that OHAM is reliable for use to solve strongly nonlinear problems in heat transfer phenomena.

Optimal Homotopy Asymptotic Method for Flow and Heat Transfer of a Viscoelastic Fluid in an Axisymmetric Channel with a Porous Wall

PubMed Central

Mabood, Fazle; Khan, Waqar A.; Ismail, Ahmad Izani

2013-01-01

In this article, an approximate analytical solution of flow and heat transfer for a viscoelastic fluid in an axisymmetric channel with porous wall is presented. The solution is obtained through the use of a powerful method known as Optimal Homotopy Asymptotic Method (OHAM). We obtained the approximate analytical solution for dimensionless velocity and temperature for various parameters. The influence and effect of different parameters on dimensionless velocity, temperature, friction factor, and rate of heat transfer are presented graphically. We also compared our solution with those obtained by other methods and it is found that OHAM solution is better than the other methods considered. This shows that OHAM is reliable for use to solve strongly nonlinear problems in heat transfer phenomena. PMID:24376722

Small-x asymptotics of the quark helicity distribution: Analytic results

DOE PAGES

Kovchegov, Yuri V.; Pitonyak, Daniel; Sievert, Matthew D.

2017-06-15

In this Letter, we analytically solve the evolution equations for the small-x asymptotic behavior of the (flavor singlet) quark helicity distribution in the large- N c limit. Here, these evolution equations form a set of coupled integro-differential equations, which previously could only be solved numerically. This approximate numerical solution, however, revealed simplifying properties of the small-x asymptotics, which we exploit here to obtain an analytic solution.

Semi-analytical solutions of the Schnakenberg model of a reaction-diffusion cell with feedback

NASA Astrophysics Data System (ADS)

Al Noufaey, K. S.

2018-06-01

This paper considers the application of a semi-analytical method to the Schnakenberg model of a reaction-diffusion cell. The semi-analytical method is based on the Galerkin method which approximates the original governing partial differential equations as a system of ordinary differential equations. Steady-state curves, bifurcation diagrams and the region of parameter space in which Hopf bifurcations occur are presented for semi-analytical solutions and the numerical solution. The effect of feedback control, via altering various concentrations in the boundary reservoirs in response to concentrations in the cell centre, is examined. It is shown that increasing the magnitude of feedback leads to destabilization of the system, whereas decreasing this parameter to negative values of large magnitude stabilizes the system. The semi-analytical solutions agree well with numerical solutions of the governing equations.

a Bounded Finite-Difference Discretization of a Two-Dimensional Diffusion Equation with Logistic Nonlinear Reaction

NASA Astrophysics Data System (ADS)

Macías-Díaz, J. E.

In the present manuscript, we introduce a finite-difference scheme to approximate solutions of the two-dimensional version of Fisher's equation from population dynamics, which is a model for which the existence of traveling-wave fronts bounded within (0,1) is a well-known fact. The method presented here is a nonstandard technique which, in the linear regime, approximates the solutions of the original model with a consistency of second order in space and first order in time. The theory of M-matrices is employed here in order to elucidate conditions under which the method is able to preserve the positivity and the boundedness of solutions. In fact, our main result establishes relatively flexible conditions under which the preservation of the positivity and the boundedness of new approximations is guaranteed. Some simulations of the propagation of a traveling-wave solution confirm the analytical results derived in this work; moreover, the experiments evince a good agreement between the numerical result and the analytical solutions.

Atmospheric guidance law for planar skip trajectories

NASA Technical Reports Server (NTRS)

Mease, K. D.; Mccreary, F. A.

1985-01-01

The applicability of an approximate, closed-form, analytical solution to the equations of motion, as a basis for a deterministic guidance law for controlling the in-plane motion during a skip trajectory, is investigated. The derivation of the solution by the method of matched asymptotic expansions is discussed. Specific issues that arise in the application of the solution to skip trajectories are addressed. Based on the solution, an explicit formula for the approximate energy loss due to an atmospheric pass is derived. A guidance strategy is proposed that illustrates the use of the approximate solution. A numerical example shows encouraging performance.

Analytical approximations for the oscillators with anti-symmetric quadratic nonlinearity

NASA Astrophysics Data System (ADS)

Alal Hosen, Md.; Chowdhury, M. S. H.; Yeakub Ali, Mohammad; Faris Ismail, Ahmad

2017-12-01

A second-order ordinary differential equation involving anti-symmetric quadratic nonlinearity changes sign. The behaviour of the oscillators with an anti-symmetric quadratic nonlinearity is assumed to oscillate different in the positive and negative directions. In this reason, Harmonic Balance Method (HBM) cannot be directly applied. The main purpose of the present paper is to propose an analytical approximation technique based on the HBM for obtaining approximate angular frequencies and the corresponding periodic solutions of the oscillators with anti-symmetric quadratic nonlinearity. After applying HBM, a set of complicated nonlinear algebraic equations is found. Analytical approach is not always fruitful for solving such kinds of nonlinear algebraic equations. In this article, two small parameters are found, for which the power series solution produces desired results. Moreover, the amplitude-frequency relationship has also been determined in a novel analytical way. The presented technique gives excellent results as compared with the corresponding numerical results and is better than the existing ones.

Padé approximant for normal stress differences in large-amplitude oscillatory shear flow

NASA Astrophysics Data System (ADS)

Poungthong, P.; Saengow, C.; Giacomin, A. J.; Kolitawong, C.; Merger, D.; Wilhelm, M.

2018-04-01

Analytical solutions for the normal stress differences in large-amplitude oscillatory shear flow (LAOS), for continuum or molecular models, normally take the inexact form of the first few terms of a series expansion in the shear rate amplitude. Here, we improve the accuracy of these truncated expansions by replacing them with rational functions called Padé approximants. The recent advent of exact solutions in LAOS presents an opportunity to identify accurate and useful Padé approximants. For this identification, we replace the truncated expansion for the corotational Jeffreys fluid with its Padé approximants for the normal stress differences. We uncover the most accurate and useful approximant, the [3,4] approximant, and then test its accuracy against the exact solution [C. Saengow and A. J. Giacomin, "Normal stress differences from Oldroyd 8-constant framework: Exact analytical solution for large-amplitude oscillatory shear flow," Phys. Fluids 29, 121601 (2017)]. We use Ewoldt grids to show the stunning accuracy of our [3,4] approximant in LAOS. We quantify this accuracy with an objective function and then map it onto the Pipkin space. Our two applications illustrate how to use our new approximant reliably. For this, we use the Spriggs relations to generalize our best approximant to multimode, and then, we compare with measurements on molten high-density polyethylene and on dissolved polyisobutylene in isobutylene oligomer.

Non-Schwarzschild black-hole metric in four dimensional higher derivative gravity: Analytical approximation

NASA Astrophysics Data System (ADS)

Kokkotas, K. D.; Konoplya, R. A.; Zhidenko, A.

2017-09-01

Higher derivative extensions of Einstein gravity are important within the string theory approach to gravity and as alternative and effective theories of gravity. H. Lü, A. Perkins, C. Pope, and K. Stelle [Phys. Rev. Lett. 114, 171601 (2015), 10.1103/PhysRevLett.114.171601] found a numerical solution describing a spherically symmetric non-Schwarzschild asymptotically flat black hole in Einstein gravity with added higher derivative terms. Using the general and quickly convergent parametrization in terms of the continued fractions, we represent this numerical solution in the analytical form, which is accurate not only near the event horizon or far from the black hole, but in the whole space. Thereby, the obtained analytical form of the metric allows one to study easily all the further properties of the black hole, such as thermodynamics, Hawking radiation, particle motion, accretion, perturbations, stability, quasinormal spectrum, etc. Thus, the found analytical approximate representation can serve in the same way as an exact solution.

Approximate analytical description of the elastic strain field due to an inclusion in a continuous medium with cubic anisotropy

NASA Astrophysics Data System (ADS)

Nenashev, A. V.; Koshkarev, A. A.; Dvurechenskii, A. V.

2018-03-01

We suggest an approach to the analytical calculation of the strain distribution due to an inclusion in elastically anisotropic media for the case of cubic anisotropy. The idea consists in the approximate reduction of the anisotropic problem to a (simpler) isotropic problem. This gives, for typical semiconductors, an improvement in accuracy by an order of magnitude, compared to the isotropic approximation. Our method allows using, in the case of elastically anisotropic media, analytical solutions obtained for isotropic media only, such as analytical formulas for the strain due to polyhedral inclusions. The present work substantially extends the applicability of analytical results, making them more suitable for describing real systems, such as epitaxial quantum dots.

Fall with Linear Drag and Wien's Displacement Law: Approximate Solution and Lambert Function

ERIC Educational Resources Information Center

Vial, Alexandre

2012-01-01

We present an approximate solution for the downward time of travel in the case of a mass falling with a linear drag force. We show how a quasi-analytical solution implying the Lambert function can be found. We also show that solving the previous problem is equivalent to the search for Wien's displacement law. These results can be of interest for…

Convergence and approximate calculation of average degree under different network sizes for decreasing random birth-and-death networks

NASA Astrophysics Data System (ADS)

Long, Yin; Zhang, Xiao-Jun; Wang, Kui

2018-05-01

In this paper, convergence and approximate calculation of average degree under different network sizes for decreasing random birth-and-death networks (RBDNs) are studied. First, we find and demonstrate that the average degree is convergent in the form of power law. Meanwhile, we discover that the ratios of the back items to front items of convergent reminder are independent of network link number for large network size, and we theoretically prove that the limit of the ratio is a constant. Moreover, since it is difficult to calculate the analytical solution of the average degree for large network sizes, we adopt numerical method to obtain approximate expression of the average degree to approximate its analytical solution. Finally, simulations are presented to verify our theoretical results.

Analytical treatment of self-phase-modulation beyond the slowly varying envelope approximation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Syrchin, M.S.; Zheltikov, A.M.; International Laser Center, M.V. Lomonosov Moscow State University, 119899 Moscow

Analytical treatment of the self-phase-modulation of an ultrashort light pulse is extended beyond the slowly varying envelope approximation. The resulting wave equation is modified to include corrections to self-phase-modulation due to higher-order spatial and temporal derivatives. Analytical solutions are found in the limiting regimes of high nonlinearities and very short pulses. Our results reveal features that can significantly impact both pulse shape and the evolution of the phase.

Analytical solutions for the profile of two-dimensional droplets with finite-length precursor films

NASA Astrophysics Data System (ADS)

Perazzo, Carlos Alberto; Mac Intyre, J. R.; Gomba, J. M.

2017-12-01

By means of the lubrication approximation we obtain the full family of static bidimensional profiles of a liquid resting on a substrate under partial-wetting conditions imposed by a disjoining-conjoining pressure. We show that for a set of quite general disjoining-conjoining pressure potentials, the free surface can adopt only five nontrivial static patterns; in particular, we find solutions when the height goes to zero which describe satisfactorily the complete free surface for a finite amount of fluid deposited on a substrate. To test the extension of the applicability of our solutions, we compare them with those obtained when the lubrication approximations are not employed and under conditions where the lubrication hypothesis are not strictly valid, and also with axisymmetric solutions. For a given disjoining-conjoining potential, we report a new analytical solution that accounts for all the five possible solutions.

Analytical approximation schemes for solving exact renormalization group equations in the local potential approximation

NASA Astrophysics Data System (ADS)

Bervillier, C.; Boisseau, B.; Giacomini, H.

2008-02-01

The relation between the Wilson-Polchinski and the Litim optimized ERGEs in the local potential approximation is studied with high accuracy using two different analytical approaches based on a field expansion: a recently proposed genuine analytical approximation scheme to two-point boundary value problems of ordinary differential equations, and a new one based on approximating the solution by generalized hypergeometric functions. A comparison with the numerical results obtained with the shooting method is made. A similar accuracy is reached in each case. Both two methods appear to be more efficient than the usual field expansions frequently used in the current studies of ERGEs (in particular for the Wilson-Polchinski case in the study of which they fail).

An analytic, approximate method for modeling steady, three-dimensional flow to partially penetrating wells

NASA Astrophysics Data System (ADS)

Bakker, Mark

2001-05-01

An analytic, approximate solution is derived for the modeling of three-dimensional flow to partially penetrating wells. The solution is written in terms of a correction on the solution for a fully penetrating well and is obtained by dividing the aquifer up, locally, in a number of aquifer layers. The resulting system of differential equations is solved by application of the theory for multiaquifer flow. The presented approach has three major benefits. First, the solution may be applied to any groundwater model that can simulate flow to a fully penetrating well; the solution may be superimposed onto the solution for the fully penetrating well to simulate the local three-dimensional drawdown and flow field. Second, the approach is applicable to isotropic, anisotropic, and stratified aquifers and to both confined and unconfined flow. Third, the solution extends over a small area around the well only; outside this area the three-dimensional effect of the partially penetrating well is negligible, and no correction to the fully penetrating well is needed. A number of comparisons are made to existing three-dimensional, analytic solutions, including radial confined and unconfined flow and a well in a uniform flow field. It is shown that a subdivision in three layers is accurate for many practical cases; very accurate solutions are obtained with more layers.

Analytical solutions for benchmarking cold regions subsurface water flow and energy transport models: one-dimensional soil thaw with conduction and advection

USGS Publications Warehouse

Kurylyk, Barret L.; McKenzie, Jeffrey M; MacQuarrie, Kerry T. B.; Voss, Clifford I.

2014-01-01

Numerous cold regions water flow and energy transport models have emerged in recent years. Dissimilarities often exist in their mathematical formulations and/or numerical solution techniques, but few analytical solutions exist for benchmarking flow and energy transport models that include pore water phase change. This paper presents a detailed derivation of the Lunardini solution, an approximate analytical solution for predicting soil thawing subject to conduction, advection, and phase change. Fifteen thawing scenarios are examined by considering differences in porosity, surface temperature, Darcy velocity, and initial temperature. The accuracy of the Lunardini solution is shown to be proportional to the Stefan number. The analytical solution results obtained for soil thawing scenarios with water flow and advection are compared to those obtained from the finite element model SUTRA. Three problems, two involving the Lunardini solution and one involving the classic Neumann solution, are recommended as standard benchmarks for future model development and testing.

Analytical Solution for Transport with Bimolecular Reactions in Fracture-Matrix Systems with Application to In-Situ Chemical Oxidation

NASA Astrophysics Data System (ADS)

Rajaram, H.; Arshadi, M.

2016-12-01

In-situ chemical oxidation (ISCO) is an effective strategy for remediation of DNAPL contamination in fractured rock. During ISCO, an oxidant (e.g. permanganate) is typically injected through fractures and is consumed by bimolecular reactions with DNAPLs such as TCE and natural organic matter in the fracture and the adjacent rock matrix. Under these conditions, moving reaction fronts form and propagate along the fracture and into the rock matrix. The propagation of these reaction fronts is strongly influenced by the heterogeneity/discontinuity across the fracture-matrix interface (advective transport dominates in the fractures, while diffusive transport dominates in the rock matrix). We present analytical solutions for the concentrations of the oxidant, TCE and natural organic matter; and the propagation of the reaction fronts in a fracture-matrix system. Our approximate analytical solutions assume advection and reaction dominate over diffusion/dispersion in the fracture and neglect the latter. Diffusion and reaction with both TCE and immobile natural organic matter in the rock matrix are considered. The behavior of the reaction-diffusion equations in the rock matrix is posed as a Stefan problem where the diffusing oxidant reacts with both diffusing (TCE) and immobile (natural organic matter) reductants. Our analytical solutions establish that the reaction fronts propagate diffusively (i.e. as the square root of time) in both the matrix and the fracture. Our analytical solutions agree very well with numerical simulations for the case of uniform advection in the fracture. We also present extensions of our analytical solutions to non-uniform flows in the fracture by invoking a travel-time transformation. The non-uniform flow solutions are relevant to field applications of ISCO. The approximate analytical solutions are relevant to a broad class of reactive transport problems in fracture-matrix systems where moving reaction fronts occur.

Reactive silica transport in fractured porous media: Analytical solutions for a system of parallel fractures

NASA Astrophysics Data System (ADS)

Yang, Jianwen

2012-04-01

A general analytical solution is derived by using the Laplace transformation to describe transient reactive silica transport in a conceptualized 2-D system involving a set of parallel fractures embedded in an impermeable host rock matrix, taking into account of hydrodynamic dispersion and advection of silica transport along the fractures, molecular diffusion from each fracture to the intervening rock matrix, and dissolution of quartz. A special analytical solution is also developed by ignoring the longitudinal hydrodynamic dispersion term but remaining other conditions the same. The general and special solutions are in the form of a double infinite integral and a single infinite integral, respectively, and can be evaluated using Gauss-Legendre quadrature technique. A simple criterion is developed to determine under what conditions the general analytical solution can be approximated by the special analytical solution. It is proved analytically that the general solution always lags behind the special solution, unless a dimensionless parameter is less than a critical value. Several illustrative calculations are undertaken to demonstrate the effect of fracture spacing, fracture aperture and fluid flow rate on silica transport. The analytical solutions developed here can serve as a benchmark to validate numerical models that simulate reactive mass transport in fractured porous media.

Weakly nonlinear behavior of a plate thickness-mode piezoelectric transformer.

PubMed

Yang, Jiashi; Chen, Ziguang; Hu, Yuantai; Jiang, Shunong; Guo, Shaohua

2007-04-01

We analyzed the weakly nonlinear behavior of a plate thickness-shear mode piezoelectric transformer near resonance. An approximate analytical solution was obtained. Numerical results based on the analytical solution are presented. It is shown that on one side of the resonant frequency the input-output relation becomes nonlinear, and on the other side the output voltage experiences jumps.

Aircraft Range Optimization Using Singular Perturbations

NASA Technical Reports Server (NTRS)

Oconnor, Joseph Taffe

1973-01-01

An approximate analytic solution is developed for the problem of maximizing the range of an aircraft for a fixed end state. The problem is formulated as a singular perturbation and solved by matched inner and outer asymptotic expansions and the minimum principle of Pontryagin. Cruise in the stratosphere, and on transition to and from cruise at constant Mach number are discussed. The state vector includes altitude, flight path angle, and mass. Specific fuel consumption becomes a linear function of power approximating that of the cruise values. Cruise represents the outer solution; altitude and flight path angle are constants, and only mass changes. Transitions between cruise and the specified initial and final conditions correspond to the inner solutions. The mass is constant and altitude and velocity vary. A solution is developed which is valid for cruise but which is not for the initial and final conditions. Transforming of the independent variable near the initial and final conditions result in solutions which are valid for the two inner solutions but not for cruise. The inner solutions can not be obtained without simplifying the state equations. The singular perturbation approach overcomes this difficulty. A quadratic approximation of the state equations is made. The resulting problem is solved analytically, and the two inner solutions are matched to the outer solution.

Radiative transfer in falling snow: A two-stream approximation

NASA Astrophysics Data System (ADS)

Koh, Gary

1989-04-01

Light transmission measurements through falling snow have produced results unexplainable by single scattering arguments. A two-stream approximation to radiative transfer is used to derive an analytical expression that describes the effects of multiple scattering as a function of the snow optical depth and the snow asymmetry parameter. The approximate solution is simple and it may be as accurate as the exact solution for describing the transmission measurements within the limits of experimental uncertainties.

Approximate analytical solutions in the analysis of thin elastic plates

NASA Astrophysics Data System (ADS)

Goloskokov, Dmitriy P.; Matrosov, Alexander V.

2018-05-01

Two approaches to the construction of approximate analytical solutions for bending of a rectangular thin plate are presented: the superposition method based on the method of initial functions (MIF) and the one built using the Green's function in the form of orthogonal series. Comparison of two approaches is carried out by analyzing a square plate clamped along its contour. Behavior of the moment and the shear force in the neighborhood of the corner points is discussed. It is shown that both solutions give identical results at all points of the plate except for the neighborhoods of the corner points. There are differences in the values of bending moments and generalized shearing forces in the neighborhoods of the corner points.

An approximate analytic solution of a set of nonlinear model alpha-omega-dynamo equations for marginally unstable systems

NASA Technical Reports Server (NTRS)

Hinata, S.

1989-01-01

An approximate analytic solution of a set of nonlinear model alpha-omega-dynamo equations is obtained. The reaction of the Lorentz force on the velocity shear which stretches and, hence, amplifies the magnetic field is incorporated into the model. To single out the effect of the Lorentz force on the omega-effect, the effect of the Lorentz force on the alpha-effect is neglected in this study. The solution represents a nonlinear oscillation with the amplitude and period determined by the dynamo number N. The amplitude is proportional to N - 1, while the period is almost exactly the same as the dissipation time of the unstable mode (proportional to N).

A Model for Axial Magnetic Bearings Including Eddy Currents

NASA Technical Reports Server (NTRS)

Kucera, Ladislav; Ahrens, Markus

1996-01-01

This paper presents an analytical method of modelling eddy currents inside axial bearings. The problem is solved by dividing an axial bearing into elementary geometric forms, solving the Maxwell equations for these simplified geometries, defining boundary conditions and combining the geometries. The final result is an analytical solution for the flux, from which the impedance and the force of an axial bearing can be derived. Several impedance measurements have shown that the analytical solution can fit the measured data with a precision of approximately 5%.

An approximate analytical solution for describing surface runoff and sediment transport over hillslope

NASA Astrophysics Data System (ADS)

Tao, Wanghai; Wang, Quanjiu; Lin, Henry

2018-03-01

Soil and water loss from farmland causes land degradation and water pollution, thus continued efforts are needed to establish mathematical model for quantitative analysis of relevant processes and mechanisms. In this study, an approximate analytical solution has been developed for overland flow model and sediment transport model, offering a simple and effective means to predict overland flow and erosion under natural rainfall conditions. In the overland flow model, the flow regime was considered to be transitional with the value of parameter β (in the kinematic wave model) approximately two. The change rate of unit discharge with distance was assumed to be constant and equal to the runoff rate at the outlet of the plane. The excess rainfall was considered to be constant under uniform rainfall conditions. The overland flow model developed can be further applied to natural rainfall conditions by treating excess rainfall intensity as constant over a small time interval. For the sediment model, the recommended values of the runoff erosion calibration constant (cr) and the splash erosion calibration constant (cf) have been given in this study so that it is easier to use the model. These recommended values are 0.15 and 0.12, respectively. Comparisons with observed results were carried out to validate the proposed analytical solution. The results showed that the approximate analytical solution developed in this paper closely matches the observed data, thus providing an alternative method of predicting runoff generation and sediment yield, and offering a more convenient method of analyzing the quantitative relationships between variables. Furthermore, the model developed in this study can be used as a theoretical basis for developing runoff and erosion control methods.

A new frequency domain analytical solution of a cascade of diffusive channels for flood routing

NASA Astrophysics Data System (ADS)

Cimorelli, Luigi; Cozzolino, Luca; Della Morte, Renata; Pianese, Domenico; Singh, Vijay P.

2015-04-01

Simplified flood propagation models are often employed in practical applications for hydraulic and hydrologic analyses. In this paper, we present a new numerical method for the solution of the Linear Parabolic Approximation (LPA) of the De Saint Venant equations (DSVEs), accounting for the space variation of model parameters and the imposition of appropriate downstream boundary conditions. The new model is based on the analytical solution of a cascade of linear diffusive channels in the Laplace Transform domain. The time domain solutions are obtained using a Fourier series approximation of the Laplace Inversion formula. The new Inverse Laplace Transform Diffusive Flood Routing model (ILTDFR) can be used as a building block for the construction of real-time flood forecasting models or in optimization models, because it is unconditionally stable and allows fast and fairly precise computation.

GENERAL: The Analytic Solution of Schrödinger Equation with Potential Function Superposed by Six Terms with Positive-power and Inverse-power Potentials

NASA Astrophysics Data System (ADS)

Hu, Xian-Quan; Luo, Guang; Cui, Li-Peng; Li, Fang-Yu; Niu, Lian-Bin

2009-03-01

The analytic solution of the radial Schrödinger equation is studied by using the tight coupling condition of several positive-power and inverse-power potential functions in this article. Furthermore, the precisely analytic solutions and the conditions that decide the existence of analytic solution have been searched when the potential of the radial Schrödinger equation is V(r) = α1r8 + α2r3 + α3r2 + β3r-1 + β2r-3 + β1r-4. Generally speaking, there is only an approximate solution, but not analytic solution for Schrödinger equation with several potentials' superposition. However, the conditions that decide the existence of analytic solution have been found and the analytic solution and its energy level structure are obtained for the Schrödinger equation with the potential which is motioned above in this paper. According to the single-value, finite and continuous standard of wave function in a quantum system, the authors firstly solve the asymptotic solution through the radial coordinate r → and r → 0; secondly, they make the asymptotic solutions combining with the series solutions nearby the neighborhood of irregular singularities; and then they compare the power series coefficients, deduce a series of analytic solutions of the stationary state wave function and corresponding energy level structure by tight coupling among the coefficients of potential functions for the radial Schrödinger equation; and lastly, they discuss the solutions and make conclusions.

Laplace transform homotopy perturbation method for the approximation of variational problems.

PubMed

Filobello-Nino, U; Vazquez-Leal, H; Rashidi, M M; Sedighi, H M; Perez-Sesma, A; Sandoval-Hernandez, M; Sarmiento-Reyes, A; Contreras-Hernandez, A D; Pereyra-Diaz, D; Hoyos-Reyes, C; Jimenez-Fernandez, V M; Huerta-Chua, J; Castro-Gonzalez, F; Laguna-Camacho, J R

2016-01-01

This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.

Exact Solution of Gas Dynamics Equations Through Reduced Differential Transform and Sumudu Transform Linked with Pades Approximants

NASA Astrophysics Data System (ADS)

Rao, T. R. Ramesh

2018-04-01

In this paper, we study the analytical method based on reduced differential transform method coupled with sumudu transform through Pades approximants. The proposed method may be considered as alternative approach for finding exact solution of Gas dynamics equation in an effective manner. This method does not require any discretization, linearization and perturbation.

Modified equations of finite-size layered plates made of orthotropic material. Comparison of the results of numerical calculations with analytical solutions

NASA Astrophysics Data System (ADS)

Volchkov, Yu. M.

2017-09-01

This paper describes the modified bending equations of layered orthotropic plates in the first approximation. The approximation of the solution of the equation of the three-dimensional theory of elasticity by the Legendre polynomial segments is used to obtain differential equations of the elastic layer. For the approximation of equilibrium equations and boundary conditions of three-dimensional theory of elasticity, several approximations of each desired function (stresses and displacements) are used. The stresses at the internal points of the plate are determined from the defining equations for the orthotropic material, averaged with respect to the plate thickness. The construction of the bending equations of layered plates for each layer is carried out with the help of the elastic layer equations and the conjugation conditions on the boundaries between layers, which are conditions for the continuity of normal stresses and displacements. The numerical solution of the problem of bending of the rectangular layered plate obtained with the help of modified equations is compared with an analytical solution. It is determined that the maximum error in determining the stresses does not exceed 3 %.

Application of geometric approximation to the CPMG experiment: Two- and three-site exchange.

PubMed

Chao, Fa-An; Byrd, R Andrew

2017-04-01

The Carr-Purcell-Meiboom-Gill (CPMG) experiment is one of the most classical and well-known relaxation dispersion experiments in NMR spectroscopy, and it has been successfully applied to characterize biologically relevant conformational dynamics in many cases. Although the data analysis of the CPMG experiment for the 2-site exchange model can be facilitated by analytical solutions, the data analysis in a more complex exchange model generally requires computationally-intensive numerical analysis. Recently, a powerful computational strategy, geometric approximation, has been proposed to provide approximate numerical solutions for the adiabatic relaxation dispersion experiments where analytical solutions are neither available nor feasible. Here, we demonstrate the general potential of geometric approximation by providing a data analysis solution of the CPMG experiment for both the traditional 2-site model and a linear 3-site exchange model. The approximate numerical solution deviates less than 0.5% from the numerical solution on average, and the new approach is computationally 60,000-fold more efficient than the numerical approach. Moreover, we find that accurate dynamic parameters can be determined in most cases, and, for a range of experimental conditions, the relaxation can be assumed to follow mono-exponential decay. The method is general and applicable to any CPMG RD experiment (e.g. N, C', C α , H α , etc.) The approach forms a foundation of building solution surfaces to analyze the CPMG experiment for different models of 3-site exchange. Thus, the geometric approximation is a general strategy to analyze relaxation dispersion data in any system (biological or chemical) if the appropriate library can be built in a physically meaningful domain. Published by Elsevier Inc.

Analytic solution of magnetic induction distribution of ideal hollow spherical field sources

NASA Astrophysics Data System (ADS)

Xu, Xiaonong; Lu, Dingwei; Xu, Xibin; Yu, Yang; Gu, Min

2017-12-01

The Halbach type hollow spherical permanent magnet arrays (HSPMA) are volume compacted, energy efficient field sources, and capable of producing multi-Tesla field in the cavity of the array, which have attracted intense interests in many practical applications. Here, we present analytical solutions of magnetic induction to the ideal HSPMA in entire space, outside of array, within the cavity of array, and in the interior of the magnet. We obtain solutions using concept of magnetic charge to solve the Poisson's and Laplace's equations for the HSPMA. Using these analytical field expressions inside the material, a scalar demagnetization function is defined to approximately indicate the regions of magnetization reversal, partial demagnetization, and inverse magnetic saturation. The analytical field solution provides deeper insight into the nature of HSPMA and offer guidance in designing optimized one.

Analytical approximation for the Einstein-dilaton-Gauss-Bonnet black hole metric

NASA Astrophysics Data System (ADS)

Kokkotas, K. D.; Konoplya, R. A.; Zhidenko, A.

2017-09-01

We construct an analytical approximation for the numerical black hole metric of P. Kanti et al. [Phys. Rev. D 54, 5049 (1996), 10.1103/PhysRevD.54.5049] in the four-dimensional Einstein-dilaton-Gauss-Bonnet (EdGB) theory. The continued fraction expansion in terms of a compactified radial coordinate, used here, converges slowly when the dilaton coupling approaches its extremal values, but for a black hole far from the extremal state, the analytical formula has a maximal relative error of a fraction of one percent already within the third order of the continued fraction expansion. The suggested analytical representation of the numerical black hole metric is relatively compact and a good approximation in the whole space outside the black hole event horizon. Therefore, it can serve in the same way as an exact solution when analyzing particles' motion, perturbations, quasinormal modes, Hawking radiation, accreting disks, and many other problems in the vicinity of a black hole. In addition, we construct the approximate analytical expression for the dilaton field.

An explicit analytical solution for sound propagation in a three-dimensional penetrable wedge with small apex angle.

PubMed

Petrov, Pavel S; Sturm, Frédéric

2016-03-01

A problem of sound propagation in a shallow-water waveguide with a weakly sloping penetrable bottom is considered. The adiabatic mode parabolic equations are used to approximate the solution of the three-dimensional (3D) Helmholtz equation by modal decomposition of the acoustic pressure field. The mode amplitudes satisfy parabolic equations that admit analytical solutions in the special case of the 3D wedge. Using the analytical formula for modal amplitudes, an explicit and remarkably simple expression for the acoustic pressure in the wedge is obtained. The proposed solution is validated by the comparison with a solution of the 3D penetrable wedge problem obtained using a fully 3D parabolic equation that includes a leading-order cross term correction.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Liemert, André, E-mail: andre.liemert@ilm.uni-ulm.de; Kienle, Alwin

Purpose: Explicit solutions of the monoenergetic radiative transport equation in the P{sub 3} approximation have been derived which can be evaluated with nearly the same computational effort as needed for solving the standard diffusion equation (DE). In detail, the authors considered the important case of a semi-infinite medium which is illuminated by a collimated beam of light. Methods: A combination of the classic spherical harmonics method and the recently developed method of rotated reference frames is used for solving the P{sub 3} equations in closed form. Results: The derived solutions are illustrated and compared to exact solutions of the radiativemore » transport equation obtained via the Monte Carlo (MC) method as well as with other approximated analytical solutions. It is shown that for the considered cases which are relevant for biomedical optics applications, the P{sub 3} approximation is close to the exact solution of the radiative transport equation. Conclusions: The authors derived exact analytical solutions of the P{sub 3} equations under consideration of boundary conditions for defining a semi-infinite medium. The good agreement to Monte Carlo simulations in the investigated domains, for example, in the steady-state and time domains, as well as the short evaluation time needed suggests that the derived equations can replace the often applied solutions of the diffusion equation for the homogeneous semi-infinite medium.« less

Reproduction of exact solutions of Lipkin model by nonlinear higher random-phase approximation

NASA Astrophysics Data System (ADS)

Terasaki, J.; Smetana, A.; Šimkovic, F.; Krivoruchenko, M. I.

2017-10-01

It is shown that the random-phase approximation (RPA) method with its nonlinear higher generalization, which was previously considered as approximation except for a very limited case, reproduces the exact solutions of the Lipkin model. The nonlinear higher RPA is based on an equation nonlinear on eigenvectors and includes many-particle-many-hole components in the creation operator of the excited states. We demonstrate the exact character of solutions analytically for the particle number N = 2 and numerically for N = 8. This finding indicates that the nonlinear higher RPA is equivalent to the exact Schrödinger equation.

Continuum calculations of continental deformation in transcurrent environments

NASA Technical Reports Server (NTRS)

Sonder, L. J.; England, P. C.; Houseman, G. A.

1986-01-01

A thin viscous sheet approximation is used to investigate continental deformation near a strike-slip boundary. The vertically averaged velocity field is calculated for a medium characterized by a power law rheology with stress exponent n. Driving stresses include those applied along boundaries of the sheet and those arising from buoyancy forces related to lateral differences in crustal thickness. Exact and approximate analytic solutions for a region with a sinusoidal strike-slip boundary condition are compared with solutions for more geologically relevant boundary conditions obtained using a finite element technique. The across-strike length scale of the deformation is approximately 1/4pi x sq rt n times the dominant wavelength of the imposed strike-slip boundary condition for both the analytic and the numerical solutions; this result is consistent with length scales observed in continental regions of large-scale transcurrent faulting. An approximate, linear relationship between displacement and rotation is found that depends only on the deformation length scale and the rheology. Calculated displacements, finite rotations, and distribution of crustal thicknesses are consistent with those observed in the region of the Pacific-North America plate boundary in California.

Analytical inversions in remote sensing of particle size distributions. IV - Comparison of Fymat and Box-McKellar solutions in the anomalous diffraction approximation

NASA Technical Reports Server (NTRS)

Fymat, A. L.; Smith, C. B.

1979-01-01

It is shown that the inverse analytical solutions, provided separately by Fymat and Box-McKellar, for reconstructing particle size distributions from remote spectral transmission measurements under the anomalous diffraction approximation can be derived using a cosine and a sine transform, respectively. Sufficient conditions of validity of the two formulas are established. Their comparison shows that the former solution is preferable to the latter in that it requires less a priori information (knowledge of the particle number density is not needed) and has wider applicability. For gamma-type distributions, and either a real or a complex refractive index, explicit expressions are provided for retrieving the distribution parameters; such expressions are, interestingly, proportional to the geometric area of the polydispersion.

SOLUTIONS APPROXIMATING SOLUTE TRANSPORT IN A LEAKY AQUIFER RECEIVING WASTEWATER INJECTION

EPA Science Inventory

A mathematical model amenable to analytical solution techniques is developed for the investigation of contaminant transport from an injection well into a leaky aquifer system, which comprises a pumped and an unpumped aquifer connected to each other by an aquitard. A steady state ...

Geometric model of pseudo-distance measurement in satellite location systems

NASA Astrophysics Data System (ADS)

Panchuk, K. L.; Lyashkov, A. A.; Lyubchinov, E. V.

2018-04-01

The existing mathematical model of pseudo-distance measurement in satellite location systems does not provide a precise solution of the problem, but rather an approximate one. The existence of such inaccuracy, as well as bias in measurement of distance from satellite to receiver, results in inaccuracy level of several meters. Thereupon, relevance of refinement of the current mathematical model becomes obvious. The solution of the system of quadratic equations used in the current mathematical model is based on linearization. The objective of the paper is refinement of current mathematical model and derivation of analytical solution of the system of equations on its basis. In order to attain the objective, geometric analysis is performed; geometric interpretation of the equations is given. As a result, an equivalent system of equations, which allows analytical solution, is derived. An example of analytical solution implementation is presented. Application of analytical solution algorithm to the problem of pseudo-distance measurement in satellite location systems allows to improve the accuracy such measurements.

AN ANALYTIC MODEL OF DUSTY, STRATIFIED, SPHERICAL H ii REGIONS

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rodríguez-Ramírez, J. C.; Raga, A. C.; Lora, V.

2016-12-20

We study analytically the effect of radiation pressure (associated with photoionization processes and with dust absorption) on spherical, hydrostatic H ii regions. We consider two basic equations, one for the hydrostatic balance between the radiation-pressure components and the gas pressure, and another for the balance among the recombination rate, the dust absorption, and the ionizing photon rate. Based on appropriate mathematical approximations, we find a simple analytic solution for the density stratification of the nebula, which is defined by specifying the radius of the external boundary, the cross section of dust absorption, and the luminosity of the central star. Wemore » compare the analytic solution with numerical integrations of the model equations of Draine, and find a wide range of the physical parameters for which the analytic solution is accurate.« less

Strength conditions for the elastic structures with a stress error

NASA Astrophysics Data System (ADS)

Matveev, A. D.

2017-10-01

As is known, the constraints (strength conditions) for the safety factor of elastic structures and design details of a particular class, e.g. aviation structures are established, i.e. the safety factor values of such structures should be within the given range. It should be noted that the constraints are set for the safety factors corresponding to analytical (exact) solutions of elasticity problems represented for the structures. Developing the analytical solutions for most structures, especially irregular shape ones, is associated with great difficulties. Approximate approaches to solve the elasticity problems, e.g. the technical theories of deformation of homogeneous and composite plates, beams and shells, are widely used for a great number of structures. Technical theories based on the hypotheses give rise to approximate (technical) solutions with an irreducible error, with the exact value being difficult to be determined. In static calculations of the structural strength with a specified small range for the safety factors application of technical (by the Theory of Strength of Materials) solutions is difficult. However, there are some numerical methods for developing the approximate solutions of elasticity problems with arbitrarily small errors. In present paper, the adjusted reference (specified) strength conditions for the structural safety factor corresponding to approximate solution of the elasticity problem have been proposed. The stress error estimation is taken into account using the proposed strength conditions. It has been shown that, to fulfill the specified strength conditions for the safety factor of the given structure corresponding to an exact solution, the adjusted strength conditions for the structural safety factor corresponding to an approximate solution are required. The stress error estimation which is the basis for developing the adjusted strength conditions has been determined for the specified strength conditions. The adjusted strength conditions presented by allowable stresses are suggested. Adjusted strength conditions make it possible to determine the set of approximate solutions, whereby meeting the specified strength conditions. Some examples of the specified strength conditions to be satisfied using the technical (by the Theory of Strength of Materials) solutions and strength conditions have been given, as well as the examples of stress conditions to be satisfied using approximate solutions with a small error.

Estimation on nonlinear damping in second order distributed parameter systems

NASA Technical Reports Server (NTRS)

Banks, H. T.; Reich, Simeon; Rosen, I. G.

1989-01-01

An approximation and convergence theory for the identification of nonlinear damping in abstract wave equations is developed. It is assumed that the unknown dissipation mechanism to be identified can be described by a maximal monotone operator acting on the generalized velocity. The stiffness is assumed to be linear and symmetric. Functional analytic techniques are used to establish that solutions to a sequence of finite dimensional (Galerkin) approximating identification problems in some sense approximate a solution to the original infinite dimensional inverse problem.

Formulation and application of optimal homotopty asymptotic method to coupled differential-difference equations.

PubMed

Ullah, Hakeem; Islam, Saeed; Khan, Ilyas; Shafie, Sharidan; Fiza, Mehreen

2015-01-01

In this paper we applied a new analytic approximate technique Optimal Homotopy Asymptotic Method (OHAM) for treatment of coupled differential-difference equations (DDEs). To see the efficiency and reliability of the method, we consider Relativistic Toda coupled nonlinear differential-difference equation. It provides us a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier and explicit.

Formulation and Application of Optimal Homotopty Asymptotic Method to Coupled Differential - Difference Equations

PubMed Central

Ullah, Hakeem; Islam, Saeed; Khan, Ilyas; Shafie, Sharidan; Fiza, Mehreen

2015-01-01

In this paper we applied a new analytic approximate technique Optimal Homotopy Asymptotic Method (OHAM) for treatment of coupled differential- difference equations (DDEs). To see the efficiency and reliability of the method, we consider Relativistic Toda coupled nonlinear differential-difference equation. It provides us a convenient way to control the convergence of approximate solutions when it is compared with other methods of solution found in the literature. The obtained solutions show that OHAM is effective, simpler, easier and explicit. PMID:25874457

Modified harmonic balance method for the solution of nonlinear jerk equations

NASA Astrophysics Data System (ADS)

Rahman, M. Saifur; Hasan, A. S. M. Z.

2018-03-01

In this paper, a second approximate solution of nonlinear jerk equations (third order differential equation) can be obtained by using modified harmonic balance method. The method is simpler and easier to carry out the solution of nonlinear differential equations due to less number of nonlinear equations are required to solve than the classical harmonic balance method. The results obtained from this method are compared with those obtained from the other existing analytical methods that are available in the literature and the numerical method. The solution shows a good agreement with the numerical solution as well as the analytical methods of the available literature.

Modeling Three-Dimensional Flow in Confined Aquifers by Superposition of Both Two- and Three-Dimensional Analytic Functions

NASA Astrophysics Data System (ADS)

Haitjema, Henk M.

1985-10-01

A technique is presented to incorporate three-dimensional flow in a Dupuit-Forchheimer model. The method is based on superposition of approximate analytic solutions to both two- and three-dimensional flow features in a confined aquifer of infinite extent. Three-dimensional solutions are used in the domain of interest, while farfield conditions are represented by two-dimensional solutions. Approximate three- dimensional solutions have been derived for a partially penetrating well and a shallow creek. Each of these solutions satisfies the condition that no flow occurs across the confining layers of the aquifer. Because of this condition, the flow at some distance of a three-dimensional feature becomes nearly horizontal. Consequently, remotely from a three-dimensional feature, its three-dimensional solution is replaced by a corresponding two-dimensional one. The latter solution is trivial as compared to its three-dimensional counterpart, and its use greatly enhances the computational efficiency of the model. As an example, the flow is modeled between a partially penetrating well and a shallow creek that occur in a regional aquifer system.

Direct application of Padé approximant for solving nonlinear differential equations.

PubMed

Vazquez-Leal, Hector; Benhammouda, Brahim; Filobello-Nino, Uriel; Sarmiento-Reyes, Arturo; Jimenez-Fernandez, Victor Manuel; Garcia-Gervacio, Jose Luis; Huerta-Chua, Jesus; Morales-Mendoza, Luis Javier; Gonzalez-Lee, Mario

2014-01-01

This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant. 34L30.

Finite analytic numerical solution of heat transfer and flow past a square channel cavity

NASA Technical Reports Server (NTRS)

Chen, C.-J.; Obasih, K.

1982-01-01

A numerical solution of flow and heat transfer characteristics is obtained by the finite analytic method for a two dimensional laminar channel flow over a two-dimensional square cavity. The finite analytic method utilizes the local analytic solution in a small element of the problem region to form the algebraic equation relating an interior nodal value with its surrounding nodal values. Stable and rapidly converged solutions were obtained for Reynolds numbers ranging to 1000 and Prandtl number to 10. Streamfunction, vorticity and temperature profiles are solved. Local and mean Nusselt number are given. It is found that the separation streamlines between the cavity and channel flow are concave into the cavity at low Reynolds number and convex at high Reynolds number (Re greater than 100) and for square cavity the mean Nusselt number may be approximately correlated with Peclet number as Nu(m) = 0.365 Pe exp 0.2.

Physics of heat pipe rewetting

NASA Technical Reports Server (NTRS)

Chan, S. H.

1992-01-01

Although several studies have been made to determine the rewetting characteristics of liquid films on heated rods, tubes, and flat plates, no solutions are yet available to describe the rewetting process of a hot plate subjected to a uniform heating. A model is presented to analyze the rewetting process of such plates with and without grooves. Approximate analytical solutions are presented for the prediction of the rewetting velocity and the transient temperature profiles of the plates. It is shown that the present rewetting velocity solution reduces correctly to the existing solution for the rewetting of an initially hot isothermal plate without heating from beneath the plate. Numerical solutions have also been obtained to validate the analytical solutions.

A quantitative approach to aquifer vulnerability mapping

NASA Astrophysics Data System (ADS)

Connell, L. D.; Daele, Gerd van den

2003-05-01

This paper presents a procedure for calculating the transport to groundwater of surface-released contaminants. The approach is derived from a series of analytical and semi-analytical solutions to the advection-dispersion equation that include root zone and unsaturated water movement effects on the transport process. The steady-state form of these equations provides an efficient means of calculating the maximum concentration at the watertable and therefore has potential for use in vulnerability mapping. A two-layer approach is used in the solutions to represent the unsaturated profile, with the root zone corresponding to the upper layer where evapotranspiration can occur and transport properties can be in contrast to the rest of the profile. A novel transformation is applied to the advection-dispersion equation that considerably simplifies the way in which water movement is represented. To provide a combined flow and transport model an approximate procedure for water movement, using averages of the infiltration and transpiration rates with a novel, simple, quasi-steady state solution, is presented that can be used in conjunction with the solutions to the advection-dispersion equation. This quasi-steady state approximation for water movement allows for layering in the soil profile and root water uptake. Results from the combined quasi-steady state water movement and semi-analytical solute transport procedure compare well with numerical solutions to the coupled unsaturated flow and solute transport equations in a series of hypothetical simulations.

Development and application of accurate analytical models for single active electron potentials

NASA Astrophysics Data System (ADS)

Miller, Michelle; Jaron-Becker, Agnieszka; Becker, Andreas

2015-05-01

The single active electron (SAE) approximation is a theoretical model frequently employed to study scenarios in which inner-shell electrons may productively be treated as frozen spectators to a physical process of interest, and accurate analytical approximations for these potentials are sought as a useful simulation tool. Density function theory is often used to construct a SAE potential, requiring that a further approximation for the exchange correlation functional be enacted. In this study, we employ the Krieger, Li, and Iafrate (KLI) modification to the optimized-effective-potential (OEP) method to reduce the complexity of the problem to the straightforward solution of a system of linear equations through simple arguments regarding the behavior of the exchange-correlation potential in regions where a single orbital dominates. We employ this method for the solution of atomic and molecular potentials, and use the resultant curve to devise a systematic construction for highly accurate and useful analytical approximations for several systems. Supported by the U.S. Department of Energy (Grant No. DE-FG02-09ER16103), and the U.S. National Science Foundation (Graduate Research Fellowship, Grants No. PHY-1125844 and No. PHY-1068706).

A note on the accuracy of spectral method applied to nonlinear conservation laws

NASA Technical Reports Server (NTRS)

Shu, Chi-Wang; Wong, Peter S.

1994-01-01

Fourier spectral method can achieve exponential accuracy both on the approximation level and for solving partial differential equations if the solutions are analytic. For a linear partial differential equation with a discontinuous solution, Fourier spectral method produces poor point-wise accuracy without post-processing, but still maintains exponential accuracy for all moments against analytic functions. In this note we assess the accuracy of Fourier spectral method applied to nonlinear conservation laws through a numerical case study. We find that the moments with respect to analytic functions are no longer very accurate. However the numerical solution does contain accurate information which can be extracted by a post-processing based on Gegenbauer polynomials.

Radiation-driven winds of hot stars. VI - Analytical solutions for wind models including the finite cone angle effect

NASA Technical Reports Server (NTRS)

Kudritzki, R. P.; Pauldrach, A.; Puls, J.; Abbott, D. C.

1989-01-01

Analytical solutions for radiation-driven winds of hot stars including the important finite cone angle effect (see Pauldrach et al., 1986; Friend and Abbott, 1986) are derived which approximate the detailed numerical solutions of the exact wind equation of motion very well. They allow a detailed discussion of the finite cone angle effect and provide for given line force parameters k, alpha, delta definite formulas for mass-loss rate M and terminal velocity v-alpha as function of stellar parameters.

Stability of sequences generated by nonlinear differential systems. [for analysis of glider jet aircraft motion

NASA Technical Reports Server (NTRS)

Brown, R. L.

1979-01-01

A local stability analysis is presented for both the analytic and numerical solutions of the initial value problem for a system of ordinary differential equations. It is shown that, using a proper choice of Liapunov function, a connected region of stable initial values of both the analytic solution and the one-leg k-step numerical solution can be approximated. Attention is given to the example of the two-dimensional problem involving the stability of the longitudinal equations of motion of a gliding jet aircraft.

Solving the Helmholtz equation in conformal mapped ARROW structures using homotopy perturbation method.

PubMed

Reck, Kasper; Thomsen, Erik V; Hansen, Ole

2011-01-31

The scalar wave equation, or Helmholtz equation, describes within a certain approximation the electromagnetic field distribution in a given system. In this paper we show how to solve the Helmholtz equation in complex geometries using conformal mapping and the homotopy perturbation method. The solution of the mapped Helmholtz equation is found by solving an infinite series of Poisson equations using two dimensional Fourier series. The solution is entirely based on analytical expressions and is not mesh dependent. The analytical results are compared to a numerical (finite element method) solution.

A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach

NASA Astrophysics Data System (ADS)

Wu, Dongmei; Wang, Zhongcheng

2006-03-01

According to Mickens [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563], the general HB (harmonic balance) method is an approximation to the convergent Fourier series representation of the periodic solution of a nonlinear oscillator and not an approximation to an expansion in terms of a small parameter. Consequently, for a nonlinear undamped Duffing equation with a driving force Bcos(ωx), to find a periodic solution when the fundamental frequency is identical to ω, the corresponding Fourier series can be written as y˜(x)=∑n=1m acos[(2n-1)ωx]. How to calculate the coefficients of the Fourier series efficiently with a computer program is still an open problem. For HB method, by substituting approximation y˜(x) into force equation, expanding the resulting expression into a trigonometric series, then letting the coefficients of the resulting lowest-order harmonic be zero, one can obtain approximate coefficients of approximation y˜(x) [R.E. Mickens, Comments on a Generalized Galerkin's method for non-linear oscillators, J. Sound Vib. 118 (1987) 563]. But for nonlinear differential equations such as Duffing equation, it is very difficult to construct higher-order analytical approximations, because the HB method requires solving a set of algebraic equations for a large number of unknowns with very complex nonlinearities. To overcome the difficulty, forty years ago, Urabe derived a computational method for Duffing equation based on Galerkin procedure [M. Urabe, A. Reiter, Numerical computation of nonlinear forced oscillations by Galerkin's procedure, J. Math. Anal. Appl. 14 (1966) 107-140]. Dooren obtained an approximate solution of the Duffing oscillator with a special set of parameters by using Urabe's method [R. van Dooren, Stabilization of Cowell's classic finite difference method for numerical integration, J. Comput. Phys. 16 (1974) 186-192]. In this paper, in the frame of the general HB method, we present a new iteration algorithm to calculate the coefficients of the Fourier series. By using this new method, the iteration procedure starts with a(x)cos(ωx)+b(x)sin(ωx), and the accuracy may be improved gradually by determining new coefficients a,a,… will be produced automatically in an one-by-one manner. In all the stage of calculation, we need only to solve a cubic equation. Using this new algorithm, we develop a Mathematica program, which demonstrates following main advantages over the previous HB method: (1) it avoids solving a set of associate nonlinear equations; (2) it is easier to be implemented into a computer program, and produces a highly accurate solution with analytical expression efficiently. It is interesting to find that, generally, for a given set of parameters, a nonlinear Duffing equation can have three independent oscillation modes. For some sets of the parameters, it can have two modes with complex displacement and one with real displacement. But in some cases, it can have three modes, all of them having real displacement. Therefore, we can divide the parameters into two classes, according to the solution property: there is only one mode with real displacement and there are three modes with real displacement. This program should be useful to study the dynamically periodic behavior of a Duffing oscillator and can provide an approximate analytical solution with high-accuracy for testing the error behavior of newly developed numerical methods with a wide range of parameters. Program summaryTitle of program:AnalyDuffing.nb Catalogue identifier:ADWR_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADWR_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Licensing provisions:none Computer for which the program is designed and others on which it has been tested:the program has been designed for a microcomputer and been tested on the microcomputer. Computers:IBM PC Installations:the address(es) of your computer(s) Operating systems under which the program has been tested:Windows XP Programming language used:Software Mathematica 4.2, 5.0 and 5.1 No. of lines in distributed program, including test data, etc.:23 663 No. of bytes in distributed program, including test data, etc.:152 321 Distribution format:tar.gz Memory required to execute with typical data:51 712 Bytes No. of bits in a word: No. of processors used:1 Has the code been vectorized?:no Peripherals used:no Program Library subprograms used:no Nature of physical problem:To find an approximate solution with analytical expressions for the undamped nonlinear Duffing equation with periodic driving force when the fundamental frequency is identical to the driving force. Method of solution:In the frame of the general HB method, by using a new iteration algorithm to calculate the coefficients of the Fourier series, we can obtain an approximate analytical solution with high-accuracy efficiently. Restrictions on the complexity of the problem:For problems, which have a large driving frequency, the convergence may be a little slow, because more iterative times are needed. Typical running time:several seconds Unusual features of the program:For an undamped Duffing equation, it can provide all the solutions or the oscillation modes with real displacement for any interesting parameters, for the required accuracy, efficiently. The program can be used to study the dynamically periodic behavior of a nonlinear oscillator, and can provide a high-accurate approximate analytical solution for developing high-accurate numerical method.

Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM.

PubMed

Singh, Brajesh K; Srivastava, Vineet K

2015-04-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM

PubMed Central

Singh, Brajesh K.; Srivastava, Vineet K.

2015-01-01

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations. PMID:26064639

An invariant asymptotic formula for solutions of second-order linear ODE's

NASA Technical Reports Server (NTRS)

Gingold, H.

1988-01-01

An invariant-matrix technique for the approximate solution of second-order ordinary differential equations (ODEs) of form y-double-prime = phi(x)y is developed analytically and demonstrated. A set of linear transformations for the companion matrix differential system is proposed; the diagonalization procedure employed in the final stage of the asymptotic decomposition is explained; and a scalar formulation of solutions for the ODEs is obtained. Several typical ODEs are analyzed, and it is shown that the Liouville-Green or WKB approximation is a special case of the present formula, which provides an approximation which is valid for the entire interval (0, infinity).

An analytical solution for the elastic response to surface loads imposed on a layered, transversely isotropic and self-gravitating Earth

NASA Astrophysics Data System (ADS)

Pan, E.; Chen, J. Y.; Bevis, M.; Bordoni, A.; Barletta, V. R.; Molavi Tabrizi, A.

2015-12-01

We present an analytical solution for the elastic deformation of an elastic, transversely isotropic, layered and self-gravitating Earth by surface loads. We first introduce the vector spherical harmonics to express the physical quantities in the layered Earth. This reduces the governing equations to a linear system of equations for the expansion coefficients. We then solve for the expansion coefficients analytically under the assumption (i.e. approximation) that in the mantle, the density in each layer varies as 1/r (where r is the radial coordinate) while the gravity is constant and that in the core the gravity in each layer varies linearly in r with constant density. These approximations dramatically simplify the subsequent mathematical analysis and render closed-form expressions for the expansion coefficients. We implement our solution in a MATLAB code and perform a benchmark which shows both the correctness of our solution and the implementation. We also calculate the load Love numbers (LLNs) of the PREM Earth for different degrees of the Legendre function for both isotropic and transversely isotropic, layered mantles with different core models, demonstrating for the first time the effect of Earth anisotropy on the LLNs.

Exact and approximate solutions for transient squeezing flow

NASA Astrophysics Data System (ADS)

Lang, Ji; Santhanam, Sridhar; Wu, Qianhong

2017-10-01

In this paper, we report two novel theoretical approaches to examine a fast-developing flow in a thin fluid gap, which is widely observed in industrial applications and biological systems. The problem is featured by a very small Reynolds number and Strouhal number, making the fluid convective acceleration negligible, while its local acceleration is not. We have developed an exact solution for this problem which shows that the flow starts with an inviscid limit when the viscous effect has no time to appear and is followed by a subsequent developing flow, in which the viscous effect continues to penetrate into the entire fluid gap. An approximate solution is also developed using a boundary layer integral method. This solution precisely captures the general behavior of the transient fluid flow process and agrees very well with the exact solution. We also performed numerical simulation using Ansys-CFX. Excellent agreement between the analytical and the numerical solutions is obtained, indicating the validity of the analytical approaches. The study presented herein fills the gap in the literature and will have a broad impact on industrial and biomedical applications.

Constructing analytic solutions on the Tricomi equation

NASA Astrophysics Data System (ADS)

Ghiasi, Emran Khoshrouye; Saleh, Reza

2018-04-01

In this paper, homotopy analysis method (HAM) and variational iteration method (VIM) are utilized to derive the approximate solutions of the Tricomi equation. Afterwards, the HAM is optimized to accelerate the convergence of the series solution by minimizing its square residual error at any order of the approximation. It is found that effect of the optimal values of auxiliary parameter on the convergence of the series solution is not negligible. Furthermore, the present results are found to agree well with those obtained through a closed-form equation available in the literature. To conclude, it is seen that the two are effective to achieve the solution of the partial differential equations.

Analysis of THG modes for femtosecond laser pulse

NASA Astrophysics Data System (ADS)

Trofimov, Vyacheslav A.; Sidorov, Pavel S.

2017-05-01

THG is used nowadays in many practical applications such as a substance diagnostics, and biological objects imaging, and etc. With developing of new materials and technology (for example, photonic crystal) an attention to THG process analysis grow. Therefore, THG features understanding are a modern problem. Early we have developed new analytical approach based on using the problem invariant for analytical solution construction of the THG process. It should be stressed that we did not use a basic wave non-depletion approximation. Nevertheless, a long pulse duration approximation and plane wave approximation has applied. The analytical solution demonstrates, in particular, an optical bistability property (and may other regimes of frequency tripling) for the third harmonic generation process. But, obviously, this approach does not reflect an influence of a medium dispersion on the frequency tripling. Therefore, in this paper we analyze THG efficiency of a femtosecond laser pulse taking into account a second order dispersion affect as well as self- and crossmodulation of the interacting waves affect on the frequency conversion process. Analysis is made using a computer simulation on the base of Schrödinger equations describing the process under consideration.

Steady-state protein focusing in carrier ampholyte based isoelectric focusing: Part I-Analytical solution.

PubMed

Shim, Jaesool; Yoo, Kisoo; Dutta, Prashanta

2017-03-01

The determination of an analytical solution to find the steady-state protein concentration distribution in IEF is very challenging due to the nonlinear coupling between mass and charge conservation equations. In this study, approximate analytical solutions are obtained for steady-state protein distribution in carrier ampholyte based IEF. Similar to the work of Svensson, the final concentration profile for proteins is assumed to be Gaussian, but appropriate expressions are presented in order to obtain the effective electric field and pH gradient in the focused protein band region. Analytical results are found from iterative solutions of a system of coupled algebraic equations using only several iterations for IEF separation of three plasma proteins: albumin, cardiac troponin I, and hemoglobin. The analytical results are compared with numerically predicted results for IEF, showing excellent agreement. Analytically obtained electric field and ionic conductivity distributions show significant deviation from their nominal values, which is essential in finding the protein focusing behavior at isoelectric points. These analytical solutions can be used to determine steady-state protein concentration distribution for experiment design of IEF considering any number of proteins and ampholytes. Moreover, the model presented herein can be used to find the conductivity, electric field, and pH field. © 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Sample distribution in peak mode isotachophoresis

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rubin, Shimon; Schwartz, Ortal; Bercovici, Moran, E-mail: mberco@technion.ac.il

We present an analytical study of peak mode isotachophoresis (ITP), and provide closed form solutions for sample distribution and electric field, as well as for leading-, trailing-, and counter-ion concentration profiles. Importantly, the solution we present is valid not only for the case of fully ionized species, but also for systems of weak electrolytes which better represent real buffer systems and for multivalent analytes such as proteins and DNA. The model reveals two major scales which govern the electric field and buffer distributions, and an additional length scale governing analyte distribution. Using well-controlled experiments, and numerical simulations, we verify andmore » validate the model and highlight its key merits as well as its limitations. We demonstrate the use of the model for determining the peak concentration of focused sample based on known buffer and analyte properties, and show it differs significantly from commonly used approximations based on the interface width alone. We further apply our model for studying reactions between multiple species having different effective mobilities yet co-focused at a single ITP interface. We find a closed form expression for an effective-on rate which depends on reactants distributions, and derive the conditions for optimizing such reactions. Interestingly, the model reveals that maximum reaction rate is not necessarily obtained when the concentration profiles of the reacting species perfectly overlap. In addition to the exact solutions, we derive throughout several closed form engineering approximations which are based on elementary functions and are simple to implement, yet maintain the interplay between the important scales. Both the exact and approximate solutions provide insight into sample focusing and can be used to design and optimize ITP-based assays.« less

Solution of magnetic field and eddy current problem induced by rotating magnetic poles (abstract)

NASA Astrophysics Data System (ADS)

Liu, Z. J.; Low, T. S.

1996-04-01

The magnetic field and eddy current problems induced by rotating permanent magnet poles occur in electromagnetic dampers, magnetic couplings, and many other devices. Whereas numerical techniques, for example, finite element methods can be exploited to study various features of these problems, such as heat generation and drag torque development, etc., the analytical solution is always of interest to the designers since it helps them to gain the insight into the interdependence of the parameters involved and provides an efficient tool for designing. Some of the previous work showed that the solution of the eddy current problem due to the linearly moving magnet poles can give satisfactory approximation for the eddy current problem due to rotating fields. However, in many practical cases, especially when the number of magnet poles is small, there is significant effect of flux focusing due to the geometry. The above approximation can therefore lead to marked errors in the theoretical predictions of the device performance. Bernot et al. recently described an analytical solution in a polar coordinate system where the radial field is excited by a time-varying source. A discussion of an analytical solution of the magnetic field and eddy current problems induced by moving magnet poles in radial field machines will be given in this article. The theoretical predictions obtained from this method is compared with the results obtained from finite element calculations. The validity of the method is also checked by the comparison of the theoretical predictions and the measurements from a test machine. It is shown that the introduced solution leads to a significant improvement in the air gap field prediction as compared with the results obtained from the analytical solution that models the eddy current problems induced by linearly moving magnet poles.

Siewert solutions of transcendental equations, generalized Lambert functions and physical applications

NASA Astrophysics Data System (ADS)

Barsan, Victor

2018-05-01

Several classes of transcendental equations, mainly eigenvalue equations associated to non-relativistic quantum mechanical problems, are analyzed. Siewert's systematic approach of such equations is discussed from the perspective of the new results recently obtained in the theory of generalized Lambert functions and of algebraic approximations of various special or elementary functions. Combining exact and approximate analytical methods, quite precise analytical outputs are obtained for apparently untractable problems. The results can be applied in quantum and classical mechanics, magnetism, elasticity, solar energy conversion, etc.

Demonstration/Validation of the Snap Sampler Passive Ground Water Sampling Device for Sampling Inorganic Analytes at the Former Pease Air Force Base

DTIC Science & Technology

2009-07-01

viii Unit Conversion Factors...sampler is also an economic alternative for sampling for inorganic analytes. ERDC/CRREL TR-09-12 xii Unit Conversion Factors Multiply By To Obtain...head- space and then covered with two layers of tightly fitting aluminum foil. To dissolve the analytes, the solutions were stirred for approximately

Hydrodynamics beyond Navier-Stokes: the slip flow model.

PubMed

Yudistiawan, Wahyu P; Ansumali, Santosh; Karlin, Iliya V

2008-07-01

Recently, analytical solutions for the nonlinear Couette flow demonstrated the relevance of the lattice Boltzmann (LB) models to hydrodynamics beyond the continuum limit [S. Ansumali, Phys. Rev. Lett. 98, 124502 (2007)]. In this paper, we present a systematic study of the simplest LB kinetic equation-the nine-bit model in two dimensions--in order to quantify it as a slip flow approximation. Details of the aforementioned analytical solution are presented, and results are extended to include a general shear- and force-driven unidirectional flow in confined geometry. Exact solutions for the velocity, as well as for pertinent higher-order moments of the distribution functions, are obtained in both Couette and Poiseuille steady-state flows for all values of rarefaction parameter (Knudsen number). Results are compared with the slip flow solution by Cercignani, and a good quantitative agreement is found for both flow situations. Thus, the standard nine-bit LB model is characterized as a valid and self-consistent slip flow model for simulations beyond the Navier-Stokes approximation.

Analytic Approximations to the Free Boundary and Multi-dimensional Problems in Financial Derivatives Pricing

NASA Astrophysics Data System (ADS)

Lau, Chun Sing

This thesis studies two types of problems in financial derivatives pricing. The first type is the free boundary problem, which can be formulated as a partial differential equation (PDE) subject to a set of free boundary condition. Although the functional form of the free boundary condition is given explicitly, the location of the free boundary is unknown and can only be determined implicitly by imposing continuity conditions on the solution. Two specific problems are studied in details, namely the valuation of fixed-rate mortgages and CEV American options. The second type is the multi-dimensional problem, which involves multiple correlated stochastic variables and their governing PDE. One typical problem we focus on is the valuation of basket-spread options, whose underlying asset prices are driven by correlated geometric Brownian motions (GBMs). Analytic approximate solutions are derived for each of these three problems. For each of the two free boundary problems, we propose a parametric moving boundary to approximate the unknown free boundary, so that the original problem transforms into a moving boundary problem which can be solved analytically. The governing parameter of the moving boundary is determined by imposing the first derivative continuity condition on the solution. The analytic form of the solution allows the price and the hedging parameters to be computed very efficiently. When compared against the benchmark finite-difference method, the computational time is significantly reduced without compromising the accuracy. The multi-stage scheme further allows the approximate results to systematically converge to the benchmark results as one recasts the moving boundary into a piecewise smooth continuous function. For the multi-dimensional problem, we generalize the Kirk (1995) approximate two-asset spread option formula to the case of multi-asset basket-spread option. Since the final formula is in closed form, all the hedging parameters can also be derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark prices obtained by numerical integration or Monte Carlo simulation. By exploiting an explicit relationship between the option price and the underlying probability distribution, we further derive an approximate distribution function for the general basket-spread variable. It can be used to approximate the transition probability distribution of any linear combination of correlated GBMs. Finally, an implicit perturbation is applied to reduce the pricing errors by factors of up to 100. When compared against the existing methods, the basket-spread option formula coupled with the implicit perturbation turns out to be one of the most robust and accurate approximation methods.

Approximation solution of Schrodinger equation for Q-deformed Rosen-Morse using supersymmetry quantum mechanics (SUSY QM)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Alemgadmi, Khaled I. K., E-mail: azozkied@yahoo.com; Suparmi; Cari

2015-09-30

The approximate analytical solution of Schrodinger equation for Q-Deformed Rosen-Morse potential was investigated using Supersymmetry Quantum Mechanics (SUSY QM) method. The approximate bound state energy is given in the closed form and the corresponding approximate wave function for arbitrary l-state given for ground state wave function. The first excited state obtained using upper operator and ground state wave function. The special case is given for the ground state in various number of q. The existence of Rosen-Morse potential reduce energy spectra of system. The larger value of q, the smaller energy spectra of system.

Overshooting thunderstorm cloud top dynamics as approximated by a linear Lagrangian parcel model with analytic exact solutions

NASA Technical Reports Server (NTRS)

Schlesinger, Robert E.

1990-01-01

Results are presented from a linear Lagrangian entraining parcel model of an overshooting thunderstorm cloud top. The model, which is similar to that of Adler and Mack (1986), gives analytic exact solutions for vertical velocity and temperature by representing mixing with Rayleigh damping instead of nonlinearly. Model results are presented for various combinations of stratospheric lapse rate, drag intensity, and mixing strength. The results are compared to those of Adler and Mack.

Unsteady fluid flow in a slightly curved pipe: A comparative study of a matched asymptotic expansions solution with a single analytical solution

DOE Office of Scientific and Technical Information (OSTI.GOV)

Messaris, Gerasimos A. T., E-mail: messaris@upatras.gr; School of Science and Technology, Hellenic Open University, 11 Sahtouri Street, GR 262 22 Patras; Hadjinicolaou, Maria

The present work is motivated by the fact that blood flow in the aorta and the main arteries is governed by large finite values of the Womersley number α and for such values of α there is not any analytical solution in the literature. The existing numerical solutions, although accurate, give limited information about the factors that affect the flow, whereas an analytical approach has an advantage in that it can provide physical insight to the flow mechanism. Having this in mind, we seek analytical solution to the equations of the fluid flow driven by a sinusoidal pressure gradient inmore » a slightly curved pipe of circular cross section when the Womersley number varies from small finite to infinite values. Initially the equations of motion are expanded in terms of the curvature ratio δ and the resulting linearized equations are solved analytically in two ways. In the first, we match the solution for the main core to that for the Stokes boundary layer. This solution is valid for very large values of α. In the second, we derive a straightforward single solution valid to the entire flow region and for 8 ≤ α < ∞, a range which includes the values of α that refer to the physiological flows. Each solution contains expressions for the axial velocity, the stream function, and the wall stresses and is compared to the analogous forms presented in other studies. The two solutions give identical results to each other regarding the axial flow but differ in the secondary flow and the circumferential wall stress, due to the approximations employed in the matched asymptotic expansion process. The results on the stream function from the second solution are in agreement with analogous results from other numerical solutions. The second solution predicts that the atherosclerotic plaques may develop in any location around the cross section of the aortic wall unlike to the prescribed locations predicted by the first solution. In addition, it gives circumferential wall stresses augmented by approximately 100% with respect to the matched asymptotic expansions, a factor that may contribute jointly with other pathological factors to the faster aging of the arterial system and the possible malfunction of the aorta.« less

Unsteady fluid flow in a slightly curved pipe: A comparative study of a matched asymptotic expansions solution with a single analytical solution

NASA Astrophysics Data System (ADS)

Messaris, Gerasimos A. T.; Hadjinicolaou, Maria; Karahalios, George T.

2016-08-01

The present work is motivated by the fact that blood flow in the aorta and the main arteries is governed by large finite values of the Womersley number α and for such values of α there is not any analytical solution in the literature. The existing numerical solutions, although accurate, give limited information about the factors that affect the flow, whereas an analytical approach has an advantage in that it can provide physical insight to the flow mechanism. Having this in mind, we seek analytical solution to the equations of the fluid flow driven by a sinusoidal pressure gradient in a slightly curved pipe of circular cross section when the Womersley number varies from small finite to infinite values. Initially the equations of motion are expanded in terms of the curvature ratio δ and the resulting linearized equations are solved analytically in two ways. In the first, we match the solution for the main core to that for the Stokes boundary layer. This solution is valid for very large values of α. In the second, we derive a straightforward single solution valid to the entire flow region and for 8 ≤ α < ∞, a range which includes the values of α that refer to the physiological flows. Each solution contains expressions for the axial velocity, the stream function, and the wall stresses and is compared to the analogous forms presented in other studies. The two solutions give identical results to each other regarding the axial flow but differ in the secondary flow and the circumferential wall stress, due to the approximations employed in the matched asymptotic expansion process. The results on the stream function from the second solution are in agreement with analogous results from other numerical solutions. The second solution predicts that the atherosclerotic plaques may develop in any location around the cross section of the aortic wall unlike to the prescribed locations predicted by the first solution. In addition, it gives circumferential wall stresses augmented by approximately 100% with respect to the matched asymptotic expansions, a factor that may contribute jointly with other pathological factors to the faster aging of the arterial system and the possible malfunction of the aorta.

Exploring inductive linearization for pharmacokinetic-pharmacodynamic systems of nonlinear ordinary differential equations.

PubMed

Hasegawa, Chihiro; Duffull, Stephen B

2018-02-01

Pharmacokinetic-pharmacodynamic systems are often expressed with nonlinear ordinary differential equations (ODEs). While there are numerous methods to solve such ODEs these methods generally rely on time-stepping solutions (e.g. Runge-Kutta) which need to be matched to the characteristics of the problem at hand. The primary aim of this study was to explore the performance of an inductive approximation which iteratively converts nonlinear ODEs to linear time-varying systems which can then be solved algebraically or numerically. The inductive approximation is applied to three examples, a simple nonlinear pharmacokinetic model with Michaelis-Menten elimination (E1), an integrated glucose-insulin model and an HIV viral load model with recursive feedback systems (E2 and E3, respectively). The secondary aim of this study was to explore the potential advantages of analytically solving linearized ODEs with two examples, again E3 with stiff differential equations and a turnover model of luteinizing hormone with a surge function (E4). The inductive linearization coupled with a matrix exponential solution provided accurate predictions for all examples with comparable solution time to the matched time-stepping solutions for nonlinear ODEs. The time-stepping solutions however did not perform well for E4, particularly when the surge was approximated by a square wave. In circumstances when either a linear ODE is particularly desirable or the uncertainty in matching the integrator to the ODE system is of potential risk, then the inductive approximation method coupled with an analytical integration method would be an appropriate alternative.

One-dimensional model and solutions for creeping gas flows in the approximation of uniform pressure

NASA Astrophysics Data System (ADS)

Vedernikov, A.; Balapanov, D.

2016-11-01

A model, along with analytical and numerical solutions, is presented to describe a wide variety of one-dimensional slow flows of compressible heat-conductive fluids. The model is based on the approximation of uniform pressure valid for the flows, in which the sound propagation time is much shorter than the duration of any meaningful density variation in the system. The energy balance is described by the heat equation that is solved independently. This approach enables the explicit solution for the fluid velocity to be obtained. Interfacial and volumetric heat and mass sources as well as boundary motion are considered as possible sources of density variation in the fluid. A set of particular tasks is analyzed for different motion sources in planar, axial, and central symmetries in the quasistationary limit of heat conduction (i.e., for large Fourier number). The analytical solutions are in excellent agreement with corresponding numerical solutions of the whole system of the Navier-Stokes equations. This work deals with the ideal gas. The approach is also valid for other equations of state.

Loglinear Approximate Solutions to Real-Business-Cycle Models: Some Observations

ERIC Educational Resources Information Center

Lau, Sau-Him Paul; Ng, Philip Hoi-Tak

2007-01-01

Following the analytical approach suggested in Campbell, the authors consider a baseline real-business-cycle (RBC) model with endogenous labor supply. They observe that the coefficients in the loglinear approximation of the dynamic equations characterizing the equilibrium are related to the fundamental parameters in a relatively simple manner.…

Thin airfoil theory based on approximate solution of the transonic flow equation

NASA Technical Reports Server (NTRS)

Spreiter, John R; Alksne, Alberta Y

1957-01-01

A method is presented for the approximate solution of the nonlinear equations transonic flow theory. Solutions are found for two-dimensional flows at a Mach number of 1 and for purely subsonic and purely supersonic flows. Results are obtained in closed analytic form for a large and significant class of nonlifting airfoils. At a Mach number of 1 general expressions are given for the pressure distribution on an airfoil of specified geometry and for the shape of an airfoil having a prescribed pressure distribution. Extensive comparisons are made with available data, particularly for a Mach number of 1, and with existing solutions.

Hydraulic modeling of riverbank filtration systems with curved boundaries using analytic elements and series solutions

NASA Astrophysics Data System (ADS)

Bakker, Mark

2010-08-01

A new analytic solution approach is presented for the modeling of steady flow to pumping wells near rivers in strip aquifers; all boundaries of the river and strip aquifer may be curved. The river penetrates the aquifer only partially and has a leaky stream bed. The water level in the river may vary spatially. Flow in the aquifer below the river is semi-confined while flow in the aquifer adjacent to the river is confined or unconfined and may be subject to areal recharge. Analytic solutions are obtained through superposition of analytic elements and Fourier series. Boundary conditions are specified at collocation points along the boundaries. The number of collocation points is larger than the number of coefficients in the Fourier series and a solution is obtained in the least squares sense. The solution is analytic while boundary conditions are met approximately. Very accurate solutions are obtained when enough terms are used in the series. Several examples are presented for domains with straight and curved boundaries, including a well pumping near a meandering river with a varying water level. The area of the river bottom where water infiltrates into the aquifer is delineated and the fraction of river water in the well water is computed for several cases.

Comparison between numeric and approximate analytic solutions for the prediction of soil metal uptake by roots. Example of cadmium.

PubMed

Schneider, André; Lin, Zhongbing; Sterckeman, Thibault; Nguyen, Christophe

2018-04-01

The dissociation of metal complexes in the soil solution can increase the availability of metals for root uptake. When it is accounted for in models of bioavailability of soil metals, the number of partial differential equations (PDEs) increases and the computation time to numerically solve these equations may be problematic when a large number of simulations are required, for example for sensitivity analyses or when considering root architecture. This work presents analytical solutions for the set of PDEs describing the bioavailability of soil metals including the kinetics of complexation for three scenarios where the metal complex in solution was fully inert, fully labile, or partially labile. The analytical solutions are only valid i) at steady-state when the PDEs become ordinary differential equations, the transient phase being not covered, ii) when diffusion is the major mechanism of transport and therefore, when convection is negligible, iii) when there is no between-root competition. The formulation of the analytical solutions is for cylindrical geometry but the solutions rely on the spread of the depletion profile around the root, which was modelled assuming a planar geometry. The analytical solutions were evaluated by comparison with the corresponding PDEs for cadmium in the case of the French agricultural soils. Provided that convection was much lower than diffusion (Péclet's number<0.02), the cumulative uptakes calculated from the analytic solutions were in very good agreement with those calculated from the PDEs, even in the case of a partially labile complex. The analytic solutions can be used instead of the PDEs to predict root uptake of metals. The analytic solutions were also used to build an indicator of the contribution of a complex to the uptake of the metal by roots, which can be helpful to predict the effect of soluble organic matter on the bioavailability of soil metals. Copyright © 2017 Elsevier B.V. All rights reserved.

Approximate analytical solution to diurnal atmospheric boundary-layer growth under well-watered conditions

USDA-ARS?s Scientific Manuscript database

The system of governing equations of a simplified slab model of the uniformly-mixed, purely convective, diurnal atmospheric boundary layer (ABL) is shown to allow immediate solutions for the potential temperature and specific humidity as functions of the ABL height and net radiation when expressed i...

Piecewise linear emulator of the nonlinear Schrödinger equation and the resulting analytic solutions for Bose-Einstein condensates.

PubMed

Theodorakis, Stavros

2003-06-01

We emulate the cubic term Psi(3) in the nonlinear Schrödinger equation by a piecewise linear term, thus reducing the problem to a set of uncoupled linear inhomogeneous differential equations. The resulting analytic expressions constitute an excellent approximation to the exact solutions, as is explicitly shown in the case of the kink, the vortex, and a delta function trap. Such a piecewise linear emulation can be used for any differential equation where the only nonlinearity is a Psi(3) one. In particular, it can be used for the nonlinear Schrödinger equation in the presence of harmonic traps, giving analytic Bose-Einstein condensate solutions that reproduce very accurately the numerically calculated ones in one, two, and three dimensions.

A screening tool for delineating subregions of steady recharge within groundwater models

USGS Publications Warehouse

Dickinson, Jesse; Ferré, T.P.A.; Bakker, Mark; Crompton, Becky

2014-01-01

We have developed a screening method for simplifying groundwater models by delineating areas within the domain that can be represented using steady-state groundwater recharge. The screening method is based on an analytical solution for the damping of sinusoidal infiltration variations in homogeneous soils in the vadose zone. The damping depth is defined as the depth at which the flux variation damps to 5% of the variation at the land surface. Groundwater recharge may be considered steady where the damping depth is above the depth of the water table. The analytical solution approximates the vadose zone diffusivity as constant, and we evaluated when this approximation is reasonable. We evaluated the analytical solution through comparison of the damping depth computed by the analytic solution with the damping depth simulated by a numerical model that allows variable diffusivity. This comparison showed that the screening method conservatively identifies areas of steady recharge and is more accurate when water content and diffusivity are nearly constant. Nomograms of the damping factor (the ratio of the flux amplitude at any depth to the amplitude at the land surface) and the damping depth were constructed for clay and sand for periodic variations between 1 and 365 d and flux means and amplitudes from nearly 0 to 1 × 10−3 m d−1. We applied the screening tool to Central Valley, California, to identify areas of steady recharge. A MATLAB script was developed to compute the damping factor for any soil and any sinusoidal flux variation.

Approximate solution to the Callan-Giddings-Harvey-Strominger field equations for two-dimensional evaporating black holes

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ori, Amos

2010-11-15

Callan, Giddings, Harvey, and Strominger (CGHS) previously introduced a two-dimensional semiclassical model of gravity coupled to a dilaton and to matter fields. Their model yields a system of field equations which may describe the formation of a black hole in gravitational collapse as well as its subsequent evaporation. Here we present an approximate analytical solution to the semiclassical CGHS field equations. This solution is constructed using the recently introduced formalism of flux-conserving hyperbolic systems. We also explore the asymptotic behavior at the horizon of the evaporating black hole.

Padé approximations for Painlevé I and II transcendents

NASA Astrophysics Data System (ADS)

Novokshenov, V. Yu.

2009-06-01

We use a version of the Fair-Luke algorithm to find the Padé approximate solutions of the Painlevé I and II equations. We find the distributions of poles for the well-known Ablowitz-Segur and Hastings-McLeod solutions of the Painlevé II equation. We show that the Boutroux tritronquée solution of the Painleé I equation has poles only in the critical sector of the complex plane. The algorithm allows checking other analytic properties of the Painlevé transcendents, such as the asymptotic behavior at infinity in the complex plane.

Landau-Zener extension of the Tavis-Cummings model: Structure of the solution

DOE PAGES

Sun, Chen; Sinitsyn, Nikolai A.

2016-09-07

We explore the recently discovered solution of the driven Tavis-Cummings model (DTCM). It describes interaction of an arbitrary number of two-level systems with a bosonic mode that has linearly time-dependent frequency. We derive compact and tractable expressions for transition probabilities in terms of the well-known special functions. In this form, our formulas are suitable for fast numerical calculations and analytical approximations. As an application, we obtain the semiclassical limit of the exact solution and compare it to prior approximations. Furthermore, we also reveal connection between DTCM and q-deformed binomial statistics.

Eigen solutions and entropic system for Hellmann potential in the presence of the Schrödinger equation

NASA Astrophysics Data System (ADS)

Onate, C. A.; Onyeaju, M. C.; Ikot, A. N.; Ebomwonyi, O.

2017-11-01

By using the supersymmetric approach, we studied the approximate analytic solutions of the three-dimensional Schrödinger equation with the Hellmann potential by applying a suitable approximation scheme to the centrifugal term. The solutions of other useful potentials, such as Coulomb potential and Yukawa potential, are obtained by transformation of variables from the Hellmann potential. Finally, we calculated the Tsallis entropy and Rényi entropy both in position and momentum spaces under the Hellmann potential using integral method. The effects of these entropies on the angular momentum quantum number are investigated in detail.

Approximate method of variational Bayesian matrix factorization/completion with sparse prior

NASA Astrophysics Data System (ADS)

Kawasumi, Ryota; Takeda, Koujin

2018-05-01

We derive the analytical expression of a matrix factorization/completion solution by the variational Bayes method, under the assumption that the observed matrix is originally the product of low-rank, dense and sparse matrices with additive noise. We assume the prior of a sparse matrix is a Laplace distribution by taking matrix sparsity into consideration. Then we use several approximations for the derivation of a matrix factorization/completion solution. By our solution, we also numerically evaluate the performance of a sparse matrix reconstruction in matrix factorization, and completion of a missing matrix element in matrix completion.

Approximate bound-state solutions of the Dirac equation for the generalized yukawa potential plus the generalized tensor interaction

NASA Astrophysics Data System (ADS)

Ikot, Akpan N.; Maghsoodi, Elham; Hassanabadi, Hassan; Obu, Joseph A.

2014-05-01

In this paper, we obtain the approximate analytical bound-state solutions of the Dirac particle with the generalized Yukawa potential within the framework of spin and pseudospin symmetries for the arbitrary к state with a generalized tensor interaction. The generalized parametric Nikiforov-Uvarov method is used to obtain the energy eigenvalues and the corresponding wave functions in closed form. We also report some numerical results and present figures to show the effect of the tensor interaction.

Periodized Daubechies wavelets

DOE Office of Scientific and Technical Information (OSTI.GOV)

Restrepo, J.M.; Leaf, G.K.; Schlossnagle, G.

1996-03-01

The properties of periodized Daubechies wavelets on [0,1] are detailed and counterparts which form a basis for L{sup 2}(R). Numerical examples illustrate the analytical estimates for convergence and demonstrated by comparison with Fourier spectral methods the superiority of wavelet projection methods for approximations. The analytical solution to inner products of periodized wavelets and their derivatives, which are known as connection coefficients, is presented, and their use ius illustrated in the approximation of two commonly used differential operators. The periodization of the connection coefficients in Galerkin schemes is presented in detail.

Nonlinear analysis for dual-frequency concurrent energy harvesting

NASA Astrophysics Data System (ADS)

Yan, Zhimiao; Lei, Hong; Tan, Ting; Sun, Weipeng; Huang, Wenhu

2018-05-01

The dual-frequency responses of the hybrid energy harvester undergoing the base excitation and galloping were analyzed numerically. In this work, an approximate dual-frequency analytical method is proposed for the nonlinear analysis of such a system. To obtain the approximate analytical solutions of the full coupled distributed-parameter model, the forcing interactions is first neglected. Then, the electromechanical decoupled governing equation is developed using the equivalent structure method. The hybrid mechanical response is finally separated to be the self-excited and forced responses for deriving the analytical solutions, which are confirmed by the numerical simulations of the full coupled model. The forced response has great impacts on the self-excited response. The boundary of Hopf bifurcation is analytically determined by the onset wind speed to galloping, which is linearly increased by the electrical damping. Quenching phenomenon appears when the increasing base excitation suppresses the galloping. The theoretical quenching boundary depends on the forced mode velocity. The quenching region increases with the base acceleration and electrical damping, but decreases with the wind speed. Superior to the base-excitation-alone case, the existence of the aerodynamic force protects the hybrid energy harvester at resonance from damages caused by the excessive large displacement. From the view of the harvested power, the hybrid system surpasses the base-excitation-alone system or the galloping-alone system. This study advances our knowledge on intrinsic nonlinear dynamics of the dual-frequency energy harvesting system by taking advantage of the analytical solutions.

Modelling of interaction of the large disrupted meteoroid with the Earth atmosphere

NASA Astrophysics Data System (ADS)

Brykina, Irina G.

2018-05-01

The model of atmospheric fragmentation of large meteoroids to the cloud of fragments is proposed. The comparison with similar models used in the literature is made. The approximate analytical solution of meteor physics equations is obtained for the mass loss of the disrupted meteoroid, the energy deposition and for the light curve normalized to the maximum brightness. This solution is applied to modelling of interaction of the Chelyabinsk meteoroid with the atmosphere. The influence of uncertainty of initial parameters of the meteoroid on characteristics of its interaction with the atmosphere is estimated. Comparison of the analytical solution with the observational data is made.

Feynman Path Integral Approach to Electron Diffraction for One and Two Slits: Analytical Results

ERIC Educational Resources Information Center

Beau, Mathieu

2012-01-01

In this paper we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered). In addition to exact formulae, various approximations of the electron distribution are shown which facilitate the interpretation of the results.…

Petermann I and II spot size: Accurate semi analytical description involving Nelder-Mead method of nonlinear unconstrained optimization and three parameter fundamental modal field

NASA Astrophysics Data System (ADS)

Roy Choudhury, Raja; Roy Choudhury, Arundhati; Kanti Ghose, Mrinal

2013-01-01

A semi-analytical model with three optimizing parameters and a novel non-Gaussian function as the fundamental modal field solution has been proposed to arrive at an accurate solution to predict various propagation parameters of graded-index fibers with less computational burden than numerical methods. In our semi analytical formulation the optimization of core parameter U which is usually uncertain, noisy or even discontinuous, is being calculated by Nelder-Mead method of nonlinear unconstrained minimizations as it is an efficient and compact direct search method and does not need any derivative information. Three optimizing parameters are included in the formulation of fundamental modal field of an optical fiber to make it more flexible and accurate than other available approximations. Employing variational technique, Petermann I and II spot sizes have been evaluated for triangular and trapezoidal-index fibers with the proposed fundamental modal field. It has been demonstrated that, the results of the proposed solution identically match with the numerical results over a wide range of normalized frequencies. This approximation can also be used in the study of doped and nonlinear fiber amplifier.

From bead to rod: Comparison of theories by measuring translational drag coefficients of micron-sized magnetic bead-chains in Stokes flow

PubMed Central

Lu, Chen; Zhao, Xiaodan; Kawamura, Ryo

2017-01-01

Frictional drag force on an object in Stokes flow follows a linear relationship with the velocity of translation and a translational drag coefficient. This drag coefficient is related to the size, shape, and orientation of the object. For rod-like objects, analytical solutions of the drag coefficients have been proposed based on three rough approximations of the rod geometry, namely the bead model, ellipsoid model, and cylinder model. These theories all agree that translational drag coefficients of rod-like objects are functions of the rod length and aspect ratio, but differ among one another on the correction factor terms in the equations. By tracking the displacement of the particles through stationary fluids of calibrated viscosity in magnetic tweezers setup, we experimentally measured the drag coefficients of micron-sized beads and their bead-chain formations with chain length of 2 to 27. We verified our methodology with analytical solutions of dimers of two touching beads, and compared our measured drag coefficient values of rod-like objects with theoretical calculations. Our comparison reveals several analytical solutions that used more appropriate approximation and derived formulae that agree with our measurement better. PMID:29145447

Analytical Solutions for Radiative Transfer: Implications for Giant Planet Formation by Disk Instability

NASA Astrophysics Data System (ADS)

Boss, Alan P.

2009-03-01

The disk instability mechanism for giant planet formation is based on the formation of clumps in a marginally gravitationally unstable protoplanetary disk, which must lose thermal energy through a combination of convection and radiative cooling if they are to survive and contract to become giant protoplanets. While there is good observational support for forming at least some giant planets by disk instability, the mechanism has become theoretically contentious, with different three-dimensional radiative hydrodynamics codes often yielding different results. Rigorous code testing is required to make further progress. Here we present two new analytical solutions for radiative transfer in spherical coordinates, suitable for testing the code employed in all of the Boss disk instability calculations. The testing shows that the Boss code radiative transfer routines do an excellent job of relaxing to and maintaining the analytical results for the radial temperature and radiative flux profiles for a spherical cloud with high or moderate optical depths, including the transition from optically thick to optically thin regions. These radial test results are independent of whether the Eddington approximation, diffusion approximation, or flux-limited diffusion approximation routines are employed. The Boss code does an equally excellent job of relaxing to and maintaining the analytical results for the vertical (θ) temperature and radiative flux profiles for a disk with a height proportional to the radial distance. These tests strongly support the disk instability mechanism for forming giant planets.

Comparison of approximate solutions to the phonon Boltzmann transport equation with the relaxation time approximation: Spherical harmonics expansions and the discrete ordinates method

NASA Astrophysics Data System (ADS)

Christenson, J. G.; Austin, R. A.; Phillips, R. J.

2018-05-01

The phonon Boltzmann transport equation is used to analyze model problems in one and two spatial dimensions, under transient and steady-state conditions. New, explicit solutions are obtained by using the P1 and P3 approximations, based on expansions in spherical harmonics, and are compared with solutions from the discrete ordinates method. For steady-state energy transfer, it is shown that analytic expressions derived using the P1 and P3 approximations agree quantitatively with the discrete ordinates method, in some cases for large Knudsen numbers, and always for Knudsen numbers less than unity. However, for time-dependent energy transfer, the PN solutions differ qualitatively from converged solutions obtained by the discrete ordinates method. Although they correctly capture the wave-like behavior of energy transfer at short times, the P1 and P3 approximations rely on one or two wave velocities, respectively, yielding abrupt, step-changes in temperature profiles that are absent when the angular dependence of the phonon velocities is captured more completely. It is shown that, with the gray approximation, the P1 approximation is formally equivalent to the so-called "hyperbolic heat equation." Overall, these results support the use of the PN approximation to find solutions to the phonon Boltzmann transport equation for steady-state conditions. Such solutions can be useful in the design and analysis of devices that involve heat transfer at nanometer length scales, where continuum-scale approaches become inaccurate.

Benchmark solutions for the galactic heavy-ion transport equations with energy and spatial coupling

NASA Technical Reports Server (NTRS)

Ganapol, Barry D.; Townsend, Lawrence W.; Lamkin, Stanley L.; Wilson, John W.

1991-01-01

Nontrivial benchmark solutions are developed for the galactic heavy ion transport equations in the straightahead approximation with energy and spatial coupling. Analytical representations of the ion fluxes are obtained for a variety of sources with the assumption that the nuclear interaction parameters are energy independent. The method utilizes an analytical LaPlace transform inversion to yield a closed form representation that is computationally efficient. The flux profiles are then used to predict ion dose profiles, which are important for shield design studies.

High Rayleigh number convection in rectangular enclosures with differentially heated vertical walls and aspect ratios between zero and unity

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kassemi, S.A.

1988-04-01

High Rayleigh number convection in a rectangular cavity with insulated horizontal surfaces and differentially heated vertical walls was analyzed for an arbitrary aspect ratio smaller than or equal to unity. Unlike previous analytical studies, a systematic method of solution based on linearization technique and analytical iteration procedure was developed to obtain approximate closed-form solutions for a wide range of aspect ratios. The predicted velocity and temperature fields are shown to be in excellent agreement with available experimental and numerical data.

High Rayleigh number convection in rectangular enclosures with differentially heated vertical walls and aspect ratios between zero and unity

NASA Technical Reports Server (NTRS)

Kassemi, Siavash A.

1988-01-01

High Rayleigh number convection in a rectangular cavity with insulated horizontal surfaces and differentially heated vertical walls was analyzed for an arbitrary aspect ratio smaller than or equal to unity. Unlike previous analytical studies, a systematic method of solution based on linearization technique and analytical iteration procedure was developed to obtain approximate closed-form solutions for a wide range of aspect ratios. The predicted velocity and temperature fields are shown to be in excellent agreement with available experimental and numerical data.

Satellite recovery - Attitude dynamics of the targets

NASA Technical Reports Server (NTRS)

Cochran, J. E., Jr.; Lahr, B. S.

1986-01-01

The problems of categorizing and modeling the attitude dynamics of uncontrolled artificial earth satellites which may be targets in recovery attempts are addressed. Methods of classification presented are based on satellite rotational kinetic energy, rotational angular momentum and orbit and on the type of control present prior to the benign failure of the control system. The use of approximate analytical solutions and 'exact' numerical solutions to the equations governing satellite attitude motions to predict uncontrolled attitude motion is considered. Analytical and numerical results are presented for the evolution of satellite attitude motions after active control termination.

Thermoelastic analysis of spent fuel and high level radioactive waste repositories in salt. A semi-analytical solution. [JUDITH

DOE Office of Scientific and Technical Information (OSTI.GOV)

St. John, C.M.

1977-04-01

An underground repository containing heat generating, High Level Waste or Spent Unreprocessed Fuel may be approximated as a finite number of heat sources distributed across the plane of the repository. The resulting temperature, displacement and stress changes may be calculated using analytical solutions, providing linear thermoelasticity is assumed. This report documents a computer program based on this approach and gives results that form the basis for a comparison between the effects of disposing of High Level Waste and Spent Unreprocessed Fuel.

Solution of the mean spherical approximation for polydisperse multi-Yukawa hard-sphere fluid mixture using orthogonal polynomial expansions

NASA Astrophysics Data System (ADS)

Kalyuzhnyi, Yurij V.; Cummings, Peter T.

2006-03-01

The Blum-Høye [J. Stat. Phys. 19 317 (1978)] solution of the mean spherical approximation for a multicomponent multi-Yukawa hard-sphere fluid is extended to a polydisperse multi-Yukawa hard-sphere fluid. Our extension is based on the application of the orthogonal polynomial expansion method of Lado [Phys. Rev. E 54, 4411 (1996)]. Closed form analytical expressions for the structural and thermodynamic properties of the model are presented. They are given in terms of the parameters that follow directly from the solution. By way of illustration the method of solution is applied to describe the thermodynamic properties of the one- and two-Yukawa versions of the model.

Numerical simulation of KdV equation by finite difference method

NASA Astrophysics Data System (ADS)

Yokus, A.; Bulut, H.

2018-05-01

In this study, the numerical solutions to the KdV equation with dual power nonlinearity by using the finite difference method are obtained. Discretize equation is presented in the form of finite difference operators. The numerical solutions are secured via the analytical solution to the KdV equation with dual power nonlinearity which is present in the literature. Through the Fourier-Von Neumann technique and linear stable, we have seen that the FDM is stable. Accuracy of the method is analyzed via the L2 and L_{∞} norm errors. The numerical, exact approximations and absolute error are presented in tables. We compare the numerical solutions with the exact solutions and this comparison is supported with the graphic plots. Under the choice of suitable values of parameters, the 2D and 3D surfaces for the used analytical solution are plotted.

Approximate analytical solutions in the analysis of elastic structures of complex geometry

NASA Astrophysics Data System (ADS)

Goloskokov, Dmitriy P.; Matrosov, Alexander V.

2018-05-01

A method of analytical decomposition for analysis plane structures of a complex configuration is presented. For each part of the structure in the form of a rectangle all the components of the stress-strain state are constructed by the superposition method. The method is based on two solutions derived in the form of trigonometric series with unknown coefficients using the method of initial functions. The coefficients are determined from the system of linear algebraic equations obtained while satisfying the boundary conditions and the conditions for joining the structure parts. The components of the stress-strain state of a bent plate with holes are calculated using the analytical decomposition method.

Closed-form analytical solutions of high-temperature heat pipe startup and frozen startup limitation

NASA Technical Reports Server (NTRS)

Cao, Y.; Faghri, A.

1992-01-01

Previous numerical and experimental studies indicate that the high-temperature heat pipe startup process is characterized by a moving hot zone with relatively sharp fronts. Based on the above observation, a flat-front model for an approximate analytical solution is proposed. A closed-form solution related to the temperature distribution in the hot zone and the hot zone length as a function of time are obtained. The analytical results agree well with the corresponding experimental data, and provide a quick prediction method for the heat pipe startup performance. Finally, a heat pipe limitation related to the frozen startup process is identified, and an explicit criterion for the high-temperature heat pipe startup is derived. The frozen startup limit identified in this paper provides a fundamental guidance for high-temperature heat pipe design.

Methods for analysis of cracks in three-dimensional solids

NASA Technical Reports Server (NTRS)

Raju, I. S.; Newman, J. C., Jr.

1984-01-01

Various analytical and numerical methods used to evaluate the stress intensity factors for cracks in three-dimensional (3-D) solids are reviewed. Classical exact solutions and many of the approximate methods used in 3-D analyses of cracks are reviewed. The exact solutions for embedded elliptic cracks in infinite solids are discussed. The approximate methods reviewed are the finite element methods, the boundary integral equation (BIE) method, the mixed methods (superposition of analytical and finite element method, stress difference method, discretization-error method, alternating method, finite element-alternating method), and the line-spring model. The finite element method with singularity elements is the most widely used method. The BIE method only needs modeling of the surfaces of the solid and so is gaining popularity. The line-spring model appears to be the quickest way to obtain good estimates of the stress intensity factors. The finite element-alternating method appears to yield the most accurate solution at the minimum cost.

Space-charge-limited currents for cathodes with electric field enhanced geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lai, Dingguo, E-mail: laidingguo@nint.ac.cn; Qiu, Mengtong; Xu, Qifu

This paper presents the approximate analytic solutions of current density for annulus and circle cathodes. The current densities of annulus and circle cathodes are derived approximately from first principles, which are in agreement with simulation results. The large scaling laws can predict current densities of high current vacuum diodes including annulus and circle cathodes in practical applications. In order to discuss the relationship between current density and electric field on cathode surface, the existing analytical solutions of currents for concentric cylinder and sphere diodes are fitted from existing solutions relating with electric field enhancement factors. It is found that themore » space-charge-limited current density for the cathode with electric-field enhanced geometry can be written in a general form of J = g(β{sub E}){sup 2}J{sub 0}, where J{sub 0} is the classical (1D) Child-Langmuir current density, β{sub E} is the electric field enhancement factor, and g is the geometrical correction factor depending on the cathode geometry.« less

Approximate analytical solutions of a pair of coupled anharmonic oscillators

NASA Astrophysics Data System (ADS)

Alam, Nasir; Mandal, Swapan; Öhberg, Patrik

2015-02-01

The Hamiltonian and the corresponding equations of motion involving the field operators of two quartic anharmonic oscillators indirectly coupled via a linear oscillator are constructed. The approximate analytical solutions of the coupled differential equations involving the non-commuting field operators are solved up to the second order in the anharmonic coupling. In the absence of nonlinearity these solutions are used to calculate the second order variances and hence the squeezing in pure and in mixed modes. The higher order quadrature squeezing and the amplitude squared squeezing of various field modes are also investigated where the squeezing in pure and in mixed modes are found to be suppressed. Moreover, the absence of a nonlinearity prohibits the higher order quadrature and higher ordered amplitude squeezing of the input coherent states. It is established that the mere coupling of two oscillators through a third one is unable to produce any squeezing effects of input coherent light, but the presence of a nonlinear interaction may provide squeezed states and other nonclassical phenomena.

Analytical mass formula and nuclear surface properties in the ETF approximation. Part II: asymmetric nuclei

NASA Astrophysics Data System (ADS)

Aymard, François; Gulminelli, Francesca; Margueron, Jérôme

2016-08-01

We have recently addressed the problem of the determination of the nuclear surface energy for symmetric nuclei in the framework of the extended Thomas-Fermi (ETF) approximation using Skyrme functionals. We presently extend this formalism to the case of asymmetric nuclei and the question of the surface symmetry energy. We propose an approximate expression for the diffuseness and the surface energy. These quantities are analytically related to the parameters of the energy functional. In particular, the influence of the different equation of state parameters can be explicitly quantified. Detailed analyses of the different energy components (local/non-local, isoscalar/isovector, surface/curvature and higher order) are also performed. Our analytical solution of the ETF integral improves previous models and leads to a precision of better than 200 keV per nucleon in the determination of the nuclear binding energy for dripline nuclei.

Application of the optimal homotopy asymptotic method to nonlinear Bingham fluid dampers

NASA Astrophysics Data System (ADS)

Marinca, Vasile; Ene, Remus-Daniel; Bereteu, Liviu

2017-10-01

Dynamic response time is an important feature for determining the performance of magnetorheological (MR) dampers in practical civil engineering applications. The objective of this paper is to show how to use the Optimal Homotopy Asymptotic Method (OHAM) to give approximate analytical solutions of the nonlinear differential equation of a modified Bingham model with non-viscous exponential damping. Our procedure does not depend upon small parameters and provides us with a convenient way to optimally control the convergence of the approximate solutions. OHAM is very efficient in practice for ensuring very rapid convergence of the solution after only one iteration and with a small number of steps.

Fourier series expansion for nonlinear Hamiltonian oscillators.

PubMed

Méndez, Vicenç; Sans, Cristina; Campos, Daniel; Llopis, Isaac

2010-06-01

The problem of nonlinear Hamiltonian oscillators is one of the classical questions in physics. When an analytic solution is not possible, one can resort to obtaining a numerical solution or using perturbation theory around the linear problem. We apply the Fourier series expansion to find approximate solutions to the oscillator position as a function of time as well as the period-amplitude relationship. We compare our results with other recent approaches such as variational methods or heuristic approximations, in particular the Ren-He's method. Based on its application to the Duffing oscillator, the nonlinear pendulum and the eardrum equation, it is shown that the Fourier series expansion method is the most accurate.

Regularization of moving boundaries in a laplacian field by a mixed Dirichlet-Neumann boundary condition: exact results.

PubMed

Meulenbroek, Bernard; Ebert, Ute; Schäfer, Lothar

2005-11-04

The dynamics of ionization fronts that generate a conducting body are in the simplest approximation equivalent to viscous fingering without regularization. Going beyond this approximation, we suggest that ionization fronts can be modeled by a mixed Dirichlet-Neumann boundary condition. We derive exact uniformly propagating solutions of this problem in 2D and construct a single partial differential equation governing small perturbations of these solutions. For some parameter value, this equation can be solved analytically, which shows rigorously that the uniformly propagating solution is linearly convectively stable and that the asymptotic relaxation is universal and exponential in time.

Analytical Phase Equilibrium Function for Mixtures Obeying Raoult's and Henry's Laws

NASA Astrophysics Data System (ADS)

Hayes, Robert

When a mixture of two substances exists in both the liquid and gas phase at equilibrium, Raoults and Henry's laws (ideal solution and ideal dilute solution approximations) can be used to estimate the gas and liquid mole fractions at the extremes of either very little solute or solvent. By assuming that a cubic polynomial can reasonably approximate the intermediate values to these extremes as a function of mole fraction, the cubic polynomial is solved and presented. A closed form equation approximating the pressure dependence on mole fraction of the constituents is thereby obtained. As a first approximation, this is a very simple and potentially useful means to estimate gas and liquid mole fractions of equilibrium mixtures. Mixtures with an azeotrope require additional attention if this type of approach is to be utilized. This work supported in part by federal Grant NRC-HQ-84-14-G-0059.

Approximate analytical solutions to the double-stance dynamics of the lossy spring-loaded inverted pendulum.

PubMed

Shahbazi, Mohammad; Saranlı, Uluç; Babuška, Robert; Lopes, Gabriel A D

2016-12-05

This paper introduces approximate time-domain solutions to the otherwise non-integrable double-stance dynamics of the 'bipedal' spring-loaded inverted pendulum (B-SLIP) in the presence of non-negligible damping. We first introduce an auxiliary system whose behavior under certain conditions is approximately equivalent to the B-SLIP in double-stance. Then, we derive approximate solutions to the dynamics of the new system following two different methods: (i) updated-momentum approach that can deal with both the lossy and lossless B-SLIP models, and (ii) perturbation-based approach following which we only derive a solution to the lossless case. The prediction performance of each method is characterized via a comprehensive numerical analysis. The derived representations are computationally very efficient compared to numerical integrations, and, hence, are suitable for online planning, increasing the autonomy of walking robots. Two application examples of walking gait control are presented. The proposed solutions can serve as instrumental tools in various fields such as control in legged robotics and human motion understanding in biomechanics.

Similarity solution of the Boussinesq equation

NASA Astrophysics Data System (ADS)

Lockington, D. A.; Parlange, J.-Y.; Parlange, M. B.; Selker, J.

Similarity transforms of the Boussinesq equation in a semi-infinite medium are available when the boundary conditions are a power of time. The Boussinesq equation is reduced from a partial differential equation to a boundary-value problem. Chen et al. [Trans Porous Media 1995;18:15-36] use a hodograph method to derive an integral equation formulation of the new differential equation which they solve by numerical iteration. In the present paper, the convergence of their scheme is improved such that numerical iteration can be avoided for all practical purposes. However, a simpler analytical approach is also presented which is based on Shampine's transformation of the boundary value problem to an initial value problem. This analytical approximation is remarkably simple and yet more accurate than the analytical hodograph approximations.

Applying the method of fundamental solutions to harmonic problems with singular boundary conditions

NASA Astrophysics Data System (ADS)

Valtchev, Svilen S.; Alves, Carlos J. S.

2017-07-01

The method of fundamental solutions (MFS) is known to produce highly accurate numerical results for elliptic boundary value problems (BVP) with smooth boundary conditions, posed in analytic domains. However, due to the analyticity of the shape functions in its approximation basis, the MFS is usually disregarded when the boundary functions possess singularities. In this work we present a modification of the classical MFS which can be applied for the numerical solution of the Laplace BVP with Dirichlet boundary conditions exhibiting jump discontinuities. In particular, a set of harmonic functions with discontinuous boundary traces is added to the MFS basis. The accuracy of the proposed method is compared with the results form the classical MFS.

Unimolecular diffusion-mediated reactions with a nonrandom time-modulated absorbing barrier

NASA Technical Reports Server (NTRS)

Bashford, D.; Weaver, D. L.

1986-01-01

A diffusion-reaction model with time-dependent reactivity is formulated and applied to unimolecular reactions. The model is solved exactly numerically and approximately analytically for the unreacted fraction as a function of time. It is shown that the approximate analytical solution is valid even when the system is far from equilibrium, and when the reactivity probability is more complicated than a square-wave function of time. A discussion is also given of an approach to problems of this type using a stochastically fluctuating reactivity, and the first-passage time for a particular example is derived.

Adiabatic model of field reversal by fast ions in an axisymmetric open trap

DOE Office of Scientific and Technical Information (OSTI.GOV)

Tsidulko, Yu. A., E-mail: tsidulko@mail.ru

2016-06-15

A model of field reversal by fast ions has been developed under the assumption of preservation of fast-ion adiabatic invariants. Analytical solutions obtained in the approximation of a narrow fast-ion layer and numerical solutions to the evolutionary problem are presented. The solutions demonstrate the process of formation of a field reversed configuration with parameters close to those of the planned experiment.

Exact and Approximate Solutions for Transient Squeezing Flow

NASA Astrophysics Data System (ADS)

Lang, Ji; Santhanam, Sridhar; Wu, Qianhong

2017-11-01

In this paper, we report two novel theoretical approaches to examine a fast-developing flow in a thin fluid gap, which is widely observed in industrial applications and biological systems. The problem is featured by a very small Reynolds number and Strouhal number, making the fluid convective acceleration is negligible, while its local acceleration is not. We have developed an exact solution for this problem which shows that the flow starts with an inviscid limit when the viscous effect has no time to appear, and is followed by a subsequent developing flow, in which the viscous effect continues to penetrate into the entire fluid gap. An approximate solution is also developed using a boundary layer integral method. This solution precisely captures the general behavior of the transient fluid flow process, and agrees very well with the exact solution. We also performed numerical simulation using Ansys-CFX. Excellent agreement between the analytical and the numerical solutions is obtained, indicating the validity of the analytical approaches. The study presented herein fills the gap in the literature, and will have a broad impact in industrial and biomedical applications. This work is supported by National Science Foundation CBET Fluid Dynamics Program under Award #1511096, and supported by the Seed Grant from The Villanova Center for the Advancement of Sustainability in Engineering (VCASE).

Establishing the traceability of a uranyl nitrate solution to a standard reference material

DOE Office of Scientific and Technical Information (OSTI.GOV)

Jackson, C.H.; Clark, J.P.

1978-01-01

A uranyl nitrate solution for use as a Working Calibration and Test Material (WCTM) was characterized, using a statistically designed procedure to document traceability to National Bureau of Standards Reference Material (SPM-960). A Reference Calibration and Test Material (PCTM) was prepared from SRM-960 uranium metal to approximate the acid and uranium concentration of the WCTM. This solution was used in the characterization procedure. Details of preparing, handling, and packaging these solutions are covered. Two outside laboratories, each having measurement expertise using a different analytical method, were selected to measure both solutions according to the procedure for characterizing the WCTM. Twomore » different methods were also used for the in-house characterization work. All analytical results were tested for statistical agreement before the WCTM concentration and limit of error values were calculated. A concentration value was determined with a relative limit of error (RLE) of approximately 0.03% which was better than the target RLE of 0.08%. The use of this working material eliminates the expense of using SRMs to fulfill traceability requirements for uranium measurements on this type material. Several years' supply of uranyl nitrate solution with NBS traceability was produced. The cost of this material was less than 10% of an equal quantity of SRM-960 uranium metal.« less

On the arbitrary l-wave solutions of the deformed hyperbolic manning-rosen potential including an improved approximation to the orbital centrifugal term

NASA Astrophysics Data System (ADS)

Xu, Chun-Long; Zhang, Min-Cang

2017-01-01

The arbitrary l-wave solutions to the Schrödinger equation for the deformed hyperbolic Manning-Rosen potential is investigated analytically by using the Nikiforov-Uvarov method, the centrifugal term is treated with an improved Greene and Aldrich's approximation scheme. The wavefunctions depend on the deformation parameter q, which is expressed in terms of the Jocobi polynomial or the hypergeometric function. The bound state energy is obtained, and the discrete spectrum is shown to be independent of the deformation parameter q.

Approximate solutions to Mathieu's equation

NASA Astrophysics Data System (ADS)

Wilkinson, Samuel A.; Vogt, Nicolas; Golubev, Dmitry S.; Cole, Jared H.

2018-06-01

Mathieu's equation has many applications throughout theoretical physics. It is especially important to the theory of Josephson junctions, where it is equivalent to Schrödinger's equation. Mathieu's equation can be easily solved numerically, however there exists no closed-form analytic solution. Here we collect various approximations which appear throughout the physics and mathematics literature and examine their accuracy and regimes of applicability. Particular attention is paid to quantities relevant to the physics of Josephson junctions, but the arguments and notation are kept general so as to be of use to the broader physics community.

Triangular dislocation: an analytical, artefact-free solution

NASA Astrophysics Data System (ADS)

Nikkhoo, Mehdi; Walter, Thomas R.

2015-05-01

Displacements and stress-field changes associated with earthquakes, volcanoes, landslides and human activity are often simulated using numerical models in an attempt to understand the underlying processes and their governing physics. The application of elastic dislocation theory to these problems, however, may be biased because of numerical instabilities in the calculations. Here, we present a new method that is free of artefact singularities and numerical instabilities in analytical solutions for triangular dislocations (TDs) in both full-space and half-space. We apply the method to both the displacement and the stress fields. The entire 3-D Euclidean space {R}3 is divided into two complementary subspaces, in the sense that in each one, a particular analytical formulation fulfils the requirements for the ideal, artefact-free solution for a TD. The primary advantage of the presented method is that the development of our solutions involves neither numerical approximations nor series expansion methods. As a result, the final outputs are independent of the scale of the input parameters, including the size and position of the dislocation as well as its corresponding slip vector components. Our solutions are therefore well suited for application at various scales in geoscience, physics and engineering. We validate the solutions through comparison to other well-known analytical methods and provide the MATLAB codes.

A finite element analysis of viscoelastically damped sandwich plates

NASA Astrophysics Data System (ADS)

Ma, B.-A.; He, J.-F.

1992-01-01

A finite element analysis associated with an asymptotic solution method for the harmonic flexural vibration of viscoelastically damped unsymmetrical sandwich plates is given. The element formulation is based on generalization of the discrete Kirchhoff theory (DKT) element formulation. The results obtained with the first order approximation of the asymptotic solution presented here are the same as those obtained by means of the modal strain energy (MSE) method. By taking more terms of the asymptotic solution, with successive calculations and use of the Padé approximants method, accuracy can be improved. The finite element computation has been verified by comparison with an analytical exact solution for rectangular plates with simply supported edges. Results for the same plates with clamped edges are also presented.

Analytical approximation and numerical simulations for periodic travelling water waves

NASA Astrophysics Data System (ADS)

Kalimeris, Konstantinos

2017-12-01

We present recent analytical and numerical results for two-dimensional periodic travelling water waves with constant vorticity. The analytical approach is based on novel asymptotic expansions. We obtain numerical results in two different ways: the first is based on the solution of a constrained optimization problem, and the second is realized as a numerical continuation algorithm. Both methods are applied on some examples of non-constant vorticity. This article is part of the theme issue 'Nonlinear water waves'.

Comptonization in Ultra-Strong Magnetic Fields: Numerical Solution to the Radiative Transfer Problem

NASA Technical Reports Server (NTRS)

Ceccobello, C.; Farinelli, R.; Titarchuk, L.

2014-01-01

We consider the radiative transfer problem in a plane-parallel slab of thermal electrons in the presence of an ultra-strong magnetic field (B approximately greater than B(sub c) approx. = 4.4 x 10(exp 13) G). Under these conditions, the magnetic field behaves like a birefringent medium for the propagating photons, and the electromagnetic radiation is split into two polarization modes, ordinary and extraordinary, that have different cross-sections. When the optical depth of the slab is large, the ordinary-mode photons are strongly Comptonized and the photon field is dominated by an isotropic component. Aims. The radiative transfer problem in strong magnetic fields presents many mathematical issues and analytical or numerical solutions can be obtained only under some given approximations. We investigate this problem both from the analytical and numerical point of view, provide a test of the previous analytical estimates, and extend these results with numerical techniques. Methods. We consider here the case of low temperature black-body photons propagating in a sub-relativistic temperature plasma, which allows us to deal with a semi-Fokker-Planck approximation of the radiative transfer equation. The problem can then be treated with the variable separation method, and we use a numerical technique to find solutions to the eigenvalue problem in the case of a singular kernel of the space operator. The singularity of the space kernel is the result of the strong angular dependence of the electron cross-section in the presence of a strong magnetic field. Results. We provide the numerical solution obtained for eigenvalues and eigenfunctions of the space operator, and the emerging Comptonization spectrum of the ordinary-mode photons for any eigenvalue of the space equation and for energies significantly lesser than the cyclotron energy, which is on the order of MeV for the intensity of the magnetic field here considered. Conclusions. We derived the specific intensity of the ordinary photons, under the approximation of large angle and large optical depth. These assumptions allow the equation to be treated using a diffusion-like approximation.

An analytical-numerical approach for parameter determination of a five-parameter single-diode model of photovoltaic cells and modules

NASA Astrophysics Data System (ADS)

Hejri, Mohammad; Mokhtari, Hossein; Azizian, Mohammad Reza; Söder, Lennart

2016-04-01

Parameter extraction of the five-parameter single-diode model of solar cells and modules from experimental data is a challenging problem. These parameters are evaluated from a set of nonlinear equations that cannot be solved analytically. On the other hand, a numerical solution of such equations needs a suitable initial guess to converge to a solution. This paper presents a new set of approximate analytical solutions for the parameters of a five-parameter single-diode model of photovoltaic (PV) cells and modules. The proposed solutions provide a good initial point which guarantees numerical analysis convergence. The proposed technique needs only a few data from the PV current-voltage characteristics, i.e. open circuit voltage Voc, short circuit current Isc and maximum power point current and voltage Im; Vm making it a fast and low cost parameter determination technique. The accuracy of the presented theoretical I-V curves is verified by experimental data.

Approximate analytic solutions to coupled nonlinear Dirac equations

DOE PAGES

Khare, Avinash; Cooper, Fred; Saxena, Avadh

2017-01-30

Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g 1 2/2(more » $$\\bar{ψ}$$ψ) 2 + g 2 2/2($$\\bar{Φ}$$Φ) 2 + g 2 3($$\\bar{ψ}$$ψ)($$\\bar{Φ}$$Φ) as well as vector–vector interactions g 1 2/2($$\\bar{ψ}$$γμψ)($$\\bar{ψ}$$γμψ) + g 2 2/2($$\\bar{Φ}$$γμΦ)($$\\bar{Φ}$$γμΦ) + g 2 3($$\\bar{ψ}$$γμψ)($$\\bar{Φ}$$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e –iω1tR 1cosθ,R 1sinθΦ=e –iω2tR 2cosη,R 2sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 3 2/g 2 2 and g 3 2/g 1 2. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.« less

Approximate analytic solutions to coupled nonlinear Dirac equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Khare, Avinash; Cooper, Fred; Saxena, Avadh

Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g 1 2/2(more » $$\\bar{ψ}$$ψ) 2 + g 2 2/2($$\\bar{Φ}$$Φ) 2 + g 2 3($$\\bar{ψ}$$ψ)($$\\bar{Φ}$$Φ) as well as vector–vector interactions g 1 2/2($$\\bar{ψ}$$γμψ)($$\\bar{ψ}$$γμψ) + g 2 2/2($$\\bar{Φ}$$γμΦ)($$\\bar{Φ}$$γμΦ) + g 2 3($$\\bar{ψ}$$γμψ)($$\\bar{Φ}$$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e –iω1tR 1cosθ,R 1sinθΦ=e –iω2tR 2cosη,R 2sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g 3 2/g 2 2 and g 3 2/g 1 2. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.« less

Analytical solution of the transient temperature profile in gain medium of passively Q-switched microchip laser.

PubMed

Han, Xiahui; Li, Jianlang

2014-11-01

The transient temperature evolution in the gain medium of a continuous wave (CW) end-pumped passively Q-switched microchip (PQSM) laser is analyzed. By approximating the time-dependent population inversion density as a sawtooth function of time and treating the time-dependent pump absorption of a CW end-pumped PQSM laser as the superposition of an infinite series of short pumping pulses, the analytical expressions of transient temperature evolution and distribution in the gain medium for four- and three-level laser systems, respectively, are given. These analytical solutions are applied to evaluate the transient temperature evolution and distribution in the gain medium of CW end-pumped PQSM Nd:YAG and Yb:YAG lasers.

Analytical solutions of the Dirac equation under Hellmann–Frost–Musulin potential

DOE Office of Scientific and Technical Information (OSTI.GOV)

Onate, C.A., E-mail: oaclems14@physicist.net; Onyeaju, M.C.; Ikot, A.N.

2016-12-15

The approximate analytical solutions of the Dirac equation with Hellmann–Frost–Musulin potential have been studied by using the generalized parametric Nikiforov–Uvarov (NU) method for arbitrary spin–orbit quantum number k under the spin and pseudospin symmetries. The Hellmann–Frost–Musulin potential is a superposition potential that consists of Yukawa potential, Coulomb potential, and Frost–Musulin potential. As a particular case, we found the energy levels of the non-relativistic limit of the spin symmetry. The energy equation of Yukawa potential, Coulomb potential, Hellmann potential and Frost–Musulin potential are obtained. Energy values are generated for some diatomic molecules.

Analytical theory of mesoscopic Bose-Einstein condensation in an ideal gas

NASA Astrophysics Data System (ADS)

Kocharovsky, Vitaly V.; Kocharovsky, Vladimir V.

2010-03-01

We find the universal structure and scaling of the Bose-Einstein condensation (BEC) statistics and thermodynamics (Gibbs free energy, average energy, heat capacity) for a mesoscopic canonical-ensemble ideal gas in a trap with an arbitrary number of atoms, any volume, and any temperature, including the whole critical region. We identify a universal constraint-cutoff mechanism that makes BEC fluctuations strongly non-Gaussian and is responsible for all unusual critical phenomena of the BEC phase transition in the ideal gas. The main result is an analytical solution to the problem of critical phenomena. It is derived by, first, calculating analytically the universal probability distribution of the noncondensate occupation, or a Landau function, and then using it for the analytical calculation of the universal functions for the particular physical quantities via the exact formulas which express the constraint-cutoff mechanism. We find asymptotics of that analytical solution as well as its simple analytical approximations which describe the universal structure of the critical region in terms of the parabolic cylinder or confluent hypergeometric functions. The obtained results for the order parameter, all higher-order moments of BEC fluctuations, and thermodynamic quantities perfectly match the known asymptotics outside the critical region for both low and high temperature limits. We suggest two- and three-level trap models of BEC and find their exact solutions in terms of the cutoff negative binomial distribution (which tends to the cutoff gamma distribution in the continuous limit) and the confluent hypergeometric distribution, respectively. Also, we present an exactly solvable cutoff Gaussian model of BEC in a degenerate interacting gas. All these exact solutions confirm the universality and constraint-cutoff origin of the strongly non-Gaussian BEC statistics. We introduce a regular refinement scheme for the condensate statistics approximations on the basis of the infrared universality of higher-order cumulants and the method of superposition and show how to model BEC statistics in the actual traps. In particular, we find that the three-level trap model with matching the first four or five cumulants is enough to yield remarkably accurate results for all interesting quantities in the whole critical region. We derive an exact multinomial expansion for the noncondensate occupation probability distribution and find its high-temperature asymptotics (Poisson distribution) and corrections to it. Finally, we demonstrate that the critical exponents and a few known terms of the Taylor expansion of the universal functions, which were calculated previously from fitting the finite-size simulations within the phenomenological renormalization-group theory, can be easily obtained from the presented full analytical solutions for the mesoscopic BEC as certain approximations in the close vicinity of the critical point.

Analytical study of the critical behavior of the nonlinear pendulum

NASA Astrophysics Data System (ADS)

Lima, F. M. S.

2010-11-01

The dynamics of a simple pendulum consisting of a small bob and a massless rigid rod has three possible regimes depending on its total energy E: Oscillatory (when E is not enough for the pendulum to reach the top position), "perpetual ascent" when E is exactly the energy needed to reach the top, and nonoscillatory for greater energies. In the latter regime, the pendulum rotates periodically without velocity inversions. In contrast to the oscillatory regime, for which an exact analytic solution is known, the other two regimes are usually studied by solving the equation of motion numerically. By applying conservation of energy, I derive exact analytical solutions to both the perpetual ascent and nonoscillatory regimes and an exact expression for the pendulum period in the nonoscillatory regime. Based on Cromer's approximation for the large-angle pendulum period, I find a simple approximate expression for the decrease of the period with the initial velocity in the nonoscillatory regime, valid near the critical velocity. This expression is used to study the critical slowing down, which is observed near the transition between the oscillatory and nonoscillatory regimes.

User's manual for the one-dimensional hypersonic experimental aero-thermodynamic (1DHEAT) data reduction code

NASA Technical Reports Server (NTRS)

Hollis, Brian R.

1995-01-01

A FORTRAN computer code for the reduction and analysis of experimental heat transfer data has been developed. This code can be utilized to determine heat transfer rates from surface temperature measurements made using either thin-film resistance gages or coaxial surface thermocouples. Both an analytical and a numerical finite-volume heat transfer model are implemented in this code. The analytical solution is based on a one-dimensional, semi-infinite wall thickness model with the approximation of constant substrate thermal properties, which is empirically corrected for the effects of variable thermal properties. The finite-volume solution is based on a one-dimensional, implicit discretization. The finite-volume model directly incorporates the effects of variable substrate thermal properties and does not require the semi-finite wall thickness approximation used in the analytical model. This model also includes the option of a multiple-layer substrate. Fast, accurate results can be obtained using either method. This code has been used to reduce several sets of aerodynamic heating data, of which samples are included in this report.

Exact PDF equations and closure approximations for advective-reactive transport

DOE Office of Scientific and Technical Information (OSTI.GOV)

Venturi, D.; Tartakovsky, Daniel M.; Tartakovsky, Alexandre M.

2013-06-01

Mathematical models of advection–reaction phenomena rely on advective flow velocity and (bio) chemical reaction rates that are notoriously random. By using functional integral methods, we derive exact evolution equations for the probability density function (PDF) of the state variables of the advection–reaction system in the presence of random transport velocity and random reaction rates with rather arbitrary distributions. These PDF equations are solved analytically for transport with deterministic flow velocity and a linear reaction rate represented mathematically by a heterog eneous and strongly-correlated random field. Our analytical solution is then used to investigate the accuracy and robustness of the recentlymore » proposed large-eddy diffusivity (LED) closure approximation [1]. We find that the solution to the LED-based PDF equation, which is exact for uncorrelated reaction rates, is accurate even in the presence of strong correlations and it provides an upper bound of predictive uncertainty.« less

Infinite product expansion of the Fokker-Planck equation with steady-state solution.

PubMed

Martin, R J; Craster, R V; Kearney, M J

2015-07-08

We present an analytical technique for solving Fokker-Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples.

Infinite product expansion of the Fokker–Planck equation with steady-state solution

PubMed Central

Martin, R. J.; Craster, R. V.; Kearney, M. J.

2015-01-01

We present an analytical technique for solving Fokker–Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples. PMID:26346100

Analytical approximations for spiral waves

DOE Office of Scientific and Technical Information (OSTI.GOV)

Löber, Jakob, E-mail: jakob@physik.tu-berlin.de; Engel, Harald

2013-12-15

We propose a non-perturbative attempt to solve the kinematic equations for spiral waves in excitable media. From the eikonal equation for the wave front we derive an implicit analytical relation between rotation frequency Ω and core radius R{sub 0}. For free, rigidly rotating spiral waves our analytical prediction is in good agreement with numerical solutions of the linear eikonal equation not only for very large but also for intermediate and small values of the core radius. An equivalent Ω(R{sub +}) dependence improves the result by Keener and Tyson for spiral waves pinned to a circular defect of radius R{sub +}more » with Neumann boundaries at the periphery. Simultaneously, analytical approximations for the shape of free and pinned spirals are given. We discuss the reasons why the ansatz fails to correctly describe the dependence of the rotation frequency on the excitability of the medium.« less

Accurate spectral solutions for the parabolic and elliptic partial differential equations by the ultraspherical tau method

NASA Astrophysics Data System (ADS)

Doha, E. H.; Abd-Elhameed, W. M.

2005-09-01

We present a double ultraspherical spectral methods that allow the efficient approximate solution for the parabolic partial differential equations in a square subject to the most general inhomogeneous mixed boundary conditions. The differential equations with their boundary and initial conditions are reduced to systems of ordinary differential equations for the time-dependent expansion coefficients. These systems are greatly simplified by using tensor matrix algebra, and are solved by using the step-by-step method. Numerical applications of how to use these methods are described. Numerical results obtained compare favorably with those of the analytical solutions. Accurate double ultraspherical spectral approximations for Poisson's and Helmholtz's equations are also noted. Numerical experiments show that spectral approximation based on Chebyshev polynomials of the first kind is not always better than others based on ultraspherical polynomials.

Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves

NASA Astrophysics Data System (ADS)

Seadawy, Aly R.

2017-01-01

The propagation of three-dimensional nonlinear irrotational flow of an inviscid and incompressible fluid of the long waves in dispersive shallow-water approximation is analyzed. The problem formulation of the long waves in dispersive shallow-water approximation lead to fifth-order Kadomtsev-Petviashvili (KP) dynamical equation by applying the reductive perturbation theory. By using an extended auxiliary equation method, the solitary travelling-wave solutions of the two-dimensional nonlinear fifth-order KP dynamical equation are derived. An analytical as well as a numerical solution of the two-dimensional nonlinear KP equation are obtained and analyzed with the effects of external pressure flow.

Topics in elementary particle physics

NASA Astrophysics Data System (ADS)

Jin, Xiang

The author of this thesis discusses two topics in elementary particle physics: n-ary algebras and their applications to M-theory (Part I), and functional evolution and Renormalization Group flows (Part II). In part I, Lie algebra is extended to four different n-ary algebraic structure: generalized Lie algebra, Filippov algebra, Nambu algebra and Nambu-Poisson tensor; though there are still many other n-ary algebras. A natural property of Generalized Lie algebras — the Bremner identity, is studied, and proved with a totally different method from its original version. We extend Bremner identity to n-bracket cases, where n is an arbitrary odd integer. Filippov algebras do not focus on associativity, and are defined by the Fundamental identity. We add associativity to Filippov algebras, and give examples of how to construct Filippov algebras from su(2), bosonic oscillator, Virasoro algebra. We try to include fermionic charges into the ternary Virasoro-Witt algebra, but the attempt fails because fermionic charges keep generating new charges that make the algebra not closed. We also study the Bremner identity restriction on Nambu algebras and Nambu-Poisson tensors. So far, the only example 3-algebra being used in physics is the BLG model with 3-algebra A4, describing two M2-branes interactions. Its extension with Nambu algebra, BLG-NB model, is believed to describe infinite M2-branes condensation. Also, there is another propose for M2-brane interactions, the ABJM model, which is constructed by ordinary Lie algebra. We compare the symmetry properties between them, and discuss the possible approaches to include these three models into a grand unification theory. In Part II, we give an approximate solution for Schroeder's equations, based on series and conjugation methods. We use the logistic map as an example, and demonstrate that this approximate solution converges to known analytical solutions around the fixed point, around which the approximate solution is constructed. Although the closed-form solutions for Schroeder's equations can not always be approached analytically, by fitting the approximation solutions, one can still obtain closed-form solutions sometimes. Based on Schroeder's theory, approximate solutions for trajectories, velocities and potentials can also be constructed. The approximate solution is significantly useful to calculate the beta function in renormalization group trajectory. By "wrapping" the series solutions with the conjugations from different inverse functions, we generate different branches of the trajectory, and construct a counterexample for a folk theorem about limited cycles.

Padé Approximant and Minimax Rational Approximation in Standard Cosmology

NASA Astrophysics Data System (ADS)

Zaninetti, Lorenzo

2016-02-01

The luminosity distance in the standard cosmology as given by $\\Lambda$CDM and consequently the distance modulus for supernovae can be defined by the Pad\\'e approximant. A comparison with a known analytical solution shows that the Pad\\'e approximant for the luminosity distance has an error of $4\\%$ at redshift $= 10$. A similar procedure for the Taylor expansion of the luminosity distance gives an error of $4\\%$ at redshift $=0.7 $; this means that for the luminosity distance, the Pad\\'e approximation is superior to the Taylor series. The availability of an analytical expression for the distance modulus allows applying the Levenberg--Marquardt method to derive the fundamental parameters from the available compilations for supernovae. A new luminosity function for galaxies derived from the truncated gamma probability density function models the observed luminosity function for galaxies when the observed range in absolute magnitude is modeled by the Pad\\'e approximant. A comparison of $\\Lambda$CDM with other cosmologies is done adopting a statistical point of view.

Physics of cosmological cascades and observable properties

NASA Astrophysics Data System (ADS)

Fitoussi, T.; Belmont, R.; Malzac, J.; Marcowith, A.; Cohen-Tanugi, J.; Jean, P.

2017-04-01

TeV photons from extragalactic sources are absorbed in the intergalactic medium and initiate electromagnetic cascades. These cascades offer a unique tool to probe the properties of the universe at cosmological scales. We present a new Monte Carlo code dedicated to the physics of such cascades. This code has been tested against both published results and analytical approximations, and is made publicly available. Using this numerical tool, we investigate the main cascade properties (spectrum, halo extension and time delays), and study in detail their dependence on the physical parameters (extragalactic magnetic field, extragalactic background light, source redshift, source spectrum and beaming emission). The limitations of analytical solutions are emphasized. In particular, analytical approximations account only for the first generation of photons and higher branches of the cascade tree are neglected.

Analytical analysis of the temporal asymmetry between seawater intrusion and retreat

NASA Astrophysics Data System (ADS)

Rathore, Saubhagya Singh; Zhao, Yue; Lu, Chunhui; Luo, Jian

2018-01-01

The quantification of timescales associated with the movement of the seawater-freshwater interface is useful for developing effective management strategies for controlling seawater intrusion (SWI). In this study, for the first time, we derive an explicit analytical solution for the timescales of SWI and seawater retreat (SWR) in a confined, homogeneous coastal aquifer system under the quasi-steady assumption, based on a classical sharp-interface solution for approximating freshwater outflow rates into the sea. The flow continuity and hydrostatic equilibrium across the interface are identified as two primary mechanisms governing timescales of the interface movement driven by an abrupt change in discharge rates or hydraulic heads at the inland boundary. Through theoretical analysis, we quantified the dependence of interface-movement timescales on porosity, hydraulic conductivity, aquifer thickness, aquifer length, density ratio, and boundary conditions. Predictions from the analytical solution closely agreed with those from numerical simulations. In addition, we define a temporal asymmetry index (the ratio of the SWI timescale to the SWR timescale) to represent the resilience of the coastal aquifer in response to SWI. The developed analytical solutions provide a simple tool for the quick assessment of SWI and SWR timescales and reveal that the temporal asymmetry between SWI and SWR mainly relies on the initial and final values of the freshwater flux at the inland boundary, and is weakly affected by aquifer parameters. Furthermore, we theoretically examined the log-linearity relationship between the timescale and the freshwater flux at the inland boundary, and found that the relationship may be approximated by two linear functions with a slope of -2 and -1 for large changes at the boundary flux for SWI and SWR, respectively.

BLUES function method in computational physics

NASA Astrophysics Data System (ADS)

Indekeu, Joseph O.; Müller-Nedebock, Kristian K.

2018-04-01

We introduce a computational method in physics that goes ‘beyond linear use of equation superposition’ (BLUES). A BLUES function is defined as a solution of a nonlinear differential equation (DE) with a delta source that is at the same time a Green’s function for a related linear DE. For an arbitrary source, the BLUES function can be used to construct an exact solution to the nonlinear DE with a different, but related source. Alternatively, the BLUES function can be used to construct an approximate piecewise analytical solution to the nonlinear DE with an arbitrary source. For this alternative use the related linear DE need not be known. The method is illustrated in a few examples using analytical calculations and numerical computations. Areas for further applications are suggested.

Dynamical analysis of the avian-human influenza epidemic model using the semi-analytical method

NASA Astrophysics Data System (ADS)

Jabbari, Azizeh; Kheiri, Hossein; Bekir, Ahmet

2015-03-01

In this work, we present a dynamic behavior of the avian-human influenza epidemic model by using efficient computational algorithm, namely the multistage differential transform method(MsDTM). The MsDTM is used here as an algorithm for approximating the solutions of the avian-human influenza epidemic model in a sequence of time intervals. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge-Kutta method (RK4M) and differential transform method(DTM) solutions. It is shown that the MsDTM has the advantage of giving an analytical form of the solution within each time interval which is not possible in purely numerical techniques like RK4M.

Integration of the Rotation of an Earth-like Body as a Perturbed Spherical Rotor

NASA Astrophysics Data System (ADS)

Ferrer, Sebastián; Lara, Martin

2010-05-01

For rigid bodies close to a sphere, we propose an analytical solution that is free from elliptic integrals and functions, and can be fundamental for application to perturbed problems. After reordering the Hamiltonian as a perturbed spherical rotor, the Lie-series solution is generated up to an arbitrary order. Using the inertia parameters of different solar system bodies, the comparison of the approximate series solution with the exact analytical one shows that the precision reached with relatively low orders is at the same level of the observational accuracy for the Earth and Mars. Thus, for instance, the periodic errors of the mathematical solution are confined to the microarcsecond level with a simple second-order truncation for the Earth. On the contrary, higher orders are required for the mathematical solution to reach a precision at the expected level of accuracy of proposed new theories for the rotational dynamics of the Moon.

Acoustic propagation in a thermally stratified atmosphere

NASA Technical Reports Server (NTRS)

Vanmoorhem, W. K.

1988-01-01

Acoustic propagation in an atmosphere with a specific form of a temperature profile has been investigated by analytical means. The temperature profile used is representative of an actual atmospheric profile and contains three free parameters. Both lapse and inversion cases have been considered. Although ray solutions have been considered, the primary emphasis has been on solutions of the acoustic wave equation with point source where the sound speed varies with height above the ground corresponding to the assumed temperature profile. The method used to obtain the solution of the wave equation is based on Hankel transformation of the wave equation, approximate solution of the transformed equation for wavelength small compared to the scale of the temperature (or sound speed) profile, and approximate or numerical inversion of the Hankel transformed solution. The solution displays the characteristics found in experimental data but extensive comparison between the models and experimental data has not been carried out.

Acoustic propagation in a thermally stratified atmosphere

NASA Technical Reports Server (NTRS)

Vanmoorhem, W. K.

1987-01-01

Acoustic propagation in an atmosphere with a specific form of temperature profile has been investigated by analytical means. The temperature profile used is representative of an actual atmospheric profile and contains three free parameters. Both lapse and inversion cases have been considered. Although ray solution have been considered the primary emphasis has been on solutions of the acoustic wave equation with point force where the sound speed varies with height above the ground corresponding to the assumed temperature profile. The method used to obtain the solution of the wave equation is based on Hankel transformation of the wave equation, approximate solution of the transformed equation for wavelength small compared to the scale of the temperature (or sound speed) profile, and approximate or numerical inversion of the Hankel transformed solution. The solution displays the characteristics found in experimental data but extensive comparison between the models and experimental data has not been carried out.

AQUEOUS PROTONATION PROPERTIES OF AMPHOTERIC NANOPARTICLES

EPA Science Inventory

A divergence is predicted between the acidity behavior of charged sites on micron sized colloidal particles and nanoparticles. Utilizing the approximate analytical solution to the Poisson-Boltzmann equation published by Ohshima et al. (1982), findings from the work included: 1):...

Analogue solution for electrical capacity of membrane covered square cylinders in square array at high concentration.

PubMed

Cole, K S

1975-12-01

Analytical solutions of Laplace equations have given the electrical characteristics of membranes and interiors of spherical, ellipsoidal, and cylindrical cells in suspensions and tissues from impedance measurements, but the underlying assumptions may be invalid above 50% volume concentrations. However, resistance measurements on several nonconducting, close-packing forms in two and three dimensions closely predicted volume concentrations up to 100% by equations derived from Maxwell and Rayleigh. Calculations of membrane capacities of cells in suspensions and tissues from extensions of theory, as developed by Fricke and by Cole, have been useful but of unknown validity at high concentrations. A resistor analogue has been used to solve the finite difference approximation to the Laplace equation for the resistance and capacity of a square array of square cylindrical cells with surface capacity. An 11 x 11 array of resistors, simulating a quarter of the unit structure, was separated into intra- and extra-cellular regions by rows of capacitors corresponding to surface membrane areas from 3 x 3 to 11 x 11 or 7.5% to 100%. The extended Rayleigh equation predicted the cell concentrations and membrane capacities to within a few percent from boundary resistance and capacity measurements at low frequencies. This single example suggests that analytical solutions for other, similar two- and three-dimensional problems may be approximated up to near 100% concentrations and that there may be analytical justifications for such analogue solutions of Laplace equations.

A generalized analytical model for radiative transfer in vacuum thermal insulation of space vehicles

NASA Astrophysics Data System (ADS)

Krainova, Irina V.; Dombrovsky, Leonid A.; Nenarokomov, Aleksey V.; Budnik, Sergey A.; Titov, Dmitry M.; Alifanov, Oleg M.

2017-08-01

The previously developed spectral model for radiative transfer in vacuum thermal insulation of space vehicles is generalized to take into account possible thermal contact between a fibrous spacer and one of the neighboring aluminum foil layers. An approximate analytical solution based on slightly modified two-flux approximation for radiative transfer in a semi-transparent fibrous spacer is derived. It was shown that thermal contact between the spacer and adjacent foil may decrease significantly the quality of thermal insulation because of an increase in radiative flux to/from the opposite aluminum foil. Theoretical predictions are confirmed by comparison with new results of laboratory experiments.

The Potential-Well Distortion Effect and Coherent Instabilities of Electron Bunches in Storage Rings

NASA Astrophysics Data System (ADS)

Korchuganov, V. N.; Smygacheva, A. S.; Fomin, E. A.

2018-05-01

The effect of electromagnetic interaction between electron bunches and the vacuum chamber of a storage ring on the longitudinal motion of bunches is studied. Specifically, the potential-well distortion effect and the so-called coherent instabilities of coupled bunches are considered. An approximate analytical solution for the frequencies of incoherent oscillations of bunches distributed arbitrarily within the ring is obtained for a distorted potential well. A new approach to determining frequencies of coherent oscillations and an approximate analytical relation for estimating the stability of a system of bunches as a function of their distribution in the accelerator orbit are presented.

Solar neutrino masses and mixing from bilinear R-parity broken supersymmetry: Analytical versus numerical results

NASA Astrophysics Data System (ADS)

Díaz, M.; Hirsch, M.; Porod, W.; Romão, J.; Valle, J.

2003-07-01

We give an analytical calculation of solar neutrino masses and mixing at one-loop order within bilinear R-parity breaking supersymmetry, and compare our results to the exact numerical calculation. Our method is based on a systematic perturbative expansion of R-parity violating vertices to leading order. We find in general quite good agreement between the approximate and full numerical calculations, but the approximate expressions are much simpler to implement. Our formalism works especially well for the case of the large mixing angle Mikheyev-Smirnov-Wolfenstein solution, now strongly favored by the recent KamLAND reactor neutrino data.

Boundary enhanced effects on the existence of quadratic solitons

NASA Astrophysics Data System (ADS)

Chen, Manna; Zhang, Ting; Li, Wenjie; Lu, Daquan; Guo, Qi; Hu, Wei

2018-05-01

We investigate, both analytically and numerically, the boundary enhanced effects exerted on the quadratic solitons consisting of fundamental waves and oscillatory second harmonics in the presence of boundary conditions. The nonlocal analogy predicts that the soliton for fundamental wave is supported by the balance between equivalent nonlinear confinement and diffraction (or dispersion). Under Snyder and Mitchell's strongly nonlocal approximation, we obtain the analytical soliton solutions both with and without the boundary conditions to show the impact of boundary conditions. We can distinguish explicitly the nonlinear confinement between the second harmonic mutual interaction and the enhanced effects caused by remote boundaries. Those boundary enhanced effects on the existence of solitons can be positive or negative, which depend on both sample size and nonlocal parameter. The piecewise existence regime of solitons can be explained analytically. The analytical soliton solutions are verified by the numerical ones and the discrepancy between them is also discussed.

Approximate method for calculating a thickwalled cylinder with rigidly clamped ends

NASA Astrophysics Data System (ADS)

Andreev, Vladimir

2018-03-01

Numerous papers dealing with the calculations of cylindrical bodies [1 -8 and others] have shown that analytic and numerical-analytical solutions in both homogeneous and inhomogeneous thick-walled shells can be obtained quite simply, using expansions in Fourier series on trigonometric functions, if the ends are hinged movable (sliding support). It is much more difficult to solve the problem of calculating shells with builtin ends.

High-beta analytic equilibria in circular, elliptical, and D-shaped large aspect ratio axisymmetric configurations with poloidal and toroidal flows

NASA Astrophysics Data System (ADS)

López, O. E.; Guazzotto, L.

2017-03-01

The Grad-Shafranov-Bernoulli system of equations is a single fluid magnetohydrodynamical description of axisymmetric equilibria with mass flows. Using a variational perturbative approach [E. Hameiri, Phys. Plasmas 20, 024504 (2013)], analytic approximations for high-beta equilibria in circular, elliptical, and D-shaped cross sections in the high aspect ratio approximation are found, which include finite toroidal and poloidal flows. Assuming a polynomial dependence of the free functions on the poloidal flux, the equilibrium problem is reduced to an inhomogeneous Helmholtz partial differential equation (PDE) subject to homogeneous Dirichlet conditions. An application of the Green's function method leads to a closed form for the circular solution and to a series solution in terms of Mathieu functions for the elliptical case, which is valid for arbitrary elongations. To extend the elliptical solution to a D-shaped domain, a boundary perturbation in terms of the triangularity is used. A comparison with the code FLOW [L. Guazzotto et al., Phys. Plasmas 11(2), 604-614 (2004)] is presented for relevant scenarios.

Nonequilibrium thermodynamics and boundary conditions for reaction and transport in heterogeneous media

NASA Astrophysics Data System (ADS)

Gaspard, Pierre; Kapral, Raymond

2018-05-01

Nonequilibrium interfacial thermodynamics is formulated in the presence of surface reactions for the study of diffusiophoresis in isothermal systems. As a consequence of microreversibility and Onsager-Casimir reciprocal relations, diffusiophoresis, i.e., the coupling of the tangential components of the pressure tensor to the concentration gradients of solute species, has a reciprocal effect where the interfacial currents of solutes are coupled to the slip velocity. The presence of surface reactions is shown to modify the diffusiophoretic and reciprocal effects at the fluid-solid interface. The thin-layer approximation is used to describe the solution flowing near a reactive solid interface. Analytic formulas describing the diffusiophoretic and reciprocal effects are deduced in the thin-layer approximation and tested numerically for the Poiseuille flow of a solution between catalytic planar surfaces.

On the evolution of perturbations to solutions of the Kadomtsev-Petviashvilli equation using the Benney-Luke equation

NASA Astrophysics Data System (ADS)

Ablowitz, Mark J.; Curtis, Christopher W.

2011-05-01

The Benney-Luke equation, which arises as a long wave asymptotic approximation of water waves, contains the Kadomtsev-Petviashvilli (KP) equation as a leading-order maximal balanced approximation. The question analyzed is how the Benney-Luke equation modifies the so-called web solutions of the KP equation. It is found that the Benney-Luke equation introduces dispersive radiation which breaks each of the symmetric soliton-like humps well away from the interaction region of the KP web solution into a tail of multi-peaked oscillating profiles behind the main solitary hump. Computation indicates that the wave structure is modified near the center of the interaction region. Both analytical and numerical techniques are employed for working with non-periodic, non-decaying solutions on unbounded domains.

The analysis of solutions behaviour of Van der Pol Duffing equation describing local brain hemodynamics

NASA Astrophysics Data System (ADS)

Cherevko, A. A.; Bord, E. E.; Khe, A. K.; Panarin, V. A.; Orlov, K. J.

2017-10-01

This article proposes the generalized model of Van der Pol — Duffing equation for describing the relaxation oscillations in local brain hemodynamics. This equation connects the velocity and pressure of blood flow in cerebral vessels. The equation is individual for each patient, since the coefficients are unique. Each set of coefficients is built based on clinical data obtained during neurosurgical operation in Siberian Federal Biomedical Research Center named after Academician E. N. Meshalkin. The equation has solutions of different structure defined by the coefficients and right side. We investigate the equations for different patients considering peculiarities of their vessel systems. The properties of approximate analytical solutions are studied. Amplitude-frequency and phase-frequency characteristics are built for the small-dimensional solution approximations.

Benchmark problems and solutions

NASA Technical Reports Server (NTRS)

Tam, Christopher K. W.

1995-01-01

The scientific committee, after careful consideration, adopted six categories of benchmark problems for the workshop. These problems do not cover all the important computational issues relevant to Computational Aeroacoustics (CAA). The deciding factor to limit the number of categories to six was the amount of effort needed to solve these problems. For reference purpose, the benchmark problems are provided here. They are followed by the exact or approximate analytical solutions. At present, an exact solution for the Category 6 problem is not available.

Extinction efficiencies from DDA calculations solved for finite circular cylinders and disks

NASA Technical Reports Server (NTRS)

Withrow, J. R.; Cox, S. K.

1993-01-01

One of the most commonly noted uncertainties with respect to the modeling of cirrus clouds and their effect upon the planetary radiation balance is the disputed validity of the use of Mie scattering results as an approximation to the scattering results of the hexagonal plates and columns found in cirrus clouds. This approximation has historically been a kind of default, a result of the lack of an appropriate analytical solution of Maxwell's equations to particles other than infinite cylinders and spheroids. Recently, however, the use of such approximate techniques as the Discrete Dipole Approximation has made scattering solutions on such particles a computationally intensive but feasible possibility. In this study, the Discrete Dipole Approximation (DDA) developed by Flatau (1992) is used to find such solutions for homogeneous, circular cylinders and disks. This can serve to not only assess the validity of the current radiative transfer schemes which are available for the study of cirrus but also to extend the current approximation of equivalent spheres to an approximation of second order, homogeneous finite circular cylinders and disks. The results will be presented in the form of a single variable, the extinction efficiency.

Application of matched asymptotic expansions to lunar and interplanetary trajectories. Volume 1: Technical discussion

NASA Technical Reports Server (NTRS)

Lancaster, J. E.

1973-01-01

Previously published asymptotic solutions for lunar and interplanetary trajectories have been modified and combined to formulate a general analytical solution to the problem on N-bodies. The earlier first-order solutions, derived by the method of matched asymptotic expansions, have been extended to second order for the purpose of obtaining increased accuracy. The derivation of the second-order solution is summarized by showing the essential steps, some in functional form. The general asymptotic solution has been used as a basis for formulating a number of analytical two-point boundary value solutions. These include earth-to-moon, one- and two-impulse moon-to-earth, and interplanetary solutions. The results show that the accuracies of the asymptotic solutions range from an order of magnitude better than conic approximations to that of numerical integration itself. Also, since no iterations are required, the asymptotic boundary value solutions are obtained in a fraction of the time required for comparable numerically integrated solutions. The subject of minimizing the second-order error is discussed, and recommendations made for further work directed toward achieving a uniform accuracy in all applications.

Nonlinear storage models of unconfined flow through a shallow aquifer on an inclined base and their quasi-steady flow application

NASA Astrophysics Data System (ADS)

Varvaris, Ioannis; Gravanis, Elias; Koussis, Antonis; Akylas, Evangelos

2013-04-01

Hillslope processes involving flow through an inclined shallow aquifer range from subsurface stormflow to stream base flow (drought flow, or groundwater recession flow). In the case of recharge, the infiltrating water moves vertically as unsaturated flow until it reaches the saturated groundwater, where the flow is approximately parallel to the base of the aquifer. Boussinesq used the Dupuit-Forchheimer (D-F) hydraulic theory to formulate unconfined groundwater flow through a soil layer resting on an impervious inclined bed, deriving a nonlinear equation for the flow rate that consists of a linear gravity-driven component and a quadratic pressure-gradient component. Inserting that flow rate equation into the differential storage balance equation (volume conservation) Boussinesq obtained a nonlinear second-order partial differential equation for the depth. So far however, only few special solutions have been advanced for that governing equation. The nonlinearity of the equation of Boussinesq is the major obstacle to deriving a general analytical solution for the depth profile of unconfined flow on a sloping base with recharge (from which the discharges could be then determined). Henderson and Wooding (1964) were able to obtain an exact analytical solution for steady unconfined flow on a sloping base, with recharge, and their work deserves special note in the realm of solutions of the nonlinear equation of Boussinesq. However, the absence of a general solution for the transient case, which is of practical interest to hydrologists, has been the motivation for developing approximate solutions of the non-linear equation of Boussinesq. In this work, we derive the aquifer storage function by integrating analytically over the aquifer base the depth profiles resulting from the complete nonlinear Boussinesq equation for steady flow. This storage function consists of a linear and a nonlinear outflow-dependent term. Then, we use this physics-based storage function in the transient storage balance over the hillslope, obtaining analytical solutions of the outflow and the storage, for recharge and drainage, via a quasi-steady flow calculation. The hydraulically derived storage model is thus embedded in a quasi-steady approximation of transient unconfined flow in sloping aquifers. We generalise this hydrologic model of groundwater flow by modifying the storage function to be the weighted sum of the linear and the nonlinear storage terms, determining the weighting factor objectively from a known integral quantity of the flow (either an initial volume of water stored in the aquifer or a drained water volume). We demonstrate the validity of this model through comparisons with experimental data and simulation results.

A Numerical Approximation Framework for the Stochastic Linear Quadratic Regulator on Hilbert Spaces

DOE Office of Scientific and Technical Information (OSTI.GOV)

Levajković, Tijana, E-mail: tijana.levajkovic@uibk.ac.at, E-mail: t.levajkovic@sf.bg.ac.rs; Mena, Hermann, E-mail: hermann.mena@uibk.ac.at; Tuffaha, Amjad, E-mail: atufaha@aus.edu

We present an approximation framework for computing the solution of the stochastic linear quadratic control problem on Hilbert spaces. We focus on the finite horizon case and the related differential Riccati equations (DREs). Our approximation framework is concerned with the so-called “singular estimate control systems” (Lasiecka in Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators: applications to boundary and point control problems, 2004) which model certain coupled systems of parabolic/hyperbolic mixed partial differential equations with boundary or point control. We prove that the solutions of the approximate finite-dimensional DREs converge to the solutionmore » of the infinite-dimensional DRE. In addition, we prove that the optimal state and control of the approximate finite-dimensional problem converge to the optimal state and control of the corresponding infinite-dimensional problem.« less

Mackey-Glass equation driven by fractional Brownian motion

NASA Astrophysics Data System (ADS)

Nguyen, Dung Tien

2012-11-01

In this paper we introduce a fractional stochastic version of the Mackey-Glass model which is a potential candidate to model objects in biology and finance. By a semi-martingale approximate approach we find an semi-analytical expression for the solution.

Motion of a Drop on a Solid Surface Due to a Wettability Gradient

NASA Technical Reports Server (NTRS)

Subramanian, R.; Moumen, Nadjoua; McLaughlin, John B.

2005-01-01

The hydrodynamic force experienced by a spherical-cap drop moving on a solid surface is obtained from two approximate analytical solutions and used to predict the quasi-steady speed of the drop in a wettability gradient. One solution is based on approximation of the shape of the drop as a collection of wedges, and the other is based on lubrication theory. Also, asymptotic results from both approximations for small contact angles, as well as an asymptotic result from lubrication theory that is good when the length scale of the drop is large compared with the slip length, are given. The results for the hydrodynamic force also can be used to predict the quasi-steady speed of a drop sliding down an incline.

First-order analytic propagation of satellites in the exponential atmosphere of an oblate planet

NASA Astrophysics Data System (ADS)

Martinusi, Vladimir; Dell'Elce, Lamberto; Kerschen, Gaëtan

2017-04-01

The paper offers the fully analytic solution to the motion of a satellite orbiting under the influence of the two major perturbations, due to the oblateness and the atmospheric drag. The solution is presented in a time-explicit form, and takes into account an exponential distribution of the atmospheric density, an assumption that is reasonably close to reality. The approach involves two essential steps. The first one concerns a new approximate mathematical model that admits a closed-form solution with respect to a set of new variables. The second step is the determination of an infinitesimal contact transformation that allows to navigate between the new and the original variables. This contact transformation is obtained in exact form, and afterwards a Taylor series approximation is proposed in order to make all the computations explicit. The aforementioned transformation accommodates both perturbations, improving the accuracy of the orbit predictions by one order of magnitude with respect to the case when the atmospheric drag is absent from the transformation. Numerical simulations are performed for a low Earth orbit starting at an altitude of 350 km, and they show that the incorporation of drag terms into the contact transformation generates an error reduction by a factor of 7 in the position vector. The proposed method aims at improving the accuracy of analytic orbit propagation and transforming it into a viable alternative to the computationally intensive numerical methods.

Core-based intrinsic fiber-optic absorption sensor for the detection of volatile organic compounds

NASA Astrophysics Data System (ADS)

Klunder, Gregory L.; Russo, Richard E.

1995-03-01

A core-based intrinsic fiber-optic absorption sensor has been developed and tested for the detection of volatile organic compounds. The distal ends of transmitting and receiving fibers are connected by a small cylindrical section of an optically clear silicone rubber. The silicone rubber acts both as a light pipe and as a selective membrane into which the analyte molecules can diffuse. The sensor has been used to detect volatile organics (trichloroethylene, 1,1-dichloroethylene, and benzene) in both aqueous solutions and in the vapor phase or headspace. Absorption spectra obtained in the near-infrared (near-IR) provide qualitative and quantitative information about the analyte. Water, which has strong broad-band absorption in the near-IR, is excluded from the spectra because of the hydrophobic properties of the silicone rubber. The rate-limiting step is shown to be the diffusion through the Nernstian boundary layer surrounding the sensor and not the diffusion through the silicone polymer. The rate of analyte diffusion into the sensor, as measured by the t(sub 90) values (the time required for the sensor to reach 90% of the equilibrium value), is 30 min for measurements in aqueous solutions and approximately 3 min for measurements made in the headspace. The limit of detection obtained with this sensor is approximately 1.1 ppm for trichloroethylene in an aqueous solution.

Physical models for the normal YORP and diurnal Yarkovsky effects

NASA Astrophysics Data System (ADS)

Golubov, O.; Kravets, Y.; Krugly, Yu. N.; Scheeres, D. J.

2016-06-01

We propose an analytic model for the normal Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) and diurnal Yarkovsky effects experienced by a convex asteroid. Both the YORP torque and the Yarkovsky force are expressed as integrals of a universal function over the surface of an asteroid. Although in general this function can only be calculated numerically from the solution of the heat conductivity equation, approximate solutions can be obtained in quadratures for important limiting cases. We consider three such simplified models: Rubincam's approximation (zero heat conductivity), low thermal inertia limit (including the next order correction and thus valid for small heat conductivity), and high thermal inertia limit (valid for large heat conductivity). All three simplified models are compared with the exact solution.

An analytical solution of Richards' equation providing the physical basis of SCS curve number method and its proportionality relationship

NASA Astrophysics Data System (ADS)

Hooshyar, Milad; Wang, Dingbao

2016-08-01

The empirical proportionality relationship, which indicates that the ratio of cumulative surface runoff and infiltration to their corresponding potentials are equal, is the basis of the extensively used Soil Conservation Service Curve Number (SCS-CN) method. The objective of this paper is to provide the physical basis of the SCS-CN method and its proportionality hypothesis from the infiltration excess runoff generation perspective. To achieve this purpose, an analytical solution of Richards' equation is derived for ponded infiltration in shallow water table environment under the following boundary conditions: (1) the soil is saturated at the land surface; and (2) there is a no-flux boundary which moves downward. The solution is established based on the assumptions of negligible gravitational effect, constant soil water diffusivity, and hydrostatic soil moisture profile between the no-flux boundary and water table. Based on the derived analytical solution, the proportionality hypothesis is a reasonable approximation for rainfall partitioning at the early stage of ponded infiltration in areas with a shallow water table for coarse textured soils.

Algorithms and analytical solutions for rapidly approximating long-term dispersion from line and area sources

NASA Astrophysics Data System (ADS)

Barrett, Steven R. H.; Britter, Rex E.

Predicting long-term mean pollutant concentrations in the vicinity of airports, roads and other industrial sources are frequently of concern in regulatory and public health contexts. Many emissions are represented geometrically as ground-level line or area sources. Well developed modelling tools such as AERMOD and ADMS are able to model dispersion from finite (i.e. non-point) sources with considerable accuracy, drawing upon an up-to-date understanding of boundary layer behaviour. Due to mathematical difficulties associated with line and area sources, computationally expensive numerical integration schemes have been developed. For example, some models decompose area sources into a large number of line sources orthogonal to the mean wind direction, for which an analytical (Gaussian) solution exists. Models also employ a time-series approach, which involves computing mean pollutant concentrations for every hour over one or more years of meteorological data. This can give rise to computer runtimes of several days for assessment of a site. While this may be acceptable for assessment of a single industrial complex, airport, etc., this level of computational cost precludes national or international policy assessments at the level of detail available with dispersion modelling. In this paper, we extend previous work [S.R.H. Barrett, R.E. Britter, 2008. Development of algorithms and approximations for rapid operational air quality modelling. Atmospheric Environment 42 (2008) 8105-8111] to line and area sources. We introduce approximations which allow for the development of new analytical solutions for long-term mean dispersion from line and area sources, based on hypergeometric functions. We describe how these solutions can be parameterized from a single point source run from an existing advanced dispersion model, thereby accounting for all processes modelled in the more costly algorithms. The parameterization method combined with the analytical solutions for long-term mean dispersion are shown to produce results several orders of magnitude more efficiently with a loss of accuracy small compared to the absolute accuracy of advanced dispersion models near sources. The method can be readily incorporated into existing dispersion models, and may allow for additional computation time to be expended on modelling dispersion processes more accurately in future, rather than on accounting for source geometry.

Nonperturbative confinement in quantum chromodynamics. I. Study of an approximate equation of Mandelstam

NASA Astrophysics Data System (ADS)

Atkinson, D.; Drohm, J. K.; Johnson, P. W.; Stam, K.

1981-11-01

An approximated form of the Dyson-Schwinger equation for the gluon propagator in quarkless QCD is subjected to nonlinear functional and numerical analysis. It is found that solutions exist, and that these have a double pole at the origin of the square of the propagator momentum, together with an accumulation of soft branch points. This analytic structure is strongly suggestive of confinement by infrared slavery.

Viscous Rayleigh-Taylor instability in spherical geometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mikaelian, Karnig O.

We consider viscous fluids in spherical geometry, a lighter fluid supporting a heavier one. Chandrasekhar [Q. J. Mech. Appl. Math. 8, 1 (1955)] analyzed this unstable configuration providing the equations needed to find, numerically, the exact growth rates for the ensuing Rayleigh-Taylor instability. He also derived an analytic but approximate solution. We point out a weakness in his approximate dispersion relation (DR) and offer one that is to some extent improved.

Viscous Rayleigh-Taylor instability in spherical geometry

DOE PAGES

Mikaelian, Karnig O.

2016-02-08

We consider viscous fluids in spherical geometry, a lighter fluid supporting a heavier one. Chandrasekhar [Q. J. Mech. Appl. Math. 8, 1 (1955)] analyzed this unstable configuration providing the equations needed to find, numerically, the exact growth rates for the ensuing Rayleigh-Taylor instability. He also derived an analytic but approximate solution. We point out a weakness in his approximate dispersion relation (DR) and offer one that is to some extent improved.

Formation Flying through Geodesic Motion and the Different Geometrical Requirements

DTIC Science & Technology

2006-09-01

APPROXIMATE SOLUTIONS IN A CLOHESSY - WILTSHIRE -TYPE SYSTEM Despite the assumed approximation, the simplified problem (5)+(6) remains complicated for an...analytical approach. For a further simplification let us introduce a CW ( Clohessy - Wiltshire ) referential system [1], [3]. Consider that the trajectory...momentum. Figure 2: The Clohessy - Wiltshire -type referential system, CX1Y1Z1. Neglecting the second order terms, equation (9) reads: (10

The scaling of oblique plasma double layers

NASA Technical Reports Server (NTRS)

Borovsky, J. E.

1983-01-01

Strong oblique plasma double layers are investigated using three methods, i.e., electrostatic particle-in-cell simulations, numerical solutions to the Poisson-Vlasov equations, and analytical approximations to the Poisson-Vlasov equations. The solutions to the Poisson-Vlasov equations and numerical simulations show that strong oblique double layers scale in terms of Debye lengths. For very large potential jumps, theory and numerical solutions indicate that all effects of the magnetic field vanish and the oblique double layers follow the same scaling relation as the field-aligned double layers.

Semiclassical Dynamicswith Exponentially Small Error Estimates

NASA Astrophysics Data System (ADS)

Hagedorn, George A.; Joye, Alain

We construct approximate solutions to the time-dependent Schrödingerequation for small values of ħ. If V satisfies appropriate analyticity and growth hypotheses and , these solutions agree with exact solutions up to errors whose norms are bounded by for some C and γ>0. Under more restrictive hypotheses, we prove that for sufficiently small T', implies the norms of the errors are bounded by for some C', γ'>0, and σ > 0.

Analytic Description of Critical Point Nuclei in a Spherical-Axially Deformed Shape Phase Transition

DOE Office of Scientific and Technical Information (OSTI.GOV)

Iachello, F.

2001-07-30

An approximate solution at the critical point of the spherical to axially deformed shape phase transition in nuclei is presented. The eigenvalues of the Hamiltonian are expressed in terms of zeros of Bessel functions of irrational order.

Versions of the collocation and least squares method for solving biharmonic equations in non-canonical domains

NASA Astrophysics Data System (ADS)

Belyaev, V. A.; Shapeev, V. P.

2017-10-01

New versions of the collocations and least squares method of high-order accuracy are proposed and implemented for the numerical solution of the boundary value problems for the biharmonic equation in non-canonical domains. The solution of the biharmonic equation is used for simulating the stress-strain state of an isotropic plate under the action of transverse load. The differential problem is projected into a space of fourth-degree polynomials by the CLS method. The boundary conditions for the approximate solution are put down exactly on the boundary of the computational domain. The versions of the CLS method are implemented on the grids which are constructed in two different ways. It is shown that the approximate solution of problems converges with high order. Thus it matches with high accuracy with the analytical solution of the test problems in the case of known solution in the numerical experiments on the convergence of the solution of various problems on a sequence of grids.

Analysis of high-speed rotating flow inside gas centrifuge casing

NASA Astrophysics Data System (ADS)

Pradhan, Sahadev, , Dr.

2017-10-01

The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity Ω_i while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential (χ) , which is derived from the equations of motion in an axisymmetric (r - z) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified.

Analysis of high-speed rotating flow inside gas centrifuge casing

NASA Astrophysics Data System (ADS)

Pradhan, Sahadev, , Dr.

2017-09-01

The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity Ωi while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential (χ) , which is derived from the equations of motion in an axisymmetric (r - z) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified.

Analysis of high-speed rotating flow inside gas centrifuge casing

NASA Astrophysics Data System (ADS)

Pradhan, Sahadev

2017-11-01

The generalized analytical model for the radial boundary layer inside the gas centrifuge casing in which the inner cylinder is rotating at a constant angular velocity Ωi while the outer one is stationary, is formulated for studying the secondary gas flow field due to wall thermal forcing, inflow/outflow of light gas along the boundaries, as well as due to the combination of the above two external forcing. The analytical model includes the sixth order differential equation for the radial boundary layer at the cylindrical curved surface in terms of master potential (χ) , which is derived from the equations of motion in an axisymmetric (r - z) plane. The linearization approximation is used, where the equations of motion are truncated at linear order in the velocity and pressure disturbances to the base flow, which is a solid-body rotation. Additional approximations in the analytical model include constant temperature in the base state (isothermal compressible Couette flow), high aspect ratio (length is large compared to the annular gap), high Reynolds number, but there is no limitation on the Mach number. The discrete eigenvalues and eigenfunctions of the linear operators (sixth-order in the radial direction for the generalized analytical equation) are obtained. The solutions for the secondary flow is determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified.

Accurate analytical periodic solution of the elliptical Kepler equation using the Adomian decomposition method

NASA Astrophysics Data System (ADS)

Alshaery, Aisha; Ebaid, Abdelhalim

2017-11-01

Kepler's equation is one of the fundamental equations in orbital mechanics. It is a transcendental equation in terms of the eccentric anomaly of a planet which orbits the Sun. Determining the position of a planet in its orbit around the Sun at a given time depends upon the solution of Kepler's equation, which we will solve in this paper by the Adomian decomposition method (ADM). Several properties of the periodicity of the obtained approximate solutions have been proved in lemmas. Our calculations demonstrated a rapid convergence of the obtained approximate solutions which are displayed in tables and graphs. Also, it has been shown in this paper that only a few terms of the Adomian decomposition series are sufficient to achieve highly accurate numerical results for any number of revolutions of the Earth around the Sun as a consequence of the periodicity property. Numerically, the four-term approximate solution coincides with the Bessel-Fourier series solution in the literature up to seven decimal places at some values of the time parameter and nine decimal places at other values. Moreover, the absolute error approaches zero using the nine term approximate Adomian solution. In addition, the approximate Adomian solutions for the eccentric anomaly have been used to show the convergence of the approximate radial distances of the Earth from the Sun for any number of revolutions. The minimal distance (perihelion) and maximal distance (aphelion) approach 147 million kilometers and 152.505 million kilometers, respectively, and these coincide with the well known results in astronomical physics. Therefore, the Adomian decomposition method is validated as an effective tool to solve Kepler's equation for elliptical orbits.

Technical Note: Approximate solution of transient drawdown for constant-flux pumping at a partially penetrating well in a radial two-zone confined aquifer

NASA Astrophysics Data System (ADS)

Huang, C.-S.; Yang, S.-Y.; Yeh, H.-D.

2015-03-01

An aquifer consisting of a skin zone and a formation zone is considered as a two-zone aquifer. Existing solutions for the problem of constant-flux pumping (CFP) in a two-zone confined aquifer involve laborious calculation. This study develops a new approximate solution for the problem based on a mathematical model including two steady-state flow equations with different hydraulic parameters for the skin and formation zones. A partially penetrating well may be treated as the Neumann condition with a known flux along the screened part and zero flux along the unscreened part. The aquifer domain is finite with an outer circle boundary treated as the Dirichlet condition. The steady-state drawdown solution of the model is derived by the finite Fourier cosine transform. Then, an approximate transient solution is developed by replacing the radius of the boundary in the steady-state solution with an analytical expression for a dimensionless time-dependent radius of influence. The approximate solution is capable of predicting good temporal drawdown distributions over the whole pumping period except at the early stage. A quantitative criterion for the validity of neglecting the vertical flow component due to a partially penetrating well is also provided. Conventional models considering radial flow without the vertical component for the CFP have good accuracy if satisfying the criterion.

Optimal Configurations for Rotating Spacecraft Formations

NASA Technical Reports Server (NTRS)

Hughes, Steven P.; Hall, Christopher D.

2000-01-01

In this paper a new class of formations that maintain a constant shape as viewed from the Earth is introduced. An algorithm is developed to place n spacecraft in a constant shape formation spaced equally in time using the classical orbital elements. To first order, the dimensions of the formation are shown to be simple functions of orbit eccentricity and inclination. The performance of the formation is investigated over a Keplerian orbit using a performance measure based on a weighted average of the angular separations between spacecraft in formation. Analytic approximations are developed that yield optimum configurations for different values of n. The analytic approximations are shown to be in excellent agreement with the exact solutions.

Propagation of Bessel-Gaussian beams through a double-apertured fractional Fourier transform optical system.

PubMed

Tang, Bin; Jiang, Chun; Zhu, Haibin

2012-08-01

Based on the scalar diffraction theory and the fact that a hard-edged aperture function can be expanded into a finite sum of complex Gaussian functions, an approximate analytical solution for Bessel-Gaussian (BG) beams propagating through a double-apertured fractional Fourier transform (FrFT) system is derived in the cylindrical coordinate. By using the approximate analytical formulas, the propagation properties of BG beams passing through a double-apertured FrFT optical system have been studied in detail by some typical numerical examples. The results indicate that the double-apertured FrFT optical system provides a convenient way for controlling the properties of the BG beams by properly choosing the optical parameters.

Analytical solution and numerical simulation of the liquid nitrogen freezing-temperature field of a single pipe

NASA Astrophysics Data System (ADS)

Cai, Haibing; Xu, Liuxun; Yang, Yugui; Li, Longqi

2018-05-01

Artificial liquid nitrogen freezing technology is widely used in urban underground engineering due to its technical advantages, such as simple freezing system, high freezing speed, low freezing temperature, high strength of frozen soil, and absence of pollution. However, technical difficulties such as undefined range of liquid nitrogen freezing and thickness of frozen wall gradually emerge during the application process. Thus, the analytical solution of the freezing-temperature field of a single pipe is established considering the freezing temperature of soil and the constant temperature of freezing pipe wall. This solution is then applied in a liquid nitrogen freezing project. Calculation results show that the radius of freezing front of liquid nitrogen is proportional to the square root of freezing time. The radius of the freezing front also decreases with decreased the freezing temperature, and the temperature gradient of soil decreases with increased distance from the freezing pipe. The radius of cooling zone in the unfrozen area is approximately four times the radius of the freezing front. Meanwhile, the numerical simulation of the liquid nitrogen freezing-temperature field of a single pipe is conducted using the Abaqus finite-element program. Results show that the numerical simulation of soil temperature distribution law well agrees with the analytical solution, further verifies the reliability of the established analytical solution of the liquid nitrogen freezing-temperature field of a single pipe.

The Investigation of Optimal Discrete Approximations for Real Time Flight Simulations

NASA Technical Reports Server (NTRS)

Parrish, E. A.; Mcvey, E. S.; Cook, G.; Henderson, K. C.

1976-01-01

The results are presented of an investigation of discrete approximations for real time flight simulation. Major topics discussed include: (1) consideration of the particular problem of approximation of continuous autopilots by digital autopilots; (2) use of Bode plots and synthesis of transfer functions by asymptotic fits in a warped frequency domain; (3) an investigation of the various substitution formulas, including the effects of nonlinearities; (4) use of pade approximation to the solution of the matrix exponential arising from the discrete state equations; and (5) an analytical integration of the state equation using interpolated input.

Nonlinear electroelastic deformations of dielectric elastomer composites: II - Non-Gaussian elastic dielectrics

NASA Astrophysics Data System (ADS)

Lefèvre, Victor; Lopez-Pamies, Oscar

2017-02-01

This paper presents an analytical framework to construct approximate homogenization solutions for the macroscopic elastic dielectric response - under finite deformations and finite electric fields - of dielectric elastomer composites with two-phase isotropic particulate microstructures. The central idea consists in employing the homogenization solution derived in Part I of this work for ideal elastic dielectric composites within the context of a nonlinear comparison medium method - this is derived as an extension of the comparison medium method of Lopez-Pamies et al. (2013) in nonlinear elastostatics to the coupled realm of nonlinear electroelastostatics - to generate in turn a corresponding solution for composite materials with non-ideal elastic dielectric constituents. Complementary to this analytical framework, a hybrid finite-element formulation to construct homogenization solutions numerically (in three dimensions) is also presented. The proposed analytical framework is utilized to work out a general approximate homogenization solution for non-Gaussian dielectric elastomers filled with nonlinear elastic dielectric particles that may exhibit polarization saturation. The solution applies to arbitrary (non-percolative) isotropic distributions of filler particles. By construction, it is exact in the limit of small deformations and moderate electric fields. For finite deformations and finite electric fields, its accuracy is demonstrated by means of direct comparisons with finite-element solutions. Aimed at gaining physical insight into the extreme enhancement in electrostriction properties displayed by emerging dielectric elastomer composites, various cases wherein the filler particles are of poly- and mono-disperse sizes and exhibit different types of elastic dielectric behavior are discussed in detail. Contrary to an initial conjecture in the literature, it is found (inter alia) that the isotropic addition of a small volume fraction of stiff (semi-)conducting/high-permittivity particles to dielectric elastomers does not lead to the extreme electrostriction enhancements observed in experiments. It is posited that such extreme enhancements are the manifestation of interphasial phenomena.

A new analytical solar radiation pressure model for current BeiDou satellites: IGGBSPM

PubMed Central

Tan, Bingfeng; Yuan, Yunbin; Zhang, Baocheng; Hsu, Hou Ze; Ou, Jikun

2016-01-01

An analytical solar radiation pressure (SRP) model, IGGBSPM (an abbreviation for Institute of Geodesy and Geophysics BeiDou Solar Pressure Model), has been developed for three BeiDou satellite types, namely, geostationary orbit (GEO), inclined geosynchronous orbit (IGSO) and medium earth orbit (MEO), based on a ray-tracing method. The performance of IGGBSPM was assessed based on numerical integration, SLR residuals and analyses of empirical SRP parameters (except overlap computations). The numerical results show that the integrated orbit resulting from IGGBSPM differs from the precise ephemerides by approximately 5 m and 2 m for GEO and non-GEO satellites, respectively. Moreover, when IGGBSPM is used as an a priori model to enhance the ECOM (5-parameter) model with stochastic pulses, named ECOM + APR, for precise orbit determination, the SLR RMS residual improves by approximately 20–25 percent over the ECOM-only solution during the yaw-steering period and by approximately 40 percent during the yaw-fixed period. For the BeiDou GEO01 satellite, improvements of 18 and 32 percent can be achieved during the out-of-eclipse season and during the eclipse season, respectively. An investigation of the estimated ECOM D0 parameters indicated that the β-angle dependence that is evident in the ECOM-only solution is no longer present in the ECOM + APR solution. PMID:27595795

A new analytical solar radiation pressure model for current BeiDou satellites: IGGBSPM.

PubMed

Tan, Bingfeng; Yuan, Yunbin; Zhang, Baocheng; Hsu, Hou Ze; Ou, Jikun

2016-09-06

An analytical solar radiation pressure (SRP) model, IGGBSPM (an abbreviation for Institute of Geodesy and Geophysics BeiDou Solar Pressure Model), has been developed for three BeiDou satellite types, namely, geostationary orbit (GEO), inclined geosynchronous orbit (IGSO) and medium earth orbit (MEO), based on a ray-tracing method. The performance of IGGBSPM was assessed based on numerical integration, SLR residuals and analyses of empirical SRP parameters (except overlap computations). The numerical results show that the integrated orbit resulting from IGGBSPM differs from the precise ephemerides by approximately 5 m and 2 m for GEO and non-GEO satellites, respectively. Moreover, when IGGBSPM is used as an a priori model to enhance the ECOM (5-parameter) model with stochastic pulses, named ECOM + APR, for precise orbit determination, the SLR RMS residual improves by approximately 20-25 percent over the ECOM-only solution during the yaw-steering period and by approximately 40 percent during the yaw-fixed period. For the BeiDou GEO01 satellite, improvements of 18 and 32 percent can be achieved during the out-of-eclipse season and during the eclipse season, respectively. An investigation of the estimated ECOM D0 parameters indicated that the β-angle dependence that is evident in the ECOM-only solution is no longer present in the ECOM + APR solution.

Numerically stable formulas for a particle-based explicit exponential integrator

NASA Astrophysics Data System (ADS)

Nadukandi, Prashanth

2015-05-01

Numerically stable formulas are presented for the closed-form analytical solution of the X-IVAS scheme in 3D. This scheme is a state-of-the-art particle-based explicit exponential integrator developed for the particle finite element method. Algebraically, this scheme involves two steps: (1) the solution of tangent curves for piecewise linear vector fields defined on simplicial meshes and (2) the solution of line integrals of piecewise linear vector-valued functions along these tangent curves. Hence, the stable formulas presented here have general applicability, e.g. exact integration of trajectories in particle-based (Lagrangian-type) methods, flow visualization and computer graphics. The Newton form of the polynomial interpolation definition is used to express exponential functions of matrices which appear in the analytical solution of the X-IVAS scheme. The divided difference coefficients in these expressions are defined in a piecewise manner, i.e. in a prescribed neighbourhood of removable singularities their series approximations are computed. An optimal series approximation of divided differences is presented which plays a critical role in this methodology. At least ten significant decimal digits in the formula computations are guaranteed to be exact using double-precision floating-point arithmetic. The worst case scenarios occur in the neighbourhood of removable singularities found in fourth-order divided differences of the exponential function.

Heating-frequency-dependent thermal conductivity: An analytical solution from diffusive to ballistic regime and its relevance to phonon scattering measurements

NASA Astrophysics Data System (ADS)

Yang, Fan; Dames, Chris

2015-04-01

The heating-frequency dependence of the apparent thermal conductivity in a semi-infinite body with periodic planar surface heating is explained by an analytical solution to the Boltzmann transport equation. This solution is obtained using a two-flux model and gray mean free time approximation and verified numerically with a lattice Boltzmann method and numerical results from the literature. Extending the gray solution to the nongray regime leads to an integral transform and accumulation-function representation of the phonon scattering spectrum, where the natural variable is mean free time rather than mean free path, as often used in previous work. The derivation leads to an approximate cutoff conduction similar in spirit to that of Koh and Cahill [Phys. Rev. B 76, 075207 (2007), 10.1103/PhysRevB.76.075207] except that the most appropriate criterion involves the heater frequency rather than thermal diffusion length. The nongray calculations are consistent with Koh and Cahill's experimental observation that the apparent thermal conductivity shows a stronger heater-frequency dependence in a SiGe alloy than in natural Si. Finally these results are demonstrated using a virtual experiment, which fits the phase lag between surface temperature and heat flux to obtain the apparent thermal conductivity and accumulation function.

A Bayesian Hierarchical Model for Glacial Dynamics Based on the Shallow Ice Approximation and its Evaluation Using Analytical Solutions

NASA Astrophysics Data System (ADS)

Gopalan, Giri; Hrafnkelsson, Birgir; Aðalgeirsdóttir, Guðfinna; Jarosch, Alexander H.; Pálsson, Finnur

2018-03-01

Bayesian hierarchical modeling can assist the study of glacial dynamics and ice flow properties. This approach will allow glaciologists to make fully probabilistic predictions for the thickness of a glacier at unobserved spatio-temporal coordinates, and it will also allow for the derivation of posterior probability distributions for key physical parameters such as ice viscosity and basal sliding. The goal of this paper is to develop a proof of concept for a Bayesian hierarchical model constructed, which uses exact analytical solutions for the shallow ice approximation (SIA) introduced by Bueler et al. (2005). A suite of test simulations utilizing these exact solutions suggests that this approach is able to adequately model numerical errors and produce useful physical parameter posterior distributions and predictions. A byproduct of the development of the Bayesian hierarchical model is the derivation of a novel finite difference method for solving the SIA partial differential equation (PDE). An additional novelty of this work is the correction of numerical errors induced through a numerical solution using a statistical model. This error correcting process models numerical errors that accumulate forward in time and spatial variation of numerical errors between the dome, interior, and margin of a glacier.

Flow Generated by a Partially Penetrating Well in a Leaky Two-Aquifer System with a Storative Semiconfining Layer

NASA Astrophysics Data System (ADS)

Sepulveda, N.; Rohrer, K.

2008-05-01

The permeability of the semiconfining layers of the highly productive Floridan Aquifer System may be large enough to invalidate the assumptions of the leaky aquifer theory. These layers are the intermediate confining and the middle semiconfining units. The analysis of aquifer-test data with analytical solutions of the ground-water flow equation developed with the approximation of a low hydraulic conductivity ratio between the semiconfining layer and the aquifer may lead to inaccurate hydraulic parameters. An analytical solution is presented here for the flow in a confined leaky aquifer, the overlying storative semiconfining layer, and the unconfined aquifer, generated by a partially penetrating well in a two-aquifer system, and allowing vertical and lateral flow components to occur in the semiconfining layer. The equations describing flow caused by a partially penetrating production well are solved analytically to provide a method to accurately determine the hydraulic parameters in the confined aquifer, semiconfining layer, and unconfined aquifer from aquifer-test data. Analysis of the drawdown data from an aquifer test performed in central Florida showed that the flow solution presented here for the semiconfining layer provides a better match and a more unique identification of the hydraulic parameters than an analytical solution that considers only vertical flow in the semiconfining layer.

A meshless method using radial basis functions for numerical solution of the two-dimensional KdV-Burgers equation

NASA Astrophysics Data System (ADS)

Zabihi, F.; Saffarian, M.

2016-07-01

The aim of this article is to obtain the numerical solution of the two-dimensional KdV-Burgers equation. We construct the solution by using a different approach, that is based on using collocation points. The solution is based on using the thin plate splines radial basis function, which builds an approximated solution with discretizing the time and the space to small steps. We use a predictor-corrector scheme to avoid solving the nonlinear system. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

Analytical Solution for the Anisotropic Rabi Model: Effects of Counter-Rotating Terms

NASA Astrophysics Data System (ADS)

Zhang, Guofeng; Zhu, Hanjie

2015-03-01

The anisotropic Rabi model, which was proposed recently, differs from the original Rabi model: the rotating and counter-rotating terms are governed by two different coupling constants. This feature allows us to vary the counter-rotating interaction independently and explore the effects of it on some quantum properties. In this paper, we eliminate the counter-rotating terms approximately and obtain the analytical energy spectrums and wavefunctions. These analytical results agree well with the numerical calculations in a wide range of the parameters including the ultrastrong coupling regime. In the weak counter-rotating coupling limit we find out that the counter-rotating terms can be considered as the shifts to the parameters of the Jaynes-Cummings model. This modification shows the validness of the rotating-wave approximation on the assumption of near-resonance and relatively weak coupling. Moreover, the analytical expressions of several physics quantities are also derived, and the results show the break-down of the U(1)-symmetry and the deviation from the Jaynes-Cummings model.

Analytical solution for the anisotropic Rabi model: effects of counter-rotating terms.

PubMed

Zhang, Guofeng; Zhu, Hanjie

2015-03-04

The anisotropic Rabi model, which was proposed recently, differs from the original Rabi model: the rotating and counter-rotating terms are governed by two different coupling constants. This feature allows us to vary the counter-rotating interaction independently and explore the effects of it on some quantum properties. In this paper, we eliminate the counter-rotating terms approximately and obtain the analytical energy spectrums and wavefunctions. These analytical results agree well with the numerical calculations in a wide range of the parameters including the ultrastrong coupling regime. In the weak counter-rotating coupling limit we find out that the counter-rotating terms can be considered as the shifts to the parameters of the Jaynes-Cummings model. This modification shows the validness of the rotating-wave approximation on the assumption of near-resonance and relatively weak coupling. Moreover, the analytical expressions of several physics quantities are also derived, and the results show the break-down of the U(1)-symmetry and the deviation from the Jaynes-Cummings model.

Analytical Solution for the Anisotropic Rabi Model: Effects of Counter-Rotating Terms

PubMed Central

Zhang, Guofeng; Zhu, Hanjie

2015-01-01

The anisotropic Rabi model, which was proposed recently, differs from the original Rabi model: the rotating and counter-rotating terms are governed by two different coupling constants. This feature allows us to vary the counter-rotating interaction independently and explore the effects of it on some quantum properties. In this paper, we eliminate the counter-rotating terms approximately and obtain the analytical energy spectrums and wavefunctions. These analytical results agree well with the numerical calculations in a wide range of the parameters including the ultrastrong coupling regime. In the weak counter-rotating coupling limit we find out that the counter-rotating terms can be considered as the shifts to the parameters of the Jaynes-Cummings model. This modification shows the validness of the rotating-wave approximation on the assumption of near-resonance and relatively weak coupling. Moreover, the analytical expressions of several physics quantities are also derived, and the results show the break-down of the U(1)-symmetry and the deviation from the Jaynes-Cummings model. PMID:25736827

Stability of semidiscrete approximations for hyperbolic initial-boundary-value problems: Stationary modes

NASA Technical Reports Server (NTRS)

Warming, Robert F.; Beam, Richard M.

1988-01-01

Spatially discrete difference approximations for hyperbolic initial-boundary-value problems (IBVPs) require numerical boundary conditions in addition to the analytical boundary conditions specified for the differential equations. Improper treatment of a numerical boundary condition can cause instability of the discrete IBVP even though the approximation is stable for the pure initial-value or Cauchy problem. In the discrete IBVP stability literature there exists a small class of discrete approximations called borderline cases. For nondissipative approximations, borderline cases are unstable according to the theory of the Gustafsson, Kreiss, and Sundstrom (GKS) but they may be Lax-Richtmyer stable or unstable in the L sub 2 norm on a finite domain. It is shown that borderline approximation can be characterized by the presence of a stationary mode for the finite-domain problem. A stationary mode has the property that it does not decay with time and a nontrivial stationary mode leads to algebraic growth of the solution norm with mesh refinement. An analytical condition is given which makes it easy to detect a stationary mode; several examples of numerical boundary conditions are investigated corresponding to borderline cases.

Maximal liquid bridges between horizontal cylinders

NASA Astrophysics Data System (ADS)

Cooray, Himantha; Huppert, Herbert E.; Neufeld, Jerome A.

2016-08-01

We investigate two-dimensional liquid bridges trapped between pairs of identical horizontal cylinders. The cylinders support forces owing to surface tension and hydrostatic pressure that balance the weight of the liquid. The shape of the liquid bridge is determined by analytically solving the nonlinear Laplace-Young equation. Parameters that maximize the trapping capacity (defined as the cross-sectional area of the liquid bridge) are then determined. The results show that these parameters can be approximated with simple relationships when the radius of the cylinders is small compared with the capillary length. For such small cylinders, liquid bridges with the largest cross-sectional area occur when the centre-to-centre distance between the cylinders is approximately twice the capillary length. The maximum trapping capacity for a pair of cylinders at a given separation is linearly related to the separation when it is small compared with the capillary length. The meniscus slope angle of the largest liquid bridge produced in this regime is also a linear function of the separation. We additionally derive approximate solutions for the profile of a liquid bridge, using the linearized Laplace-Young equation. These solutions analytically verify the above-mentioned relationships obtained for the maximization of the trapping capacity.

Secular dynamics of multiplanetary circumbinary systems: stationary solutions and binary-planet secular resonance

NASA Astrophysics Data System (ADS)

Andrade-Ines, Eduardo; Robutel, Philippe

2018-01-01

We present an analytical formalism to study the secular dynamics of a system consisting of N-2 planets orbiting a binary star in outer orbits. We introduce a canonical coordinate system and expand the disturbing function in terms of canonical elliptic elements, combining both Legendre polynomials and Laplace coefficients, to obtain a general formalism for the secular description of this type of configuration. With a quadratic approximation of the development, we present a simplified analytical solution for the planetary orbits for both the single planet and the two-planet cases. From the two-planet model, we show that the inner planet accelerates the precession rate of the binary pericenter, which, in turn, may enter in resonance with the secular frequency of the outer planet, characterizing a secular resonance. We calculate an analytical expression for the approximate location of this resonance and apply it to known circumbinary systems, where we show that it can occur at relatively close orbits, for example at 2.4 au for the Kepler-38 system. With a more refined model, we analyse the dynamics of this secular resonance and we show that a bifurcation of the corresponding fixed points can affect the long- term evolution and stability of planetary systems. By comparing our results with complete integrations of the exact equations of motion, we verified the accuracy of our analytical model.

Pseudospin symmetry of the Dirac equation for a Möbius square plus Mie type potential with a Coulomb-like tensor interaction via SUSYQM

NASA Astrophysics Data System (ADS)

Akpan, N. Ikot; Zarrinkamar, S.; Eno, J. Ibanga; Maghsoodi, E.; Hassanabadi, H.

2014-01-01

We investigate the approximate solution of the Dirac equation for a combination of Möbius square and Mie type potentials under the pseudospin symmetry limit by using supersymmetry quantum mechanics. We obtain the bound-state energy equation and the corresponding spinor wave functions in an approximate analytical manner. We comment on the system via various useful figures and tables.

Fast-slow asymptotic for semi-analytical ignition criteria in FitzHugh-Nagumo system.

PubMed

Bezekci, B; Biktashev, V N

2017-09-01

We study the problem of initiation of excitation waves in the FitzHugh-Nagumo model. Our approach follows earlier works and is based on the idea of approximating the boundary between basins of attraction of propagating waves and of the resting state as the stable manifold of a critical solution. Here, we obtain analytical expressions for the essential ingredients of the theory by singular perturbation using two small parameters, the separation of time scales of the activator and inhibitor and the threshold in the activator's kinetics. This results in a closed analytical expression for the strength-duration curve.

Analytic saddlepoint approximation for ionization energy loss distributions

DOE PAGES

Sjue, Sky K. L.; George, Jr., Richard Neal; Mathews, David Gregory

2017-07-27

Here, we present a saddlepoint approximation for ionization energy loss distributions, valid for arbitrary relativistic velocities of the incident particle 0 < v/c < 1, provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form solution closely related to Moyal’s distribution. This distribution is intended for use in simulations with relatively low computational overhead. The approximation generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov’s κ approaches 1.

Analytic saddlepoint approximation for ionization energy loss distributions

NASA Astrophysics Data System (ADS)

Sjue, S. K. L.; George, R. N.; Mathews, D. G.

2017-09-01

We present a saddlepoint approximation for ionization energy loss distributions, valid for arbitrary relativistic velocities of the incident particle 0 < v / c < 1 , provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form solution closely related to Moyal's distribution. This distribution is intended for use in simulations with relatively low computational overhead. The approximation generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov's κ approaches 1.

Analytic saddlepoint approximation for ionization energy loss distributions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sjue, Sky K. L.; George, Jr., Richard Neal; Mathews, David Gregory

Here, we present a saddlepoint approximation for ionization energy loss distributions, valid for arbitrary relativistic velocities of the incident particle 0 < v/c < 1, provided that ionizing collisions are still the dominant energy loss mechanism. We derive a closed form solution closely related to Moyal’s distribution. This distribution is intended for use in simulations with relatively low computational overhead. The approximation generally reproduces the Vavilov most probable energy loss and full width at half maximum to better than 1% and 10%, respectively, with significantly better agreement as Vavilov’s κ approaches 1.

A Full-Maxwell Approach for Large-Angle Polar Wander of Viscoelastic Bodies

NASA Astrophysics Data System (ADS)

Hu, H.; van der Wal, W.; Vermeersen, L. L. A.

2017-12-01

For large-angle long-term true polar wander (TPW) there are currently two types of nonlinear methods which give approximated solutions: those assuming that the rotational axis coincides with the axis of maximum moment of inertia (MoI), which simplifies the Liouville equation, and those based on the quasi-fluid approximation, which approximates the Love number. Recent studies show that both can have a significant bias for certain models. Therefore, we still lack an (semi)analytical method which can give exact solutions for large-angle TPW for a model based on Maxwell rheology. This paper provides a method which analytically solves the MoI equation and adopts an extended iterative procedure introduced in Hu et al. (2017) to obtain a time-dependent solution. The new method can be used to simulate the effect of a remnant bulge or models in different hydrostatic states. We show the effect of the viscosity of the lithosphere on long-term, large-angle TPW. We also simulate models without hydrostatic equilibrium and show that the choice of the initial stress-free shape for the elastic (or highly viscous) lithosphere of a given model is as important as its thickness for obtaining a correct TPW behavior. The initial shape of the lithosphere can be an alternative explanation to mantle convection for the difference between the observed and model predicted flattening. Finally, it is concluded that based on the quasi-fluid approximation, TPW speed on Earth and Mars is underestimated, while the speed of the rotational axis approaching the end position on Venus is overestimated.

A new approximation for pore pressure accumulation in marine sediment due to water waves

NASA Astrophysics Data System (ADS)

Jeng, D.-S.; Seymour, B. R.; Li, J.

2007-01-01

The residual mechanism of wave-induced pore water pressure accumulation in marine sediments is re-examined. An analytical approximation is derived using a linear relation for pore pressure generation in cyclic loading, and mistakes in previous solutions (Int. J. Numer. Anal. Methods Geomech. 2001; 25:885-907; J. Offshore Mech. Arctic Eng. (ASME) 1989; 111(1):1-11) are corrected. A numerical scheme is then employed to solve the case with a non-linear relation for pore pressure generation. Both analytical and numerical solutions are verified with experimental data (Laboratory and field investigation of wave-sediment interaction. Joseph H. Defrees Hydraulics Laboratory, School of Civil and Environmental Engineering, Cornell University, Ithaca, NY, 1983), and provide a better prediction of pore pressure accumulation than the previous solution (J. Offshore Mech. Arctic Eng. (ASME) 1989; 111(1):1-11). The parametric study concludes that the pore pressure accumulation and use of full non-linear relation of pore pressure become more important under the following conditions: (1) large wave amplitude, (2) longer wave period, (3) shallow water, (4) shallow soil and (5) softer soils with a low consolidation coefficient. Copyright

Quantification of Water Flux in Vesicular Systems.

PubMed

Hannesschläger, Christof; Barta, Thomas; Siligan, Christine; Horner, Andreas

2018-06-04

Water transport across lipid membranes is fundamental to all forms of life and plays a major role in health and disease. However, not only typical water facilitators like aquaporins facilitate water flux, but also transporters, ion channels or receptors represent potent water pathways. The efforts directed towards a mechanistic understanding of water conductivity determinants in transmembrane proteins, the development of water flow inhibitors, and the creation of biomimetic membranes with incorporated membrane proteins or artificial water channels depend on reliable and accurate ways of quantifying water permeabilities P f . A conventional method is to subject vesicles to an osmotic gradient in a stopped-flow device: Fast recordings of scattered light intensity are converted into the time course of vesicle volume change. Even though an analytical solution accurately acquiring P f from scattered light intensities exists, approximations potentially misjudging P f by orders of magnitude are used. By means of computational and experimental data we point out that erroneous results such as that the single channel water permeability p f depends on the osmotic gradient are direct results of such approximations. Finally, we propose an empirical solution of which calculated permeability values closely match those calculated with the analytical solution in the relevant range of parameters.

Viscous damping and spring force in periodic perforated planar microstructures when the Reynolds’ equation cannot be applied

PubMed Central

Homentcovschi, Dorel; Miles, Ronald N.

2010-01-01

A model of squeeze-film behavior is developed based on Stokes’ equations for viscous, compressible isothermal flows. The flow domain is an axisymmetrical, unit cell approximation of a planar, periodic, perforated microstructure. The model is developed for cases when the lubrication approximation cannot be applied. The complex force generated by vibrations of the diaphragm driving the flow has two components: the damping force and the spring force. While for large frequencies the spring force dominates, at low (acoustical) frequencies the damping force is the most important part. The analytical approach developed here yields an explicit formula for both forces. In addition, using a finite element software package, the damping force is also obtained numerically. A comparison is made between the analytic result, numerical solution, and some experimental data found in the literature, which validates the analytic formula and provides compelling arguments about its value in designing microelectomechanical devices. PMID:20329828

Spectrum Evolution of Accelerating or Slowing down Soliton at its Propagation in a Medium with Gold Nanorods

NASA Astrophysics Data System (ADS)

Trofimov, Vyacheslav A.; Lysak, Tatiana M.

2018-04-01

We investigate both numerically and analytically the spectrum evolution of a novel type soliton - nonlinear chirped accelerating or decelerating soliton - at a femtosecond pulse propagation in a medium containing noble nanoparticles. In our consideration, we take into account one- or two-photon absorption of laser radiation by nanorods, and time-dependent nanorod aspect ratio changing due to their melting or reshaping because of laser energy absorption. The chirped solitons are formed due to the trapping of laser radiation by the nanorods reshaping fronts, if a positive or negative phase-amplitude grating is induced by laser radiation. Accelerating or slowing down chirped soliton formation is accompanied by the soliton spectrum blue or red shift. To prove our numerical results, we derived the approximate analytical law for the spectrum maximum intensity evolution along the propagation coordinate, based on earlier developed approximate analytical solutions for accelerating and decelerating solitons.

Benchmark solutions for the galactic ion transport equations: Energy and spatially dependent problems

NASA Technical Reports Server (NTRS)

Ganapol, Barry D.; Townsend, Lawrence W.; Wilson, John W.

1989-01-01

Nontrivial benchmark solutions are developed for the galactic ion transport (GIT) equations in the straight-ahead approximation. These equations are used to predict potential radiation hazards in the upper atmosphere and in space. Two levels of difficulty are considered: (1) energy independent, and (2) spatially independent. The analysis emphasizes analytical methods never before applied to the GIT equations. Most of the representations derived have been numerically implemented and compared to more approximate calculations. Accurate ion fluxes are obtained (3 to 5 digits) for nontrivial sources. For monoenergetic beams, both accurate doses and fluxes are found. The benchmarks presented are useful in assessing the accuracy of transport algorithms designed to accommodate more complex radiation protection problems. In addition, these solutions can provide fast and accurate assessments of relatively simple shield configurations.

An analytical solution for predicting the transient seepage from a subsurface drainage system

NASA Astrophysics Data System (ADS)

Xin, Pei; Dan, Han-Cheng; Zhou, Tingzhang; Lu, Chunhui; Kong, Jun; Li, Ling

2016-05-01

Subsurface drainage systems have been widely used to deal with soil salinization and waterlogging problems around the world. In this paper, a mathematical model was introduced to quantify the transient behavior of the groundwater table and the seepage from a subsurface drainage system. Based on the assumption of a hydrostatic pressure distribution, the model considered the pore-water flow in both the phreatic and vadose soil zones. An approximate analytical solution for the model was derived to quantify the drainage of soils which were initially water-saturated. The analytical solution was validated against laboratory experiments and a 2-D Richards equation-based model, and found to predict well the transient water seepage from the subsurface drainage system. A saturated flow-based model was also tested and found to over-predict the time required for drainage and the total water seepage by nearly one order of magnitude, in comparison with the experimental results and the present analytical solution. During drainage, a vadose zone with a significant water storage capacity developed above the phreatic surface. A considerable amount of water still remained in the vadose zone at the steady state with the water table situated at the drain bottom. Sensitivity analyses demonstrated that effects of the vadose zone were intensified with an increased thickness of capillary fringe, capillary rise and/or burying depth of drains, in terms of the required drainage time and total water seepage. The analytical solution provides guidance for assessing the capillary effects on the effectiveness and efficiency of subsurface drainage systems for combating soil salinization and waterlogging problems.

On the Coplanar Integrable Case of the Twice-Averaged Hill Problem with Central Body Oblateness

NASA Astrophysics Data System (ADS)

Vashkov'yak, M. A.

2018-01-01

The twice-averaged Hill problem with the oblateness of the central planet is considered in the case where its equatorial plane coincides with the plane of its orbital motion relative to the perturbing body. A qualitative study of this so-called coplanar integrable case was begun by Y. Kozai in 1963 and continued by M.L. Lidov and M.V. Yarskaya in 1974. However, no rigorous analytical solution of the problem can be obtained due to the complexity of the integrals. In this paper we obtain some quantitative evolution characteristics and propose an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of satellite orbit elements. The methodical accuracy has been estimated for several orbits of artificial lunar satellites by comparison with the numerical solution of the evolution system.

Pressure-coupled combustion response model for solid propellants based on Zeldovich-Novozhilov approach

NASA Technical Reports Server (NTRS)

Harstad, K. G.; Strand, L. D.

1987-01-01

An exact analytical solution is given to the problem of long-time propellant thermal response to a specified pressure oscillation. Coupling to the gas phase is made using the quasisteady Zeldovich-Novozhilov approximation. Explicit linear and lowest order (quadratic) nonlinear expressions for propellant response are obtained from the implicit nonlinear solutions. Using these expressions, response curves are presented for an ammonium perchlorate composite propellant and HMX monopropellant.

Mathematical analysis of the boundary-integral based electrostatics estimation approximation for molecular solvation: exact results for spherical inclusions.

PubMed

Bardhan, Jaydeep P; Knepley, Matthew G

2011-09-28

We analyze the mathematically rigorous BIBEE (boundary-integral based electrostatics estimation) approximation of the mixed-dielectric continuum model of molecular electrostatics, using the analytically solvable case of a spherical solute containing an arbitrary charge distribution. Our analysis, which builds on Kirkwood's solution using spherical harmonics, clarifies important aspects of the approximation and its relationship to generalized Born models. First, our results suggest a new perspective for analyzing fast electrostatic models: the separation of variables between material properties (the dielectric constants) and geometry (the solute dielectric boundary and charge distribution). Second, we find that the eigenfunctions of the reaction-potential operator are exactly preserved in the BIBEE model for the sphere, which supports the use of this approximation for analyzing charge-charge interactions in molecular binding. Third, a comparison of BIBEE to the recent GBε theory suggests a modified BIBEE model capable of predicting electrostatic solvation free energies to within 4% of a full numerical Poisson calculation. This modified model leads to a projection-framework understanding of BIBEE and suggests opportunities for future improvements. © 2011 American Institute of Physics

Magnetohydrodynamic Jump Conditions for Oblique Relativistic Shocks with Gyrotropic Pressure

NASA Technical Reports Server (NTRS)

Double, Glen P.; Baring, Matthew G.; Jones, Frank C.; Ellison, Donald C.

2003-01-01

Shock jump conditions, i.e., the specification of the downstream parameters of the gas in terms of the upstream parameters, are obtained for steady-state, plane shocks with oblique magnetic fields and arbitrary flow speeds. This is done by combining the continuity of particle number flux and the electromagnetic boundary conditions at the shock with the magnetohydrodynamic conservation laws derived from the stress-energy tensor. For ultrarelativistic and nonrelativistic shocks, the jump conditions may be solved analytically. For mildly relativistic shocks, analytic solutions are obtained for isotropic pressure using an approximation for the adiabatic index that is valid in high sonic Mach number cases. Examples assuming isotropic pressure illustrate how the shock compression ratio depends on the shock speed and obliquity. In the more general case of gyrotropic pressure, the jump conditions cannot be solved analytically with- out additional assumptions, and the effects of gyrotropic pressure are investigated by parameterizing the distribution of pressure parallel and perpendicular to the magnetic field. Our numerical solutions reveal that relatively small departures from isotropy (e.g., approximately 20%) produce significant changes in the shock compression ratio, r , at all shock Lorentz factors, including ultrarelativistic ones, where an analytic solution with gyrotropic pressure is obtained. In particular, either dynamically important fields or significant pressure anisotropies can incur marked departures from the canonical gas dynamic value of r = 3 for a shocked ultrarelativistic flow and this may impact models of particle acceleration in gamma-ray bursts and other environments where relativistic shocks are inferred. The jump conditions presented apply directly to test-particle acceleration, and will facilitate future self-consistent numerical modeling of particle acceleration at oblique, relativistic shocks; such models include the modification of the fluid velocity profile due to the contribution of energetic particles to the momentum and energy fluxes.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Dechant, Lawrence J.

Wave packet analysis provides a connection between linear small disturbance theory and subsequent nonlinear turbulent spot flow behavior. The traditional association between linear stability analysis and nonlinear wave form is developed via the method of stationary phase whereby asymptotic (simplified) mean flow solutions are used to estimate dispersion behavior and stationary phase approximation are used to invert the associated Fourier transform. The resulting process typically requires nonlinear algebraic equations inversions that can be best performed numerically, which partially mitigates the value of the approximation as compared to a more complete, e.g. DNS or linear/nonlinear adjoint methods. To obtain a simpler,more » closed-form analytical result, the complete packet solution is modeled via approximate amplitude (linear convected kinematic wave initial value problem) and local sinusoidal (wave equation) expressions. Significantly, the initial value for the kinematic wave transport expression follows from a separable variable coefficient approximation to the linearized pressure fluctuation Poisson expression. The resulting amplitude solution, while approximate in nature, nonetheless, appears to mimic many of the global features, e.g. transitional flow intermittency and pressure fluctuation magnitude behavior. A low wave number wave packet models also recover meaningful auto-correlation and low frequency spectral behaviors.« less

A pertinent approach to solve nonlinear fuzzy integro-differential equations.

PubMed

Narayanamoorthy, S; Sathiyapriya, S P

2016-01-01

Fuzzy integro-differential equations is one of the important parts of fuzzy analysis theory that holds theoretical as well as applicable values in analytical dynamics and so an appropriate computational algorithm to solve them is in essence. In this article, we use parametric forms of fuzzy numbers and suggest an applicable approach for solving nonlinear fuzzy integro-differential equations using homotopy perturbation method. A clear and detailed description of the proposed method is provided. Our main objective is to illustrate that the construction of appropriate convex homotopy in a proper way leads to highly accurate solutions with less computational work. The efficiency of the approximation technique is expressed via stability and convergence analysis so as to guarantee the efficiency and performance of the methodology. Numerical examples are demonstrated to verify the convergence and it reveals the validity of the presented numerical technique. Numerical results are tabulated and examined by comparing the obtained approximate solutions with the known exact solutions. Graphical representations of the exact and acquired approximate fuzzy solutions clarify the accuracy of the approach.

Efficiency of unconstrained minimization techniques in nonlinear analysis

NASA Technical Reports Server (NTRS)

Kamat, M. P.; Knight, N. F., Jr.

1978-01-01

Unconstrained minimization algorithms have been critically evaluated for their effectiveness in solving structural problems involving geometric and material nonlinearities. The algorithms have been categorized as being zeroth, first, or second order depending upon the highest derivative of the function required by the algorithm. The sensitivity of these algorithms to the accuracy of derivatives clearly suggests using analytically derived gradients instead of finite difference approximations. The use of analytic gradients results in better control of the number of minimizations required for convergence to the exact solution.

Roy-Steiner equations for pion-nucleon scattering

NASA Astrophysics Data System (ADS)

Ditsche, C.; Hoferichter, M.; Kubis, B.; Meißner, U.-G.

2012-06-01

Starting from hyperbolic dispersion relations, we derive a closed system of Roy-Steiner equations for pion-nucleon scattering that respects analyticity, unitarity, and crossing symmetry. We work out analytically all kernel functions and unitarity relations required for the lowest partial waves. In order to suppress the dependence on the high energy regime we also consider once- and twice-subtracted versions of the equations, where we identify the subtraction constants with subthreshold parameters. Assuming Mandelstam analyticity we determine the maximal range of validity of these equations. As a first step towards the solution of the full system we cast the equations for the π π to overline N N partial waves into the form of a Muskhelishvili-Omnès problem with finite matching point, which we solve numerically in the single-channel approximation. We investigate in detail the role of individual contributions to our solutions and discuss some consequences for the spectral functions of the nucleon electromagnetic form factors.

Free and Forced Vibrations of Thick-Walled Anisotropic Cylindrical Shells

NASA Astrophysics Data System (ADS)

Marchuk, A. V.; Gnedash, S. V.; Levkovskii, S. A.

2017-03-01

Two approaches to studying the free and forced axisymmetric vibrations of cylindrical shell are proposed. They are based on the three-dimensional theory of elasticity and division of the original cylindrical shell with concentric cross-sectional circles into several coaxial cylindrical shells. One approach uses linear polynomials to approximate functions defined in plan and across the thickness. The other approach also uses linear polynomials to approximate functions defined in plan, but their variation with thickness is described by the analytical solution of a system of differential equations. Both approaches have approximation and arithmetic errors. When determining the natural frequencies by the semi-analytical finite-element method in combination with the divide and conqure method, it is convenient to find the initial frequencies by the finite-element method. The behavior of the shell during free and forced vibrations is analyzed in the case where the loading area is half the shell thickness

Approximate transient and long time limit solutions for the band broadening induced by the thin sidewall-layer in liquid chromatography columns.

PubMed

Broeckhoven, Ken; Desmet, Gert

2007-11-16

Using a combination of both analytical and numerical techniques, approximate analytical expressions have been established for the transient and long time limit band broadening, originating from the presence of a thin disturbed sidewall layer in liquid chromatography columns, including packed, monolithic as well as microfabricated columns. The established expressions can be used to compare the importance of a thin disturbed sidewall layer with that of other radial heterogeneity effects (such as transcolumn packing density variations due to the relief of packing stresses). The expressions are independent of the actual velocity profile inside the layer as long as the disturbed sidewall layer occupies less than 2.5% of the column width.

Interactions of bright and dark solitons with localized PT-symmetric potentials.

PubMed

Karjanto, N; Hanif, W; Malomed, B A; Susanto, H

2015-02-01

We study collisions of moving nonlinear-Schrödinger solitons with a PT-symmetric dipole embedded into the one-dimensional self-focusing or defocusing medium. Accurate analytical results are produced for bright solitons, and, in a more qualitative form, for dark ones. In the former case, an essential aspect of the approximation is that it must take into regard the intrinsic chirp of the soliton, thus going beyond the framework of the simplest quasi-particle description of the soliton's dynamics. Critical velocities separating reflection and transmission of the incident bright solitons are found by means of numerical simulations, and in the approximate semi-analytical form. An exact solution for the dark soliton pinned by the complex PT-symmetric dipole is produced too.

Finite element modeling of light propagation in fruit under illumination of continuous-wave beam

USDA-ARS?s Scientific Manuscript database

Spatially-resolved spectroscopy provides a means for measuring the optical properties of biological tissues, based on analytical solutions to diffusion approximation for semi-infinite media under the normal illumination of infinitely small size light beam. The method is, however, prone to error in m...

Finite element simulation of light transfer in turbid media under structured illumination

USDA-ARS?s Scientific Manuscript database

Spatial-frequency domain (SFD) imaging technique allows to estimate the optical properties of biological tissues in a wide field of view. The technique is, however, prone to error in measurement because the two crucial assumptions used for deriving the analytical solution to diffusion approximation ...

Determination of synthetic ferric chelates used as fertilizers by liquid chromatography-electrospray/mass spectrometry in agricultural matrices.

PubMed

Alvarez-Fernández, Ana; Orera, Irene; Abadía, Javier; Abadía, Anunciación

2007-01-01

A high-performance liquid chromatography-electrospray ionization/mass spectrometry (time of flight) method has been developed for the simultaneous determination of synthetic Fe(III)-chelates used as fertilizers. Analytes included the seven major Fe(III)-chelates used in agriculture, Fe(III)-EDTA, Fe(III)-DTPA, Fe(III)-HEDTA, Fe(III)-CDTA, Fe(III)-o,oEDDHA, Fe(III)-o,pEDDHA, and Fe(III)-EDDHMA, and the method was validated using isotope labeled (57)Fe(III)-chelates as internal standards. Calibration curves had R values in the range 0.9962-0.9997. Limits of detection and quantification were in the ranges 3-164 and 14-945 pmol, respectively. Analyte concentrations could be determined between the limits of quantification and 25 muM (racemic and meso Fe(III)-o,oEDDHA and Fe(III)-EDDHMA) or 50 muM (Fe(III)-EDTA, Fe(III)-HEDTA, Fe(III)-DTPA, Fe(III)-CDTA and Fe(III)-o,pEDDHA). The average intraday repeatability values were approximately 0.5 and 5% for retention time and peak area, respectively, whereas the interday repeatability values were approximately 0.7 and 8% for retention time and peak area, respectively. The method was validated using four different agricultural matrices, including nutrient solution, irrigation water, soil solution, and plant xylem exudates, spiked with Fe(III)-chelate standards and their stable isotope-labeled corresponding chelates. Analyte recoveries found were in the ranges 92-101% (nutrient solution), 89-102% (irrigation water), 82-100% (soil solution), and 70-111% (plant xylem exudates). Recoveries depended on the analyte, with Fe(III)-EDTA and Fe(III)-DTPA showing the lowest recoveries (average values of 87 and 88%, respectively, for all agricultural matrices used), whereas for other analytes recoveries were between 91 and 101%. The method was also used to determine the real concentrations of Fe(III)-chelates in commercial fertilizers. Furthermore, the method is also capable of resolving two more synthetic Fe(III)-chelates, Fe(III)-EDDHSA and Fe(III)-EDDCHA, whose exact quantification is not currently possible because of lack of commercial standards.

Analytical solutions for tomato peeling with combined heat flux and convective boundary conditions

NASA Astrophysics Data System (ADS)

Cuccurullo, G.; Giordano, L.; Metallo, A.

2017-11-01

Peeling of tomatoes by radiative heating is a valid alternative to steam or lye, which are expensive and pollutant methods. Suitable energy densities are required in order to realize short time operations, thus involving only a thin layer under the tomato surface. This paper aims to predict the temperature field in rotating tomatoes exposed to the source irradiation. Therefore, a 1D unsteady analytical model is presented, which involves a semi-infinite slab subjected to time dependent heating while convective heat transfer takes place on the exposed surface. In order to account for the tomato rotation, the heat source is described as the positive half-wave of a sinusoidal function. The problem being linear, the solution is derived following the Laplace Transform Method. In addition, an easy-to-handle solution for the problem at hand is presented, which assumes a differentiable function for approximating the source while neglecting convective cooling, the latter contribution turning out to be negligible for the context at hand. A satisfying agreement between the two analytical solutions is found, therefore, an easy procedure for a proper design of the dry heating system can be set up avoiding the use of numerical simulations.

Technical Note: Approximate solution of transient drawdown for constant-flux pumping at a partially penetrating well in a radial two-zone confined aquifer

NASA Astrophysics Data System (ADS)

Huang, C.-S.; Yang, S.-Y.; Yeh, H.-D.

2015-06-01

An aquifer consisting of a skin zone and a formation zone is considered as a two-zone aquifer. Existing solutions for the problem of constant-flux pumping in a two-zone confined aquifer involve laborious calculation. This study develops a new approximate solution for the problem based on a mathematical model describing steady-state radial and vertical flows in a two-zone aquifer. Hydraulic parameters in these two zones can be different but are assumed homogeneous in each zone. A partially penetrating well may be treated as the Neumann condition with a known flux along the screened part and zero flux along the unscreened part. The aquifer domain is finite with an outer circle boundary treated as the Dirichlet condition. The steady-state drawdown solution of the model is derived by the finite Fourier cosine transform. Then, an approximate transient solution is developed by replacing the radius of the aquifer domain in the steady-state solution with an analytical expression for a dimensionless time-dependent radius of influence. The approximate solution is capable of predicting good temporal drawdown distributions over the whole pumping period except at the early stage. A quantitative criterion for the validity of neglecting the vertical flow due to a partially penetrating well is also provided. Conventional models considering radial flow without the vertical component for the constant-flux pumping have good accuracy if satisfying the criterion.

Cylindrical and spherical solitary waves in an electron-acoustic plasma with vortex electron distribution

NASA Astrophysics Data System (ADS)

Demiray, Hilmi; El-Zahar, Essam R.

2018-04-01

We consider the nonlinear propagation of electron-acoustic waves in a plasma composed of a cold electron fluid, hot electrons obeying a trapped/vortex-like distribution, and stationary ions. The basic nonlinear equations of the above described plasma are re-examined in the cylindrical (spherical) coordinates by employing the reductive perturbation technique. The modified cylindrical (spherical) KdV equation with fractional power nonlinearity is obtained as the evolution equation. Due to the nature of nonlinearity, this evolution equation cannot be reduced to the conventional KdV equation. A new family of closed form analytical approximate solution to the evolution equation and a comparison with numerical solution are presented and the results are depicted in some 2D and 3D figures. The results reveal that both solutions are in good agreement and the method can be used to obtain a new progressive wave solution for such evolution equations. Moreover, the resulting closed form analytical solution allows us to carry out a parametric study to investigate the effect of the physical parameters on the solution behavior of the modified cylindrical (spherical) KdV equation.

Collisional evolution - an analytical study for the non steady-state mass distribution.

NASA Astrophysics Data System (ADS)

Vieira Martins, R.

1999-05-01

To study the collisional evolution of asteroidal groups one can use an analytical solution for the self-similar collision cascades. This solution is suitable to study the steady-state mass distribution of the collisional fragmentation. However, out of the steady-state conditions, this solution is not satisfactory for some values of the collisional parameters. In fact, for some values for the exponent of the mass distribution power law of an asteroidal group and its relation to the exponent of the function which describes "how rocks break" the author arrives at singular points for the equation which describes the collisional evolution. These singularities appear since some approximations are usually made in the laborious evaluation of many integrals that appear in the analytical calculations. They concern the cutoff for the smallest and the largest bodies. These singularities set some restrictions to the study of the analytical solution for the collisional equation. To overcome these singularities the author performed an algebraic computation considering the smallest and the largest bodies and he obtained the analytical expressions for the integrals that describe the collisional evolution without restriction on the parameters. However, the new distribution is more sensitive to the values of the collisional parameters. In particular the steady-state solution for the differential mass distribution has exponents slightly different from 11/6 for the usual parameters in the asteroid belt. The sensitivity of this distribution with respect to the parameters is analyzed for the usual values in the asteroidal groups. With an expression for the mass distribution without singularities, one can evaluate also its time evolution. The author arrives at an analytical expression given by a power series of terms constituted by a small parameter multiplied by the mass to an exponent, which depends on the initial power law distribution. This expression is a formal solution for the equation which describes the collisional evolution.

A Simple Model for Nonlinear Confocal Ultrasonic Beams

NASA Astrophysics Data System (ADS)

Zhang, Dong; Zhou, Lin; Si, Li-Sheng; Gong, Xiu-Fen

2007-01-01

A confocally and coaxially arranged pair of focused transmitter and receiver represents one of the best geometries for medical ultrasonic imaging and non-invasive detection. We develop a simple theoretical model for describing the nonlinear propagation of a confocal ultrasonic beam in biological tissues. On the basis of the parabolic approximation and quasi-linear approximation, the nonlinear Khokhlov-Zabolotskaya-Kuznetsov (KZK) equation is solved by using the angular spectrum approach. Gaussian superposition technique is applied to simplify the solution, and an analytical solution for the second harmonics in the confocal ultrasonic beam is presented. Measurements are performed to examine the validity of the theoretical model. This model provides a preliminary model for acoustic nonlinear microscopy.

Solving the Integral of Quadratic Forms of Covariance Matrices for Applications in Polarimetric Radar Imagery

NASA Astrophysics Data System (ADS)

Marino, Armando; Hajnsek, Irena

2015-04-01

In this work, the solution of quadratic forms with special application to polarimetric and interferometric covariance matrices is investigated. An analytical solution for the integral of a single quadratic form is derived. Additionally, the integral of the Pol-InSAR coherence (expressed as combination of quadratic forms) is investigated. An approximation for such integral is proposed and defined as Trace coherence. Such approximation is tested on real data to verify that the error is acceptable. The trace coherence can be used for tackle problems related to change detection. Moreover, the use of the Trace coherence in model inversion (as for the RVoG three stage inversion) will be investigated in the future.

Analysis of the dynamic response of a supersonic inlet to flow-field perturbations upstream of the normal shock

NASA Technical Reports Server (NTRS)

Cole, G. L.; Willoh, R. G.

1975-01-01

A linearized mathematical analysis is presented for determining the response of normal shock position and subsonic duct pressures to flow-field perturbations upstream of the normal shock in mixed-compression supersonic inlets. The inlet duct cross-sectional area variation is approximated by constant-area sections; this approximation results in one-dimensional wave equations. A movable normal shock separates the supersonic and subsonic flow regions, and a choked exit is assumed for the inlet exit condition. The analysis leads to a closed-form matrix solution for the shock position and pressure transfer functions. Analytical frequency response results are compared with experimental data and a method of characteristics solution.

A useful approximation for the flat surface impulse response

NASA Technical Reports Server (NTRS)

Brown, Gary S.

1989-01-01

The flat surface impulse response (FSIR) is a very useful quantity in computing the mean return power for near-nadir-oriented short-pulse radar altimeters. However, for very small antenna beamwidths and relatively large pointing angles, previous analytical descriptions become very difficult to compute accurately. An asymptotic approximation is developed to overcome these computational problems. Since accuracy is of key importance, a condition is developed under which this solution is within 2 percent of the exact answer. The asymptotic solution is shown to be in functional agreement with a conventional clutter power result and gives a 1.25-dB correction to this formula to account properly for the antenna-pattern variation over the illuminated area.

Analytical steady-state solutions for water-limited cropping systems using saline irrigation water

NASA Astrophysics Data System (ADS)

Skaggs, T. H.; Anderson, R. G.; Corwin, D. L.; Suarez, D. L.

2014-12-01

Due to the diminishing availability of good quality water for irrigation, it is increasingly important that irrigation and salinity management tools be able to target submaximal crop yields and support the use of marginal quality waters. In this work, we present a steady-state irrigated systems modeling framework that accounts for reduced plant water uptake due to root zone salinity. Two explicit, closed-form analytical solutions for the root zone solute concentration profile are obtained, corresponding to two alternative functional forms of the uptake reduction function. The solutions express a general relationship between irrigation water salinity, irrigation rate, crop salt tolerance, crop transpiration, and (using standard approximations) crop yield. Example applications are illustrated, including the calculation of irrigation requirements for obtaining targeted submaximal yields, and the generation of crop-water production functions for varying irrigation waters, irrigation rates, and crops. Model predictions are shown to be mostly consistent with existing models and available experimental data. Yet the new solutions possess advantages over available alternatives, including: (i) the solutions were derived from a complete physical-mathematical description of the system, rather than based on an ad hoc formulation; (ii) the analytical solutions are explicit and can be evaluated without iterative techniques; (iii) the solutions permit consideration of two common functional forms of salinity induced reductions in crop water uptake, rather than being tied to one particular representation; and (iv) the utilized modeling framework is compatible with leading transient-state numerical models.

Frequency-dependent laminar electroosmotic flow in a closed-end rectangular microchannel.

PubMed

Marcos; Yang, C; Ooi, K T; Wong, T N; Masliyah, J H

2004-07-15

This article presents an analysis of the frequency- and time-dependent electroosmotic flow in a closed-end rectangular microchannel. An exact solution to the modified Navier-Stokes equation governing the ac electroosmotic flow field is obtained by using the Green's function formulation in combination with a complex variable approach. An analytical expression for the induced backpressure gradient is derived. With the Debye-Hückel approximation, the electrical double-layer potential distribution in the channel is obtained by analytically solving the linearized two-dimensional Poisson-Boltzmann equation. Since the counterparts of the flow rate and the electrical current are shown to be linearly proportional to the applied electric field and the pressure gradient, Onsager's principle of reciprocity is demonstrated for transient and ac electroosmotic flows. The time evolution of the electroosmotic flow and the effect of a frequency-dependent ac electric field on the oscillating electroosmotic flow in a closed-end rectangular microchannel are examined. Specifically, the induced pressure gradient is analyzed under effects of the channel dimension and the frequency of electric field. In addition, based on the Stokes second problem, the solution of the slip velocity approximation is presented for comparison with the results obtained from the analytical scheme developed in this study. Copyright 2004 Elsevier Inc.

An advanced analytical solution for pressure build-up during CO2 injection into infinite saline aquifers: The role of compressibility

NASA Astrophysics Data System (ADS)

Wu, Haiqing; Bai, Bing; Li, Xiaochun

2018-02-01

Existing analytical or approximate solutions that are appropriate for describing the migration mechanics of CO2 and the evolution of fluid pressure in reservoirs do not consider the high compressibility of CO2, which reduces their calculation accuracy and application value. Therefore, this work first derives a new governing equation that represents the movement of complex fluids in reservoirs, based on the equation of continuity and the generalized Darcy's law. A more rigorous definition of the coefficient of compressibility of fluid is then presented, and a power function model (PFM) that characterizes the relationship between the physical properties of CO2 and the pressure is derived. Meanwhile, to avoid the difficulty of determining the saturation of fluids, a method that directly assumes the average relative permeability of each fluid phase in different fluid domains is proposed, based on the theory of gradual change. An advanced analytical solution is obtained that includes both the partial miscibility and the compressibility of CO2 and brine in evaluating the evolution of fluid pressure by integrating within different regions. Finally, two typical sample analyses are used to verify the reliability, improved nature and universality of this new analytical solution. Based on the physical characteristics and the results calculated for the examples, this work elaborates the concept and basis of partitioning for use in further work.

Approximation methods of European option pricing in multiscale stochastic volatility model

NASA Astrophysics Data System (ADS)

Ni, Ying; Canhanga, Betuel; Malyarenko, Anatoliy; Silvestrov, Sergei

2017-01-01

In the classical Black-Scholes model for financial option pricing, the asset price follows a geometric Brownian motion with constant volatility. Empirical findings such as volatility smile/skew, fat-tailed asset return distributions have suggested that the constant volatility assumption might not be realistic. A general stochastic volatility model, e.g. Heston model, GARCH model and SABR volatility model, in which the variance/volatility itself follows typically a mean-reverting stochastic process, has shown to be superior in terms of capturing the empirical facts. However in order to capture more features of the volatility smile a two-factor, of double Heston type, stochastic volatility model is more useful as shown in Christoffersen, Heston and Jacobs [12]. We consider one modified form of such two-factor volatility models in which the volatility has multiscale mean-reversion rates. Our model contains two mean-reverting volatility processes with a fast and a slow reverting rate respectively. We consider the European option pricing problem under one type of the multiscale stochastic volatility model where the two volatility processes act as independent factors in the asset price process. The novelty in this paper is an approximating analytical solution using asymptotic expansion method which extends the authors earlier research in Canhanga et al. [5, 6]. In addition we propose a numerical approximating solution using Monte-Carlo simulation. For completeness and for comparison we also implement the semi-analytical solution by Chiarella and Ziveyi [11] using method of characteristics, Fourier and bivariate Laplace transforms.

An analytical solution to assess the SH seismoelectric response of the vadose zone

NASA Astrophysics Data System (ADS)

Monachesi, L. B.; Zyserman, F. I.; Jouniaux, L.

2018-03-01

We derive an analytical solution of the seismoelectric conversions generated in the vadose zone, when this region is crossed by a pure shear horizontal (SH) wave. Seismoelectric conversions are induced by electrokinetic effects linked to relative motions between fluid and porous media. The considered model assumes a one-dimensional soil constituted by a single layer on top of a half space in contact at the water table, and a shearing force located at the earth's surface as the wave source. The water table is an interface expected to induce a seismoelectric interfacial response (IR). The top layer represents a porous rock which porous space is partially saturated by water and air, while the half-space is completely saturated with water, representing the saturated zone. The analytical expressions for the coseismic fields and the interface responses, both electric and magnetic, are derived by solving Pride's equations with proper boundary conditions. An approximate analytical expression of the solution is also obtained, which is very simple and applicable in a fairly broad set of situations. Hypothetical scenarios are proposed to study and analyse the dependence of the electromagnetic fields on various parameters of the medium. An analysis of the approximate solution is also made together with a comparison to the exact solution. The main result of the present analysis is that the amplitude of the interface response generated at the water table is found to be proportional to the jump in the electric current density, which in turn depends on the saturation contrast, poro-mechanical and electrical properties of the medium and on the amplitude of the solid displacement produced by the source. This result is in agreement with the one numerically obtained by the authors, which has been published in a recent work. We also predict the existence of an interface response located at the surface, and that the electric interface response is several orders of magnitude bigger than the electric coseismic field, whereas it is the opposite using compressional waves as shown by theoretical and experimental results. This fact should encourage the performance of field and laboratory tests to check the viability of SHTE seismoelectrics as a near surface prospecting/monitoring tool.

An analytical solution to assess the SH seismoelectric response of the vadose zone

NASA Astrophysics Data System (ADS)

Monachesi, L. B.; Zyserman, F. I.; Jouniaux, L.

2018-06-01

We derive an analytical solution of the seismoelectric conversions generated in the vadose zone, when this region is crossed by a pure shear horizontal (SH) wave. Seismoelectric conversions are induced by electrokinetic effects linked to relative motions between fluid and porous media. The considered model assumes a 1D soil constituted by a single layer on top of a half-space in contact at the water table, and a shearing force located at the earth's surface as the wave source. The water table is an interface expected to induce a seismoelectric interfacial response (IR). The top layer represents a porous rock in which porous space is partially saturated by water and air, while the half-space is completely saturated with water, representing the saturated zone. The analytical expressions for the coseismic fields and the interface responses, both electric and magnetic, are derived by solving Pride's equations with proper boundary conditions. An approximate analytical expression of the solution is also obtained, which is very simple and applicable in a fairly broad set of situations. Hypothetical scenarios are proposed to study and analyse the dependence of the electromagnetic fields on various parameters of the medium. An analysis of the approximate solution is also made together with a comparison to the exact solution. The main result of the present analysis is that the amplitude of the interface response generated at the water table is found to be proportional to the jump in the electric current density, which in turn depends on the saturation contrast, poro-mechanical and electrical properties of the medium and on the amplitude of the solid displacement produced by the source. This result is in agreement with the one numerically obtained by the authors, which has been published in a recent work. We also predict the existence of an interface response located at the surface, and that the electric interface response is several orders of magnitude bigger than the electric coseismic field, whereas it is the opposite using compressional waves as shown by theoretical and experimental results. This fact should encourage the performance of field and laboratory tests to check the viability of SHTE seismoelectrics as a near surface prospecting/monitoring tool.

Analytical evaluation of the trajectories of hypersonic projectiles launched into space

NASA Astrophysics Data System (ADS)

Stutz, John David

An equation of motion has been derived that may be solved using simple analytic functions which describes the motion of a projectile launched from the surface of the Earth into space accounting for both Newtonian gravity and aerodynamic drag. The equation of motion is based upon the Kepler equation of motion differential and variable transformations with the inclusion of a decaying angular momentum driving function and appropriate simplifying assumptions. The new equation of motion is first compared to various numerical and analytical trajectory approximations in a non-rotating Earth reference frame. The Modified Kepler solution is then corrected to include Earth rotation and compared to a rotating Earth simulation. Finally, the modified equation of motion is used to predict the apogee and trajectory of projectiles launched into space by the High Altitude Research Project from 1961 to 1967. The new equation of motion allows for the rapid equalization of projectile trajectories and intercept solutions that may be used to calculate firing solutions to enable ground launched projectiles to intercept or rendezvous with targets in low Earth orbit such as ballistic missiles.

An extension of the Derrida-Lebowitz-Speer-Spohn equation

NASA Astrophysics Data System (ADS)

Bordenave, Charles; Germain, Pierre; Trogdon, Thomas

2015-12-01

We show how the derivation of the Derrida-Lebowitz-Speer-Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy-Widom GOE distribution.

Analytical and numerical study of the electro-osmotic annular flow of viscoelastic fluids.

PubMed

Ferrás, L L; Afonso, A M; Alves, M A; Nóbrega, J M; Pinho, F T

2014-04-15

In this work we present semi-analytical solutions for the electro-osmotic annular flow of viscoelastic fluids modeled by the Linear and Exponential PTT models. The viscoelastic fluid flows in the axial direction between two concentric cylinders under the combined influences of electrokinetic and pressure forcings. The analysis invokes the Debye-Hückel approximation and includes the limit case of pure electro-osmotic flow. The solution is valid for both no slip and slip velocity at the walls and the chosen slip boundary condition is the linear Navier slip velocity model. The combined effects of fluid rheology, electro-osmotic and pressure gradient forcings on the fluid velocity distribution are also discussed. Copyright © 2013 Elsevier Inc. All rights reserved.

Analytical solution to compensate for thermal expansion change in photopolymer volume holograms using a tunable laser.

PubMed

Tanaka, Tomiji; Watanabe, Kenjiro

2008-02-20

For holographic data storage, it is necessary to adjust the wavelength and direction of the reading beam if the reading and recording temperature do not match. An analytical solution for this adjustment is derived using first-order approximations in a two-dimensional model. The optimum wavelength is a linear function of the temperature difference between recording and reading, and is independent of the direction of the reference beam. However, the optimum direction of incidence is not only a linear function of the temperature difference, but also depends on the direction of the reference beam. The retrieved image, which is produced by a diffracted beam, shrinks or expands slightly according to the temperature difference.

On the wing behaviour of the overtones of self-localized modes

NASA Astrophysics Data System (ADS)

Dusi, R.; Wagner, M.

1998-08-01

In this paper the solutions for self-localized modes in a nonlinear chain are investigated. We present a converging iteration procedure, which is based on analytical information of the wings and which takes into account higher overtones of the solitonic oscillations. The accuracy is controlled in a step by step manner by means of a Gaussian error analysis. Our numerical procedure allows for highly accurate solutions, in all anharmonicity regimes, and beyond the rotating-wave approximation (RWA). It is found that the overtone wings change their analytical behaviour at certain critical values of the energy of the self-localized mode: there is a turnover in the exponent of descent. The results are shown for a Fermi-Pasta-Ulam (FPU) chain with quartic anharmonicity.

Gaussian closure technique applied to the hysteretic Bouc model with non-zero mean white noise excitation

NASA Astrophysics Data System (ADS)

Waubke, Holger; Kasess, Christian H.

2016-11-01

Devices that emit structure-borne sound are commonly decoupled by elastic components to shield the environment from acoustical noise and vibrations. The elastic elements often have a hysteretic behavior that is typically neglected. In order to take hysteretic behavior into account, Bouc developed a differential equation for such materials, especially joints made of rubber or equipped with dampers. In this work, the Bouc model is solved by means of the Gaussian closure technique based on the Kolmogorov equation. Kolmogorov developed a method to derive probability density functions for arbitrary explicit first-order vector differential equations under white noise excitation using a partial differential equation of a multivariate conditional probability distribution. Up to now no analytical solution of the Kolmogorov equation in conjunction with the Bouc model exists. Therefore a wide range of approximate solutions, especially the statistical linearization, were developed. Using the Gaussian closure technique that is an approximation to the Kolmogorov equation assuming a multivariate Gaussian distribution an analytic solution is derived in this paper for the Bouc model. For the stationary case the two methods yield equivalent results, however, in contrast to statistical linearization the presented solution allows to calculate the transient behavior explicitly. Further, stationary case leads to an implicit set of equations that can be solved iteratively with a small number of iterations and without instabilities for specific parameter sets.

Soft x-ray continuum radiation transmitted through metallic filters: an analytical approach to fast electron temperature measurements.

PubMed

Delgado-Aparicio, L; Tritz, K; Kramer, T; Stutman, D; Finkenthal, M; Hill, K; Bitter, M

2010-10-01

A new set of analytic formulas describes the transmission of soft x-ray continuum radiation through a metallic foil for its application to fast electron temperature measurements in fusion plasmas. This novel approach shows good agreement with numerical calculations over a wide range of plasma temperatures in contrast with the solutions obtained when using a transmission approximated by a single-Heaviside function [S. von Goeler et al., Rev. Sci. Instrum. 70, 599 (1999)]. The new analytic formulas can improve the interpretation of the experimental results and thus contribute in obtaining fast temperature measurements in between intermittent Thomson scattering data.

An approximate viscous shock layer technique for calculating chemically reacting hypersonic flows about blunt-nosed bodies

NASA Technical Reports Server (NTRS)

Cheatwood, F. Mcneil; Dejarnette, Fred R.

1991-01-01

An approximate axisymmetric method was developed which can reliably calculate fully viscous hypersonic flows over blunt nosed bodies. By substituting Maslen's second order pressure expression for the normal momentum equation, a simplified form of the viscous shock layer (VSL) equations is obtained. This approach can solve both the subsonic and supersonic regions of the shock layer without a starting solution for the shock shape. The approach is applicable to perfect gas, equilibrium, and nonequilibrium flowfields. Since the method is fully viscous, the problems associated with a boundary layer solution with an inviscid layer solution are avoided. This procedure is significantly faster than the parabolized Navier-Stokes (PNS) or VSL solvers and would be useful in a preliminary design environment. Problems associated with a previously developed approximate VSL technique are addressed before extending the method to nonequilibrium calculations. Perfect gas (laminar and turbulent), equilibrium, and nonequilibrium solutions were generated for airflows over several analytic body shapes. Surface heat transfer, skin friction, and pressure predictions are comparable to VSL results. In addition, computed heating rates are in good agreement with experimental data. The present technique generates its own shock shape as part of its solution, and therefore could be used to provide more accurate initial shock shapes for higher order procedures which require starting solutions.

Investigation of hydrophobic substrates for solution residue analysis utilizing an ambient desorption liquid sampling-atmospheric pressure glow discharge microplasma.

PubMed

Paing, Htoo W; Marcus, R Kenneth

2018-03-12

A practical method for preparation of solution residue samples for analysis utilizing the ambient desorption liquid sampling-atmospheric pressure glow discharge optical emission spectroscopy (AD-LS-APGD-OES) microplasma is described. Initial efforts involving placement of solution aliquots in wells drilled into copper substrates, proved unsuccessful. A design-of-experiment (DOE) approach was carried out to determine influential factors during sample deposition including solution volume, solute concentration, number of droplets deposited, and the solution matrix. These various aspects are manifested in the mass of analyte deposited as well as the size/shape of the product residue. Statistical analysis demonstrated that only those initial attributes were significant factors towards the emission response of the analyte. Various approaches were investigated to better control the location/uniformity of the deposited sample. Three alternative substrates, a glass slide, a poly(tetrafluoro)ethylene (PTFE) sheet, and a polydimethylsiloxane (PDMS)-coated glass slide, were evaluated towards the microplasma analytical performance. Co-deposition with simple organic dyes provided an accurate means of determining the location of the analyte with only minor influence on emission responses. The PDMS-coated glass provided the best performance by virtue of its providing a uniform spatial distribution of the residue material. This uniformity yielded an improved limits of detection by approximately 22× for 20 μL and 4 x for 2 μL over the other two substrates. While they operate by fundamentally different processes, this choice of substrate is not restricted to the LS-APGD, but may also be applicable to other AD methods such as DESI, DART, or LIBS. Further developments will be directed towards a field-deployable ambient desorption OES source for quantitative analysis of microvolume solution residues of nuclear forensics importance.

Landau-Zener extension of the Tavis-Cummings model: structure of the solution

NASA Astrophysics Data System (ADS)

Sun, Chen; Sinitsyn, Nikolai

We explore the recently discovered solution of the driven Tavis-Cummings model (DTCM). It describes interaction of arbitrary number of two-level systems with a bosonic mode that has linearly time-dependent frequency. We derive compact and tractable expressions for transition probabilities in terms of the well known special functions. In the new form, our formulas are suitable for fast numerical calculations and analytical approximations. As an application, we obtain the semiclassical limit of the exact solution and compare it to prior approximations. We also reveal connection between DTCM and q-deformed binomial statistics. Under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. Authors also thank the support from the LDRD program at LANL.

Light diffusion in N-layered turbid media: steady-state domain.

PubMed

Liemert, André; Kienle, Alwin

2010-01-01

We deal with light diffusion in N-layered turbid media. The steady-state diffusion equation is solved for N-layered turbid media having a finite or an infinitely thick N'th layer. Different refractive indices are considered in the layers. The Fourier transform formalism is applied to derive analytical solutions of the fluence rate in Fourier space. The inverse Fourier transform is calculated using four different methods to test their performance and accuracy. Further, to avoid numerical errors, approximate formulas in Fourier space are derived. Fast solutions for calculation of the spatially resolved reflectance and transmittance from the N-layered turbid media ( approximately 10 ms) with small relative differences (<10(-7)) are found. Additionally, the solutions of the diffusion equation are compared to Monte Carlo simulations for turbid media having up to 20 layers.

General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Van Gorder, Robert A., E-mail: rav@knights.ucf.edu

2014-06-15

In his study of superfluid turbulence in the low-temperature limit, Svistunov [“Superfluid turbulence in the low-temperature limit,” Phys. Rev. B 52, 3647 (1995)] derived a Hamiltonian equation for the self-induced motion of a vortex filament. Under the local induction approximation (LIA), the Svistunov formulation is equivalent to a nonlinear dispersive partial differential equation. In this paper, we consider a family of rotating vortex filament solutions for the LIA reduction of the Svistunov formulation, which we refer to as the 2D LIA (since it permits a potential formulation in terms of two of the three Cartesian coordinates). This class of solutionsmore » holds the well-known Hasimoto-type planar vortex filament [H. Hasimoto, “Motion of a vortex filament and its relation to elastica,” J. Phys. Soc. Jpn. 31, 293 (1971)] as one reduction and helical solutions as another. More generally, we obtain solutions which are periodic in the space variable. A systematic analytical study of the behavior of such solutions is carried out. In the case where vortex filaments have small deviations from the axis of rotation, closed analytical forms of the filament solutions are given. A variety of numerical simulations are provided to demonstrate the wide range of rotating filament behaviors possible. Doing so, we are able to determine a number of vortex filament structures not previously studied. We find that the solution structure progresses from planar to helical, and then to more intricate and complex filament structures, possibly indicating the onset of superfluid turbulence.« less

An integrated model to simulate the scattering of ultrasounds by inclusions in steels.

PubMed

Darmon, Michel; Calmon, Pierre; Bèle, Bertrand

2004-04-01

We present a study performed to model and predict the ultrasonic response of alumina inclusions in steels. The Born and the extended quasistatic approximations have been applied and modified to improve their accuracy in the framework of this application. The modified Born approximation, called "doubly distorted wave (D(2)W) Born approximation" allowing to deal with various inclusion shapes, has been selected to be implemented in the CIVA software. The model reliability has been evaluated by comparison with Ying and Truell's exact analytical solution. In parallel, measurements have been carried out upon both natural and artificial alumina inclusions.

Scattering of focused ultrasonic beams by cavities in a solid half-space.

PubMed

Rahni, Ehsan Kabiri; Hajzargarbashi, Talieh; Kundu, Tribikram

2012-08-01

The ultrasonic field generated by a point focused acoustic lens placed in a fluid medium adjacent to a solid half-space, containing one or more spherical cavities, is modeled. The semi-analytical distributed point source method (DPSM) is followed for the modeling. This technique properly takes into account the interaction effect between the cavities placed in the focused ultrasonic field, fluid-solid interface and the lens surface. The approximate analytical solution that is available in the literature for the single cavity geometry is very restrictive and cannot handle multiple cavity problems. Finite element solutions for such problems are also prohibitively time consuming at high frequencies. Solution of this problem is necessary to predict when two cavities placed in close proximity inside a solid can be distinguished by an acoustic lens placed outside the solid medium and when such distinction is not possible.

Reducing microwave absorption with fast frequency modulation.

PubMed

Qin, Juehang; Hubler, A

2017-05-01

We study the response of a two-level quantum system to a chirp signal, using both numerical and analytical methods. The numerical method is based on numerical solutions of the Schrödinger solution of the two-level system, while the analytical method is based on an approximate solution of the same equations. We find that when two-level systems are perturbed by a chirp signal, the peak population of the initially unpopulated state exhibits a high sensitivity to frequency modulation rate. We also find that the aforementioned sensitivity depends on the strength of the forcing, and weaker forcings result in a higher sensitivity, where the frequency modulation rate required to produce the same reduction in peak population would be lower. We discuss potential applications of this result in the field of microwave power transmission, as it shows applying fast frequency modulation to transmitted microwaves used for power transmission could decrease unintended absorption of microwaves by organic tissue.

Transient well flow in leaky multiple-aquifer systems

NASA Astrophysics Data System (ADS)

Hemker, C. J.

1985-10-01

A previously developed eigenvalue analysis approach to groundwater flow in leaky multiple aquifers is used to derive exact solutions for transient well flow problems in leaky and confined systems comprising any number of aquifers. Equations are presented for the drawdown distribution in systems of infinite extent, caused by wells penetrating one or more of the aquifers completely and discharging each layer at a constant rate. Since the solution obtained may be regarded as a combined analytical-numerical technique, a type of one-dimensional modelling can be applied to find approximate solutions for several complicating conditions. Numerical evaluations are presented as time-drawdown curves and include effects of storage in the aquitard, unconfined conditions, partially penetrating wells and stratified aquifers. The outcome of calculations for relatively simple systems compares very well with published corresponding results. The proposed multilayer solution can be a valuable tool in aquifer test evaluation, as it provides the analytical expression required to enable the application of existing computer methods to the determination of aquifer characteristics.

Effect of Carreau-Yasuda rheological parameters on subcritical Lapwood convection in horizontal porous cavity saturated by shear-thinning fluid

NASA Astrophysics Data System (ADS)

Khechiba, Khaled; Mamou, Mahmoud; Hachemi, Madjid; Delenda, Nassim; Rebhi, Redha

2017-06-01

The present study is focused on Lapwood convection in isotropic porous media saturated with non-Newtonian shear thinning fluid. The non-Newtonian rheological behavior of the fluid is modeled using the general viscosity model of Carreau-Yasuda. The convection configuration consists of a shallow porous cavity with a finite aspect ratio and subject to a vertical constant heat flux, whereas the vertical walls are maintained impermeable and adiabatic. An approximate analytical solution is developed on the basis of the parallel flow assumption, and numerical solutions are obtained by solving the full governing equations. The Darcy model with the Boussinesq approximation and energy transport equations are solved numerically using a finite difference method. The results are obtained in terms of the Nusselt number and the flow fields as functions of the governing parameters. A good agreement is obtained between the analytical approximation and the numerical solution of the full governing equations. The effects of the rheological parameters of the Carreau-Yasuda fluid and Rayleigh number on the onset of subcritical convection thresholds are demonstrated. Regardless of the aspect ratio of the enclosure and thermal boundary condition type, the subcritical convective flows are seen to occur below the onset of stationary convection. Correlations are proposed to estimate the subcritical Rayleigh number for the onset of finite amplitude convection as a function of the fluid rheological parameters. Linear stability of the convective motion, predicted by the parallel flow approximation, is studied, and the onset of Hopf bifurcation, from steady convective flow to oscillatory behavior, is found to depend strongly on the rheological parameters. In general, Hopf bifurcation is triggered earlier as the fluid becomes more and more shear-thinning.

Electrolyte diodes with weak acids and bases. I. Theory and an approximate analytical solution.

PubMed

Iván, Kristóf; Simon, Péter L; Wittmann, Mária; Noszticzius, Zoltán

2005-10-22

Until now acid-base diodes and transistors applied strong mineral acids and bases exclusively. In this work properties of electrolyte diodes with weak electrolytes are studied and compared with those of diodes with strong ones to show the advantages of weak acids and bases in these applications. The theoretical model is a one dimensional piece of gel containing fixed ionizable groups and connecting reservoirs of an acid and a base. The electric current flowing through the gel is measured as a function of the applied voltage. The steady-state current-voltage characteristic (CVC) of such a gel looks like that of a diode under these conditions. Results of our theoretical, numerical, and experimental investigations are reported in two parts. In this first, theoretical part governing equations necessary to calculate the steady-state CVC of a reverse-biased electrolyte diode are presented together with an approximate analytical solution of this reaction-diffusion-ionic migration problem. The applied approximations are quasielectroneutrality and quasiequilibrium. It is shown that the gel can be divided into an alkaline and an acidic zone separated by a middle weakly acidic region. As a further approximation it is assumed that the ionization of the fixed acidic groups is complete in the alkaline zone and that it is completely suppressed in the acidic one. The general solution given here describes the CVC and the potential and ionic concentration profiles of diodes applying either strong or weak electrolytes. It is proven that previous formulas valid for a strong acid-strong base diode can be regarded as a special case of the more general formulas presented here.

COMPLEX VARIABLE BOUNDARY ELEMENT METHOD: APPLICATIONS.

USGS Publications Warehouse

Hromadka, T.V.; Yen, C.C.; Guymon, G.L.

1985-01-01

The complex variable boundary element method (CVBEM) is used to approximate several potential problems where analytical solutions are known. A modeling result produced from the CVBEM is a measure of relative error in matching the known boundary condition values of the problem. A CVBEM error-reduction algorithm is used to reduce the relative error of the approximation by adding nodal points in boundary regions where error is large. From the test problems, overall error is reduced significantly by utilizing the adaptive integration algorithm.

Numerical realization of the variational method for generating self-trapped beams

NASA Astrophysics Data System (ADS)

Duque, Erick I.; Lopez-Aguayo, Servando; Malomed, Boris A.

2018-03-01

We introduce a numerical variational method based on the Rayleigh-Ritz optimization principle for predicting two-dimensional self-trapped beams in nonlinear media. This technique overcomes the limitation of the traditional variational approximation in performing analytical Lagrangian integration and differentiation. Approximate soliton solutions of a generalized nonlinear Schr\\"odinger equation are obtained, demonstrating robustness of the beams of various types (fundamental, vortices, multipoles, azimuthons) in the course of their propagation. The algorithm offers possibilities to produce more sophisticated soliton profiles in general nonlinear models.

Approximate Green's function methods for HZE transport in multilayered materials

NASA Technical Reports Server (NTRS)

Wilson, John W.; Badavi, Francis F.; Shinn, Judy L.; Costen, Robert C.

1993-01-01

A nonperturbative analytic solution of the high charge and energy (HZE) Green's function is used to implement a computer code for laboratory ion beam transport in multilayered materials. The code is established to operate on the Langley nuclear fragmentation model used in engineering applications. Computational procedures are established to generate linear energy transfer (LET) distributions for a specified ion beam and target for comparison with experimental measurements. The code was found to be highly efficient and compared well with the perturbation approximation.

Exact and approximate solutions for the decades-old Michaelis-Menten equation: Progress-curve analysis through integrated rate equations.

PubMed

Goličnik, Marko

2011-01-01

The Michaelis-Menten rate equation can be found in most general biochemistry textbooks, where the time derivative of the substrate is a hyperbolic function of two kinetic parameters (the limiting rate V, and the Michaelis constant K(M) ) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to understand fully, or can even be misunderstood, by students when based only on the differential form of the Michaelis-Menten equation, and the variety of methods available to calculate the kinetic constants from rate versus substrate concentration "textbook data." Consequently, enzyme kinetics can be confusing if an analytical solution of the Michaelis-Menten equation is not available. Therefore, the still rarely known exact solution to the Michaelis-Menten equation is presented here through the explicit closed-form equation in terms of the Lambert W(x) function. Unfortunately, as the W(x) is not available in standard curve-fitting computer programs, the practical use of this direct solution is limited for most life-science students. Thus, the purpose of this article is to provide analytical approximations to the equation for modeling Michaelis-Menten kinetics. The elementary and explicit nature of these approximations can provide students with direct and simple estimations of kinetic parameters from raw experimental time-course data. The Michaelis-Menten kinetics studied in the latter context can provide an ideal alternative to the 100-year-old problems of data transformation, graphical visualization, and data analysis of enzyme-catalyzed reactions. Hence, the content of the course presented here could gradually become an important component of the modern biochemistry curriculum in the 21st century. Copyright © 2011 Wiley Periodicals, Inc.

Multibunch solutions of the differential-difference equation for traffic flow

PubMed

Nakanishi

2000-09-01

The Newell-Whitham type of car-following model, with a hyperbolic tangent as the optimal velocity function, has a finite number of exact steady traveling wave solutions that can be expressed in terms of elliptic theta functions. Each such solution describes a density wave with a definite number of car bunches on a circuit. In our numerical simulations, we observe a transition process from uniform flow to congested flow described by a one-bunch analytic solution, which appears to be an attractor of the system. In this process, the system exhibits a series of transitions through which it comes to assume configurations closely approximating multibunch solutions with successively fewer bunches.

Finite element modeling of light propagation in turbid media under illumination of a continuous-wave beam

USDA-ARS?s Scientific Manuscript database

Spatially-resolved spectroscopy provides a means for measuring the optical properties of biological tissues, based on analytical solutions to diffusion approximation for semi-infinite media under the normal illumination of infinitely small size light beam. The method is, however, prone to error in m...

Projected regression method for solving Fredholm integral equations arising in the analytic continuation problem of quantum physics

NASA Astrophysics Data System (ADS)

Arsenault, Louis-François; Neuberg, Richard; Hannah, Lauren A.; Millis, Andrew J.

2017-11-01

We present a supervised machine learning approach to the inversion of Fredholm integrals of the first kind as they arise, for example, in the analytic continuation problem of quantum many-body physics. The approach provides a natural regularization for the ill-conditioned inverse of the Fredholm kernel, as well as an efficient and stable treatment of constraints. The key observation is that the stability of the forward problem permits the construction of a large database of outputs for physically meaningful inputs. Applying machine learning to this database generates a regression function of controlled complexity, which returns approximate solutions for previously unseen inputs; the approximate solutions are then projected onto the subspace of functions satisfying relevant constraints. Under standard error metrics the method performs as well or better than the Maximum Entropy method for low input noise and is substantially more robust to increased input noise. We suggest that the methodology will be similarly effective for other problems involving a formally ill-conditioned inversion of an integral operator, provided that the forward problem can be efficiently solved.

Valuing options in shot noise market

NASA Astrophysics Data System (ADS)

Laskin, Nick

2018-07-01

A new exactly solvable option pricing model has been introduced and elaborated. It is assumed that a stock price follows a Geometric shot noise process. An arbitrage-free integro-differential option pricing equation has been obtained and solved. The new Greeks have been analytically calculated. It has been shown that in diffusion approximation the developed option pricing model incorporates the well-known Black-Scholes equation and its solution. The stochastic dynamic origin of the Black-Scholes volatility has been uncovered. To model the observed market stock price patterns consisting of high frequency small magnitude and low frequency large magnitude jumps, the superposition of two Geometric shot noises has been implemented. A new generalized option pricing equation has been obtained and its exact solution was found. Merton's jump-diffusion formula for option price was recovered in diffusion approximation. Despite the non-Gaussian nature of probability distributions involved, the new option pricing model has the same degree of analytical tractability as the Black-Scholes model and the Merton jump-diffusion model. This attractive feature allows one to derive exact formulas to value options and option related instruments in the market with jump-like price patterns.

Variational method enabling simplified solutions to the linearized Boltzmann equation for oscillatory gas flows

NASA Astrophysics Data System (ADS)

Ladiges, Daniel R.; Sader, John E.

2018-05-01

Nanomechanical resonators and sensors, operated in ambient conditions, often generate low-Mach-number oscillating rarefied gas flows. Cercignani [C. Cercignani, J. Stat. Phys. 1, 297 (1969), 10.1007/BF01007482] proposed a variational principle for the linearized Boltzmann equation, which can be used to derive approximate analytical solutions of steady (time-independent) flows. Here we extend and generalize this principle to unsteady oscillatory rarefied flows and thus accommodate resonating nanomechanical devices. This includes a mathematical approach that facilitates its general use and allows for systematic improvements in accuracy. This formulation is demonstrated for two canonical flow problems: oscillatory Couette flow and Stokes' second problem. Approximate analytical formulas giving the bulk velocity and shear stress, valid for arbitrary oscillation frequency, are obtained for Couette flow. For Stokes' second problem, a simple system of ordinary differential equations is derived which may be solved to obtain the desired flow fields. Using this framework, a simple and accurate formula is provided for the shear stress at the oscillating boundary, again for arbitrary frequency, which may prove useful in application. These solutions are easily implemented on any symbolic or numerical package, such as Mathematica or matlab, facilitating the characterization of flows produced by nanomechanical devices and providing insight into the underlying flow physics.

A three-dimensional semi-analytical solution for predicting drug release through the orifice of a spherical device.

PubMed

Simon, Laurent; Ospina, Juan

2016-07-25

Three-dimensional solute transport was investigated for a spherical device with a release hole. The governing equation was derived using the Fick's second law. A mixed Neumann-Dirichlet condition was imposed at the boundary to represent diffusion through a small region on the surface of the device. The cumulative percentage of drug released was calculated in the Laplace domain and represented by the first term of an infinite series of Legendre and modified Bessel functions of the first kind. Application of the Zakian algorithm yielded the time-domain closed-form expression. The first-order solution closely matched a numerical solution generated by Mathematica(®). The proposed method allowed computation of the characteristic time. A larger surface pore resulted in a smaller effective time constant. The agreement between the numerical solution and the semi-analytical method improved noticeably as the size of the orifice increased. It took four time constants for the device to release approximately ninety-eight of its drug content. Copyright © 2016 Elsevier B.V. All rights reserved.

Predator prey oscillations in a simple cascade model of drift wave turbulence

DOE Office of Scientific and Technical Information (OSTI.GOV)

Berionni, V.; Guercan, Oe. D.

2011-11-15

A reduced three shell limit of a simple cascade model of drift wave turbulence, which emphasizes nonlocal interactions with a large scale mode, is considered. It is shown to describe both the well known predator prey dynamics between the drift waves and zonal flows and to reduce to the standard three wave interaction equations. Here, this model is considered as a dynamical system whose characteristics are investigated. The analytical solutions for the purely nonlinear limit are given in terms of the Jacobi elliptic functions. An approximate analytical solution involving Jacobi elliptic functions and exponential growth is computed using scale separationmore » for the case of unstable solutions that are observed when the energy injection rate is high. The fixed points of the system are determined, and the behavior around these fixed points is studied. The system is shown to display periodic solutions corresponding to limit cycle oscillations, apparently chaotic phase space orbits, as well as unstable solutions that grow slowly while oscillating rapidly. The period doubling route to transition to chaos is examined.« less

Self-similar solutions to isothermal shock problems

NASA Astrophysics Data System (ADS)

Deschner, Stephan C.; Illenseer, Tobias F.; Duschl, Wolfgang J.

We investigate exact solutions for isothermal shock problems in different one-dimensional geometries. These solutions are given as analytical expressions if possible, or are computed using standard numerical methods for solving ordinary differential equations. We test the numerical solutions against the analytical expressions to verify the correctness of all numerical algorithms. We use similarity methods to derive a system of ordinary differential equations (ODE) yielding exact solutions for power law density distributions as initial conditions. Further, the system of ODEs accounts for implosion problems (IP) as well as explosion problems (EP) by changing the initial or boundary conditions, respectively. Taking genuinely isothermal approximations into account leads to additional insights of EPs in contrast to earlier models. We neglect a constant initial energy contribution but introduce a parameter to adjust the initial mass distribution of the system. Moreover, we show that due to this parameter a constant initial density is not allowed for isothermal EPs. Reasonable restrictions for this parameter are given. Both, the (genuinely) isothermal implosion as well as the explosion problem are solved for the first time.

Derivation of a closed form analytical expression for fluorescence recovery after photo bleaching in the case of continuous bleaching during read out

NASA Astrophysics Data System (ADS)

Endress, E.; Weigelt, S.; Reents, G.; Bayerl, T. M.

2005-01-01

Measurements of very slow diffusive processes in membranes, like the diffusion of integral membrane proteins, by fluorescence recovery after photo bleaching (FRAP) are hampered by bleaching of the probe during the read out of the fluorescence recovery. In the limit of long observation time (very slow diffusion as in the case of large membrane proteins), this bleaching may cause errors to the recovery function and thus provides error-prone diffusion coefficients. In this work we present a new approach to a two-dimensional closed form analytical solution of the reaction-diffusion equation, based on the addition of a dissipative term to the conventional diffusion equation. The calculation was done assuming (i) a Gaussian laser beam profile for bleaching the spot and (ii) that the fluorescence intensity profile emerging from the spot can be approximated by a two-dimensional Gaussian. The detection scheme derived from the analytical solution allows for diffusion measurements without the constraint of observation bleaching. Recovery curves of experimental FRAP data obtained under non-negligible read-out bleaching for native membranes (rabbit endoplasmic reticulum) on a planar solid support showed excellent agreement with the analytical solution and allowed the calculation of the lipid diffusion coefficient.

Approximate Analytical Time-Dependent Solutions to Describe Large-Amplitude Local Calcium Transients in the Presence of Buffers

PubMed Central

Mironova, Lidia A.; Mironov, Sergej L.

2008-01-01

Local Ca2+ signaling controls many neuronal functions, which is often achieved through spatial localization of Ca2+ signals. These nanodomains are formed due to combined effects of Ca2+ diffusion and binding to the cytoplasmic buffers. In this article we derived simple analytical expressions to describe Ca2+ diffusion in the presence of mobile and immobile buffers. A nonlinear character of the reaction-diffusion problem was circumvented by introducing a logarithmic approximation of the concentration term. The obtained formulas reproduce free Ca2+ levels up to 50 μM and their changes in the millisecond range. Derived equations can be useful to predict spatiotemporal profiles of large-amplitude [Ca2+] transients, which participate in various physiological processes. PMID:17872951

Qualitative methods in quantum theory

DOE Office of Scientific and Technical Information (OSTI.GOV)

Migdal, A.B.

The author feels that the solution of most problems in theoretical physics begins with the application of qualitative methods - dimensional estimates and estimates made from simple models, the investigation of limiting cases, the use of the analytic properties of physical quantities, etc. This book proceeds in this spirit, rather than in a formal, mathematical way with no traces of the sweat involved in the original work left to show. The chapters are entitled Dimensional and model approximations, Various types of perturbation theory, The quasi-classical approximation, Analytic properties of physical quantities, Methods in the many-body problem, and Qualitative methods inmore » quantum field theory. Each chapter begins with a detailed introduction, in which the physical meaning of the results obtained in that chapter is explained in a simple way. 61 figures. (RWR)« less

Pattern Storage, Bifurcations, and Groupwise Correlation Structure of an Exactly Solvable Asymmetric Neural Network Model.

PubMed

Fasoli, Diego; Cattani, Anna; Panzeri, Stefano

2018-05-01

Despite their biological plausibility, neural network models with asymmetric weights are rarely solved analytically, and closed-form solutions are available only in some limiting cases or in some mean-field approximations. We found exact analytical solutions of an asymmetric spin model of neural networks with arbitrary size without resorting to any approximation, and we comprehensively studied its dynamical and statistical properties. The network had discrete time evolution equations and binary firing rates, and it could be driven by noise with any distribution. We found analytical expressions of the conditional and stationary joint probability distributions of the membrane potentials and the firing rates. By manipulating the conditional probability distribution of the firing rates, we extend to stochastic networks the associating learning rule previously introduced by Personnaz and coworkers. The new learning rule allowed the safe storage, under the presence of noise, of point and cyclic attractors, with useful implications for content-addressable memories. Furthermore, we studied the bifurcation structure of the network dynamics in the zero-noise limit. We analytically derived examples of the codimension 1 and codimension 2 bifurcation diagrams of the network, which describe how the neuronal dynamics changes with the external stimuli. This showed that the network may undergo transitions among multistable regimes, oscillatory behavior elicited by asymmetric synaptic connections, and various forms of spontaneous symmetry breaking. We also calculated analytically groupwise correlations of neural activity in the network in the stationary regime. This revealed neuronal regimes where, statistically, the membrane potentials and the firing rates are either synchronous or asynchronous. Our results are valid for networks with any number of neurons, although our equations can be realistically solved only for small networks. For completeness, we also derived the network equations in the thermodynamic limit of infinite network size and we analytically studied their local bifurcations. All the analytical results were extensively validated by numerical simulations.

Calculation procedure for transient heat transfer to a cooled plate in a heated stream whose temperature varies arbitrarily with time. [turbine blades

NASA Technical Reports Server (NTRS)

Sucec, J.

1975-01-01

Solutions for the surface temperature and surface heat flux are found for laminar, constant property, slug flow over a plate convectively cooled from below, when the temperature of the fluid over the plate varies arbitrarily with time at the plate leading edge. A simple technique is presented for handling arbitrary fluid temperature variation with time by approximating it by a sequence of ramps or steps for which exact analytical solutions are available.

Transport methods and interactions for space radiations

NASA Technical Reports Server (NTRS)

Wilson, John W.; Townsend, Lawrence W.; Schimmerling, Walter S.; Khandelwal, Govind S.; Khan, Ferdous S.; Nealy, John E.; Cucinotta, Francis A.; Simonsen, Lisa C.; Shinn, Judy L.; Norbury, John W.

1991-01-01

A review of the program in space radiation protection at the Langley Research Center is given. The relevant Boltzmann equations are given with a discussion of approximation procedures for space applications. The interaction coefficients are related to solution of the many-body Schroedinger equation with nuclear and electromagnetic forces. Various solution techniques are discussed to obtain relevant interaction cross sections with extensive comparison with experiments. Solution techniques for the Boltzmann equations are discussed in detail. Transport computer code validation is discussed through analytical benchmarking, comparison with other codes, comparison with laboratory experiments and measurements in space. Applications to lunar and Mars missions are discussed.

Evaluation of purity with its uncertainty value in high purity lead stick by conventional and electro-gravimetric methods

PubMed Central

2013-01-01

Background A conventional gravimetry and electro-gravimetry study has been carried out for the precise and accurate purity determination of lead (Pb) in high purity lead stick and for preparation of reference standard. Reference materials are standards containing a known amount of an analyte and provide a reference value to determine unknown concentrations or to calibrate analytical instruments. A stock solution of approximate 2 kg has been prepared after dissolving approximate 2 g of Pb stick in 5% ultra pure nitric acid. From the stock solution five replicates of approximate 50 g have been taken for determination of purity by each method. The Pb has been determined as PbSO4 by conventional gravimetry, as PbO2 by electro gravimetry. The percentage purity of the metallic Pb was calculated accordingly from PbSO4 and PbO2. Results On the basis of experimental observations it has been concluded that by conventional gravimetry and electro-gravimetry the purity of Pb was found to be 99.98 ± 0.24 and 99.97 ± 0.27 g/100 g and on the basis of Pb purity the concentration of reference standard solutions were found to be 1000.88 ± 2.44 and 1000.81 ± 2.68 mg kg-1 respectively with 95% confidence level (k = 2). The uncertainty evaluation has also been carried out in Pb determination following EURACHEM/GUM guidelines. The final analytical results quantifying uncertainty fulfills this requirement and gives a measure of the confidence level of the concerned laboratory. Conclusions Gravimetry is the most reliable technique in comparison to titremetry and instrumental method and the results of gravimetry are directly traceable to SI unit. Gravimetric analysis, if methods are followed carefully, provides for exceedingly precise analysis. In classical gravimetry the major uncertainties are due to repeatability but in electro-gravimetry several other factors also affect the final results. PMID:23800080

Evaluation of purity with its uncertainty value in high purity lead stick by conventional and electro-gravimetric methods.

PubMed

Singh, Nahar; Singh, Niranjan; Tripathy, S Swarupa; Soni, Daya; Singh, Khem; Gupta, Prabhat K

2013-06-26

A conventional gravimetry and electro-gravimetry study has been carried out for the precise and accurate purity determination of lead (Pb) in high purity lead stick and for preparation of reference standard. Reference materials are standards containing a known amount of an analyte and provide a reference value to determine unknown concentrations or to calibrate analytical instruments. A stock solution of approximate 2 kg has been prepared after dissolving approximate 2 g of Pb stick in 5% ultra pure nitric acid. From the stock solution five replicates of approximate 50 g have been taken for determination of purity by each method. The Pb has been determined as PbSO4 by conventional gravimetry, as PbO2 by electro gravimetry. The percentage purity of the metallic Pb was calculated accordingly from PbSO4 and PbO2. On the basis of experimental observations it has been concluded that by conventional gravimetry and electro-gravimetry the purity of Pb was found to be 99.98 ± 0.24 and 99.97 ± 0.27 g/100 g and on the basis of Pb purity the concentration of reference standard solutions were found to be 1000.88 ± 2.44 and 1000.81 ± 2.68 mg kg-1 respectively with 95% confidence level (k = 2). The uncertainty evaluation has also been carried out in Pb determination following EURACHEM/GUM guidelines. The final analytical results quantifying uncertainty fulfills this requirement and gives a measure of the confidence level of the concerned laboratory. Gravimetry is the most reliable technique in comparison to titremetry and instrumental method and the results of gravimetry are directly traceable to SI unit. Gravimetric analysis, if methods are followed carefully, provides for exceedingly precise analysis. In classical gravimetry the major uncertainties are due to repeatability but in electro-gravimetry several other factors also affect the final results.

Stress Analysis of Beams with Shear Deformation of the Flanges

NASA Technical Reports Server (NTRS)

Kuhn, Paul

1937-01-01

This report discusses the fundamental action of shear deformation of the flanges on the basis of simplifying assumptions. The theory is developed to the point of giving analytical solutions for simple cases of beams and of skin-stringer panels under axial load. Strain-gage tests on a tension panel and on a beam corresponding to these simple cases are described and the results are compared with analytical results. For wing beams, an approximate method of applying the theory is given. As an alternative, the construction of a mechanical analyzer is advocated.

Precision of Sensitivity in the Design Optimization of Indeterminate Structures

NASA Technical Reports Server (NTRS)

Patnaik, Surya N.; Pai, Shantaram S.; Hopkins, Dale A.

2006-01-01

Design sensitivity is central to most optimization methods. The analytical sensitivity expression for an indeterminate structural design optimization problem can be factored into a simple determinate term and a complicated indeterminate component. Sensitivity can be approximated by retaining only the determinate term and setting the indeterminate factor to zero. The optimum solution is reached with the approximate sensitivity. The central processing unit (CPU) time to solution is substantially reduced. The benefit that accrues from using the approximate sensitivity is quantified by solving a set of problems in a controlled environment. Each problem is solved twice: first using the closed-form sensitivity expression, then using the approximation. The problem solutions use the CometBoards testbed as the optimization tool with the integrated force method as the analyzer. The modification that may be required, to use the stiffener method as the analysis tool in optimization, is discussed. The design optimization problem of an indeterminate structure contains many dependent constraints because of the implicit relationship between stresses, as well as the relationship between the stresses and displacements. The design optimization process can become problematic because the implicit relationship reduces the rank of the sensitivity matrix. The proposed approximation restores the full rank and enhances the robustness of the design optimization method.

Approximate Algorithms for Computing Spatial Distance Histograms with Accuracy Guarantees

PubMed Central

Grupcev, Vladimir; Yuan, Yongke; Tu, Yi-Cheng; Huang, Jin; Chen, Shaoping; Pandit, Sagar; Weng, Michael

2014-01-01

Particle simulation has become an important research tool in many scientific and engineering fields. Data generated by such simulations impose great challenges to database storage and query processing. One of the queries against particle simulation data, the spatial distance histogram (SDH) query, is the building block of many high-level analytics, and requires quadratic time to compute using a straightforward algorithm. Previous work has developed efficient algorithms that compute exact SDHs. While beating the naive solution, such algorithms are still not practical in processing SDH queries against large-scale simulation data. In this paper, we take a different path to tackle this problem by focusing on approximate algorithms with provable error bounds. We first present a solution derived from the aforementioned exact SDH algorithm, and this solution has running time that is unrelated to the system size N. We also develop a mathematical model to analyze the mechanism that leads to errors in the basic approximate algorithm. Our model provides insights on how the algorithm can be improved to achieve higher accuracy and efficiency. Such insights give rise to a new approximate algorithm with improved time/accuracy tradeoff. Experimental results confirm our analysis. PMID:24693210

Analytical Solution for Time-drawdown Response to Constant Pumping from a Homogeneous, Confined Horizontal Aquifer with Unidirectional Flow

NASA Astrophysics Data System (ADS)

Parrish, K. E.; Zhang, J.; Teasdale, E.

2007-12-01

An exact analytical solution to the ordinary one-dimensional partial differential equation is derived for transient groundwater flow in a homogeneous, confined, horizontal aquifer using Laplace transformation. The theoretical analysis is based on the assumption that the aquifer is homogeneous and one-dimensional (horizontal); confined between impermeable formations on top and bottom; and of infinite horizontal extent and constant thickness. It is also assumed that there is only a single pumping well penetrating the entire aquifer; flow is everywhere horizontal within the aquifer to the well; the well is pumping with a constant discharge rate; the well diameter is infinitesimally small; and the hydraulic head is uniform throughout the aquifer before pumping. Similar to the Theis solution, this solution is suited to determine transmissivity and storativity for a two- dimensional, vertically confined aquifer, such as a long vertically fractured zone of high permeability within low permeable rocks or a long, high-permeability trench inside a low-permeability porous media. In addition, it can be used to analyze time-drawdown responses to pumping and injection in similar settings. The solution can also be used to approximate the groundwater flow for unconfined conditions if (1) the variation of transmissivity is negligible (groundwater table variation is small in comparison to the saturated thickness); and (2) the unsaturated flow is negligible. The errors associated with the use of the solution to unconfined conditions depend on the accuracies of the above two assumptions. The solution can also be used to assess the impacts of recharge from a seasonal river or irrigation canal on the groundwater system by assuming uniform, time- constant recharge along the river or canal. This paper presents the details for derivation of the analytical solution. The analytical solution is compared to numerical simulation results with example cases. Its accuracy is also assessed and discussed for confined and unconfined conditions.

Asymptotic approximations to posterior distributions via conditional moment equations

USGS Publications Warehouse

Yee, J.L.; Johnson, W.O.; Samaniego, F.J.

2002-01-01

We consider asymptotic approximations to joint posterior distributions in situations where the full conditional distributions referred to in Gibbs sampling are asymptotically normal. Our development focuses on problems where data augmentation facilitates simpler calculations, but results hold more generally. Asymptotic mean vectors are obtained as simultaneous solutions to fixed point equations that arise naturally in the development. Asymptotic covariance matrices flow naturally from the work of Arnold & Press (1989) and involve the conditional asymptotic covariance matrices and first derivative matrices for conditional mean functions. When the fixed point equations admit an analytical solution, explicit formulae are subsequently obtained for the covariance structure of the joint limiting distribution, which may shed light on the use of the given statistical model. Two illustrations are given. ?? 2002 Biometrika Trust.

Stimulated neutrino transformation through turbulence

DOE PAGES

Patton, Kelly M.; Kneller, James P.; McLaughlin, Gail C.

2014-04-30

We derive an analytical solution for the flavor evolution of a neutrino through a turbulent density profile which is found to accurately predict the amplitude and transition wavelength of numerical solutions on a case-by-case basis. The evolution is seen to strongly depend upon those Fourier modes in the turbulence which are approximately the same as the splitting between neutrino eigenvalues. Transitions are strongly enhanced by those Fourier modes in the turbulence which are approximately the same as the splitting between neutrino eigenvalues. Lastly, we also find a suppression of transitions due to the long wavelength modes when the ratio ofmore » their amplitude and the wavenumber is of order, or greater than, the first root of the Bessel function J 0.« less

Revisiting the time until fixation of a neutral mutant in a finite population - A coalescent theory approach.

PubMed

Greenbaum, Gili

2015-09-07

Evaluation of the time scale of the fixation of neutral mutations is crucial to the theoretical understanding of the role of neutral mutations in evolution. Diffusion approximations of the Wright-Fisher model are most often used to derive analytic formulations of genetic drift, as well as for the time scales of the fixation of neutral mutations. These approximations require a set of assumptions, most notably that genetic drift is a stochastic process in a continuous allele-frequency space, an assumption appropriate for large populations. Here equivalent approximations are derived using a coalescent theory approach which relies on a different set of assumptions than the diffusion approach, and adopts a discrete allele-frequency space. Solutions for the mean and variance of the time to fixation of a neutral mutation derived from the two approaches converge for large populations but slightly differ for small populations. A Markov chain analysis of the Wright-Fisher model for small populations is used to evaluate the solutions obtained, showing that both the mean and the variance are better approximated by the coalescent approach. The coalescence approximation represents a tighter upper-bound for the mean time to fixation than the diffusion approximation, while the diffusion approximation and coalescence approximation form an upper and lower bound, respectively, for the variance. The converging solutions and the small deviations of the two approaches strongly validate the use of diffusion approximations, but suggest that coalescent theory can provide more accurate approximations for small populations. Copyright © 2015 Elsevier Ltd. All rights reserved.

Highly accurate analytic formulae for projectile motion subjected to quadratic drag

NASA Astrophysics Data System (ADS)

Turkyilmazoglu, Mustafa

2016-05-01

The classical phenomenon of motion of a projectile fired (thrown) into the horizon through resistive air charging a quadratic drag onto the object is revisited in this paper. No exact solution is known that describes the full physical event under such an exerted resistance force. Finding elegant analytical approximations for the most interesting engineering features of dynamical behavior of the projectile is the principal target. Within this purpose, some analytical explicit expressions are derived that accurately predict the maximum height, its arrival time as well as the flight range of the projectile at the highest ascent. The most significant property of the proposed formulas is that they are not restricted to the initial speed and firing angle of the object, nor to the drag coefficient of the medium. In combination with the available approximations in the literature, it is possible to gain information about the flight and complete the picture of a trajectory with high precision, without having to numerically simulate the full governing equations of motion.

Analytic quantum-interference conditions in Coulomb corrected photoelectron holography

NASA Astrophysics Data System (ADS)

Maxwell, A. S.; Al-Jawahiry, A.; Lai, X. Y.; Figueira de Morisson Faria, C.

2018-02-01

We provide approximate analytic expressions for above-threshold ionization (ATI) transition probabilities and photoelectron angular distributions. These analytic expressions are more general than those existing in the literature and include the residual binding potential in the electron continuum propagation. They successfully reproduce the ATI side lobes and specific holographic structures such as the near-threshold fan-shaped pattern and the spider-like structure that extends up to relatively high photoelectron energies. We compare such expressions with the Coulomb quantum orbit strong-field approximation (CQSFA) and the full solution of the time-dependent Schrödinger equation for different driving-field frequencies and intensities, and provide an in-depth analysis of the physical mechanisms behind specific holographic structures. Our results shed additional light on what aspects of the CQSFA must be prioritized in order to obtain the key holographic features, and highlight the importance of forward scattered trajectories. Furthermore, we find that the holographic patterns change considerably for different field parameters, even if the Keldysh parameter is kept roughly the same.

Onset of detachment in adhesive contact of an elastic half-space and flat-ended punches with non-circular shape: analytic estimates and comparison with numeric analysis

NASA Astrophysics Data System (ADS)

Li, Qiang; Argatov, Ivan; Popov, Valentin L.

2018-04-01

A recent paper by Popov, Pohrt and Li (PPL) in Friction investigated adhesive contacts of flat indenters in unusual shapes using numerical, analytical and experimental methods. Based on that paper, we analyze some special cases for which analytical solutions are known. As in the PPL paper, we consider adhesive contact in the Johnson-Kendall-Roberts approximation. Depending on the energy balance, different upper and lower estimates are obtained in terms of certain integral characteristics of the contact area. The special cases of an elliptical punch as well as a system of two circular punches are considered. Theoretical estimations for the first critical force (force at which the detachment process begins) are confirmed by numerical simulations using the adhesive boundary element method. It is shown that simpler approximations for the pull-off force, based both on the Holm radius of contact and the contact area, substantially overestimate the maximum adhesive force.

Determination of Slope Safety Factor with Analytical Solution and Searching Critical Slip Surface with Genetic-Traversal Random Method

PubMed Central

2014-01-01

In the current practice, to determine the safety factor of a slope with two-dimensional circular potential failure surface, one of the searching methods for the critical slip surface is Genetic Algorithm (GA), while the method to calculate the slope safety factor is Fellenius' slices method. However GA needs to be validated with more numeric tests, while Fellenius' slices method is just an approximate method like finite element method. This paper proposed a new method to determine the minimum slope safety factor which is the determination of slope safety factor with analytical solution and searching critical slip surface with Genetic-Traversal Random Method. The analytical solution is more accurate than Fellenius' slices method. The Genetic-Traversal Random Method uses random pick to utilize mutation. A computer automatic search program is developed for the Genetic-Traversal Random Method. After comparison with other methods like slope/w software, results indicate that the Genetic-Traversal Random Search Method can give very low safety factor which is about half of the other methods. However the obtained minimum safety factor with Genetic-Traversal Random Search Method is very close to the lower bound solutions of slope safety factor given by the Ansys software. PMID:24782679

Geometric factor and influence of sensors in the establishment of a resistivity-moisture relation in soil samples

NASA Astrophysics Data System (ADS)

López-Sánchez, M.; Mansilla-Plaza, L.; Sánchez-de-laOrden, M.

2017-10-01

Prior to field scale research, soil samples are analysed on a laboratory scale for electrical resistivity calibrations. Currently, there are a variety of field instruments to estimate the water content in soils using different physical phenomena. These instruments can be used to develop moisture-resistivity relationships on the same soil samples. This assures that measurements are performed on the same material and under the same conditions (e.g., humidity and temperature). A geometric factor is applied to the location of electrodes, in order to calculate the apparent electrical resistivity of the laboratory test cells. This geometric factor can be determined in three different ways: by means of the use of an analytical approximation, laboratory trials (experimental approximation), or by the analysis of a numerical model. The first case, the analytical approximation, is not appropriate for complex cells or arrays. And both, the experimental and numerical approximation can lead to inaccurate results. Therefore, we propose a novel approach to obtain a compromise solution between both techniques, providing a more precise determination of the geometrical factor.

Reduction of matrix effects in inductively coupled plasma mass spectrometry by flow injection with an unshielded torch.

PubMed

Gross, Cory T; McIntyre, Sally M; Houk, R S

2009-06-15

Solution samples with matrix concentrations above approximately 0.1% generally present difficulties for analysis by inductively coupled plasma mass spectrometry (ICP-MS) because of cone clogging and matrix effects. Flow injection (FI) is coupled to ICP-MS to reduce deposition from samples such as 1% sodium salts (as NaCl) and seawater (approximately 3% dissolved salts). Surprisingly, matrix effects are also less severe during flow injection, at least for some matrix elements on the particular instrument used. Sodium chloride at 1% Na and undiluted seawater cause only 2 to 29% losses of signal for typical analyte elements. A heavy matrix element (Bi) at 0.1% also induces only approximately 14% loss of analyte signal. However, barium causes a much worse matrix effect, that is, approximately 90% signal loss at 5000 ppm Na. Also, matrix effects during FI are much more severe when a grounded metal shield is inserted between the load coil and the torch, which is the most common mode of operation for the particular ICP-MS device used.

Methods for analysis of cracks in three-dimensional solids

NASA Technical Reports Server (NTRS)

Raju, I. S.; Newman, J. C., Jr.

1984-01-01

Analytical and numerical methods evaluating the stress-intensity factors for three-dimensional cracks in solids are presented, with reference to fatigue failure in aerospace structures. The exact solutions for embedded elliptical and circular cracks in infinite solids, and the approximate methods, including the finite-element, the boundary-integral equation, the line-spring models, and the mixed methods are discussed. Among the mixed methods, the superposition of analytical and finite element methods, the stress-difference, the discretization-error, the alternating, and the finite element-alternating methods are reviewed. Comparison of the stress-intensity factor solutions for some three-dimensional crack configurations showed good agreement. Thus, the choice of a particular method in evaluating the stress-intensity factor is limited only to the availability of resources and computer programs.

Analytical close-to-source investigation for an isotropic point source in an unbounded, anisotropically scattering medium

NASA Astrophysics Data System (ADS)

Rinzema, Kees; ten Bosch, Jaap J.; Ferwerda, Hedzer A.; Hoenders, Bernhard J.

1995-01-01

The diffusion approximation, which is often used to describe the propagation of light in biological tissues, is only good at a sufficient distance from sources and boundaries. Light- tissue interaction is however most intense in the region close to the source. It would therefore be interesting to study this region more closely. Although scattering in biological tissues is predominantly forward peaked, explicit solutions to the transport equation have only been obtained in the case of isotropic scattering. Particularly, for the case of an isotropic point source in an unbounded, isotropically scattering medium the solution is well known. We show that this problem can also be solved analytically if the scattering is no longer isotropic, while everything else remains the same.

A compatible high-order meshless method for the Stokes equations with applications to suspension flows

NASA Astrophysics Data System (ADS)

Trask, Nathaniel; Maxey, Martin; Hu, Xiaozhe

2018-02-01

A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a discretization that couples a staggered scheme for pressure approximation with a divergence-free velocity reconstruction to obtain an adaptive, high-order, finite difference-like discretization that can be efficiently solved with conventional algebraic multigrid techniques. We use analytic benchmarks to demonstrate equal-order convergence for both velocity and pressure when solving problems with curvilinear geometries. In order to study problems in dense suspensions, we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme. The combination of high-order accuracy with fully-implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the Stokes problem without the need to introduce sub-grid lubrication models.

Modeling of large amplitude plasma blobs in three-dimensions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Angus, Justin R.; Umansky, Maxim V.

2014-01-15

Fluctuations in fusion boundary and similar plasmas often have the form of filamentary structures, or blobs, that convectively propagate radially. This may lead to the degradation of plasma facing components as well as plasma confinement. Theoretical analysis of plasma blobs usually takes advantage of the so-called Boussinesq approximation of the potential vorticity equation, which greatly simplifies the treatment analytically and numerically. This approximation is only strictly justified when the blob density amplitude is small with respect to that of the background plasma. However, this is not the case for typical plasma blobs in the far scrape-off layer region, where themore » background density is small compared to that of the blob, and results obtained based on the Boussinesq approximation are questionable. In this report, the solution of the full vorticity equation, without the usual Boussinesq approximation, is proposed via a novel numerical approach. The method is used to solve for the evolution of 2D and 3D plasma blobs in a regime where the Boussinesq approximation is not valid. The Boussinesq solution under predicts the cross field transport in 2D. However, in 3D, for parameters typical of current tokamaks, the disparity between the radial cross field transport from the Boussinesq approximation and full solution is virtually non-existent due to the effects of the drift wave instability.« less

An experimental study of wall-injected flows in a rectangular cylinder

NASA Astrophysics Data System (ADS)

Perrotta, A.; Romano, G. P.; Favini, B.

2018-01-01

An experimental investigation of the flow inside a rectangular cylinder with air injected continuously along the wall is performed. This kind of flow is a two-dimensional approximation of what happens inside a solid rocket motor, where the lateral grain burns expelling exhaust gas or in processes with air filtration or devices to attain uniform flows. We propose a brief derivation of some analytical solutions and a comparison between these solutions and experimental data, which are obtained using the particle image velocimetry technique, to provide a global reconstruction of the flowfield. The flow, which enters orthogonal to the injecting wall, turns suddenly its direction being pushed towards the exit of the chamber. Under the incompressible and inviscid flow hypothesis, two analytical solutions are reported and compared. The first one, known as Hart-McClure solution, is irrotational and the injection velocity is non-perpendicular to the injecting wall. The other one, due to Taylor and Culick, has non-zero vorticity and constant, vertical injection velocity. The comparison with laminar solutions is useful to assess whether transition to turbulence is reached and how the disturbance thrown in by the porous injection influences and modifies those solutions.

Localized solutions of Lugiato-Lefever equations with focused pump.

PubMed

Cardoso, Wesley B; Salasnich, Luca; Malomed, Boris A

2017-12-04

Lugiato-Lefever (LL) equations in one and two dimensions (1D and 2D) accurately describe the dynamics of optical fields in pumped lossy cavities with the intrinsic Kerr nonlinearity. The external pump is usually assumed to be uniform, but it can be made tightly focused too-in particular, for building small pixels. We obtain solutions of the LL equations, with both the focusing and defocusing intrinsic nonlinearity, for 1D and 2D confined modes supported by the localized pump. In the 1D setting, we first develop a simple perturbation theory, based in the sech ansatz, in the case of weak pump and loss. Then, a family of exact analytical solutions for spatially confined modes is produced for the pump focused in the form of a delta-function, with a nonlinear loss (two-photon absorption) added to the LL model. Numerical findings demonstrate that these exact solutions are stable, both dynamically and structurally (the latter means that stable numerical solutions close to the exact ones are found when a specific condition, necessary for the existence of the analytical solution, does not hold). In 2D, vast families of stable confined modes are produced by means of a variational approximation and full numerical simulations.

Analytical solutions by squeezing to the anisotropic Rabi model in the nonperturbative deep-strong-coupling regime

NASA Astrophysics Data System (ADS)

Zhang, Yu-Yu; Chen, Xiang-You

2017-12-01

An unexplored nonperturbative deep strong coupling (npDSC) achieved in superconducting circuits has been studied in the anisotropic Rabi model by the generalized squeezing rotating-wave approximation. Energy levels are evaluated analytically from the reformulated Hamiltonian and agree well with numerical ones in a wide range of coupling strength. Such improvement ascribes to deformation effects in the displaced-squeezed state presented by the squeezed momentum variance, which are omitted in previous displaced states. The atom population dynamics confirms the validity of our approach for the npDSC strength. Our approach offers the possibility to explore interesting phenomena analytically in the npDSC regime in qubit-oscillator experiments.

Using semi-analytic solutions to approximate the area of potential impact for carbon dioxide injection

EPA Science Inventory

This study examines using the threshold critical pressure increase and the extent of the carbon dioxide (CO2) plume to delineate the area of potential impact (AoPI) for geologic CO2 storage projects. The combined area covering both the CO2 plume and the region where the pressure ...

Finite stretching of a circular plate of neo-Hookean material.

NASA Technical Reports Server (NTRS)

Biricikoglu, V.

1971-01-01

The analytical solution presented is based on the assumption that the deformed thickness of the plate is approximately constant. The nonlinear equations governing finite axisymmetric deformations of a circular plate made of neo-Hookean material are used in the analysis. The variation of circumferential extension ratio and the variation of deformed thickness are shown in graphs.

Flame characteristics for fires in southern fuels

Treesearch

Ralph M. Nelson

1980-01-01

A flame model and analytical method are used to derive forest fire flame characteristics. Approximate solutions are used to express flame lengths, angles, heights, and tip velocities of headfires and calm-air fires in terms of fire intensity. Equations are compared with data from low-intensity controlled burns in southern fuels and with data from the literature.

A Rotating Plug Model of Friction Stir Welding Heat Transfer

NASA Technical Reports Server (NTRS)

Raghulapadu J. K.; Peddieson, J.; Buchanan, G. R.; Nunes, A. C.

2006-01-01

A simplified rotating plug model is employed to study the heat transfer phenomena associated with the fiction stir welding process. An approximate analytical solution is obtained based on this idealized model and used both to demonstrate the qualitative influence of process parameters on predictions and to estimate temperatures produced in typical fiction stir welding situations.

Approximate analysis of thermal convection in a crystal-growth cell for Spacelab 3

NASA Technical Reports Server (NTRS)

Dressler, R. F.

1982-01-01

The transient and steady thermal convection in microgravity is described. The approach is applicable to many three dimensional flows in containers of various shapes with various thermal gradients imposed. The method employs known analytical solutions to two dimensional thermal flows in simpler geometries, and does not require recourse to numerical calculations by computer.

D-Dimensional Dirac Equation for Energy-Dependent Pseudoharmonic and Mie-type Potentials via SUSYQM

NASA Astrophysics Data System (ADS)

A. N., Ikot; Hassanabadi, H.; Maghsoodi, E.; Zarrinkamar, S.

2014-04-01

We investigate the approximate solution of the Dirac equation for energy-dependent pseudoharmonic and Mie-type potentials under the pseudospin and spin symmetries using the supersymmetry quantum mechanics. We obtain the bound-state energy equation in an analytical manner and comment on the system behavior via various figures and tables.

Radial distribution function for hard spheres in fractal dimensions: A heuristic approximation.

PubMed

Santos, Andrés; de Haro, Mariano López

2016-06-01

Analytic approximations for the radial distribution function, the structure factor, and the equation of state of hard-core fluids in fractal dimension d (1≤d≤3) are developed as heuristic interpolations from the knowledge of the exact and Percus-Yevick results for the hard-rod and hard-sphere fluids, respectively. In order to assess their value, such approximate results are compared with those of recent Monte Carlo simulations and numerical solutions of the Percus-Yevick equation for a fractal dimension [M. Heinen et al., Phys. Rev. Lett. 115, 097801 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.097801], a good agreement being observed.

Optimal discrete-time LQR problems for parabolic systems with unbounded input: Approximation and convergence

NASA Technical Reports Server (NTRS)

Rosen, I. G.

1988-01-01

An abstract approximation and convergence theory for the closed-loop solution of discrete-time linear-quadratic regulator problems for parabolic systems with unbounded input is developed. Under relatively mild stabilizability and detectability assumptions, functional analytic, operator techniques are used to demonstrate the norm convergence of Galerkin-based approximations to the optimal feedback control gains. The application of the general theory to a class of abstract boundary control systems is considered. Two examples, one involving the Neumann boundary control of a one-dimensional heat equation, and the other, the vibration control of a cantilevered viscoelastic beam via shear input at the free end, are discussed.

Torsional vibration of a cracked rod by variational formulation and numerical analysis

NASA Astrophysics Data System (ADS)

Chondros, T. G.; Labeas, G. N.

2007-04-01

The torsional vibration of a circumferentially cracked cylindrical shaft is studied through an "exact" analytical solution and a numerical finite element (FE) analysis. The Hu-Washizu-Barr variational formulation is used to develop the differential equation and the boundary conditions of the cracked rod. The equations of motion for a uniform cracked rod in torsional vibration are derived and solved, and the Rayleigh quotient is used to further approximate the natural frequencies of the cracked rod. Results for the problem of the torsional vibration of a cylindrical shaft with a peripheral crack are provided through an analytical solution based on variational formulation to derive the equation of motion and a numerical analysis utilizing a parametric three-dimensional (3D) solid FE model of the cracked rod. The crack is modelled as a continuous flexibility based on fracture mechanics principles. The variational formulation results are compared with the FE alternative. The sensitivity of the FE discretization with respect to the analytical results is assessed.

Applications of computer algebra to distributed parameter systems

NASA Technical Reports Server (NTRS)

Storch, Joel A.

1993-01-01

In the analysis of vibrations of continuous elastic systems, one often encounters complicated transcendental equations with roots directly related to the system's natural frequencies. Typically, these equations contain system parameters whose values must be specified before a numerical solution can be obtained. The present paper presents a method whereby the fundamental frequency can be obtained in analytical form to any desired degree of accuracy. The method is based upon truncation of rapidly converging series involving inverse powers of the system natural frequencies. A straightforward method to developing these series and summing them in closed form is presented. It is demonstrated how Computer Algebra can be exploited to perform the intricate analytical procedures which otherwise would render the technique difficult to apply in practice. We illustrate the method by developing two analytical approximations to the fundamental frequency of a vibrating cantilever carrying a rigid tip body. The results are compared to the numerical solution of the exact (transcendental) frequency equation over a range of system parameters.

Monte Carlo shock-like solutions to the Boltzmann equation with collective scattering

NASA Technical Reports Server (NTRS)

Ellison, D. C.; Eichler, D.

1984-01-01

The results of Monte Carlo simulations of steady state shocks generated by a collision operator that isotropizes the particles by means of elastic scattering in some locally defined frame of reference are presented. The simulations include both the back reaction of accelerated particles on the inflowing plasma and the free escape of high-energy particles from finite shocks. Energetic particles are found to be naturally extracted out of the background plasma by the shock process with an efficiency in good quantitative agreement with an earlier analytic approximation (Eichler, 1983 and 1984) and observations (Gosling et al., 1981) of the entire particle spectrum at a quasi-parallel interplanetary shock. The analytic approximation, which allows a self-consistent determination of the effective adiabatic index of the shocked gas, is used to calculate the overall acceleration efficiency and particle spectrum for cases where ultrarelativistic energies are obtained. It is found that shocks of the strength necessary to produce galactic cosmic rays put approximately 15 percent of the shock energy into relativistic particles.

Atomic density functional and diagram of structures in the phase field crystal model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ankudinov, V. E., E-mail: vladimir@ankudinov.org; Galenko, P. K.; Kropotin, N. V.

2016-02-15

The phase field crystal model provides a continual description of the atomic density over the diffusion time of reactions. We consider a homogeneous structure (liquid) and a perfect periodic crystal, which are constructed from the one-mode approximation of the phase field crystal model. A diagram of 2D structures is constructed from the analytic solutions of the model using atomic density functionals. The diagram predicts equilibrium atomic configurations for transitions from the metastable state and includes the domains of existence of homogeneous, triangular, and striped structures corresponding to a liquid, a body-centered cubic crystal, and a longitudinal cross section of cylindricalmore » tubes. The method developed here is employed for constructing the diagram for the homogeneous liquid phase and the body-centered iron lattice. The expression for the free energy is derived analytically from density functional theory. The specific features of approximating the phase field crystal model are compared with the approximations and conclusions of the weak crystallization and 2D melting theories.« less

Analytical approximations for effective relative permeability in the capillary limit

NASA Astrophysics Data System (ADS)

Rabinovich, Avinoam; Li, Boxiao; Durlofsky, Louis J.

2016-10-01

We present an analytical method for calculating two-phase effective relative permeability, krjeff, where j designates phase (here CO2 and water), under steady state and capillary-limit assumptions. These effective relative permeabilities may be applied in experimental settings and for upscaling in the context of numerical flow simulations, e.g., for CO2 storage. An exact solution for effective absolute permeability, keff, in two-dimensional log-normally distributed isotropic permeability (k) fields is the geometric mean. We show that this does not hold for krjeff since log normality is not maintained in the capillary-limit phase permeability field (Kj=k·krj) when capillary pressure, and thus the saturation field, is varied. Nevertheless, the geometric mean is still shown to be suitable for approximating krjeff when the variance of ln⁡k is low. For high-variance cases, we apply a correction to the geometric average gas effective relative permeability using a Winsorized mean, which neglects large and small Kj values symmetrically. The analytical method is extended to anisotropically correlated log-normal permeability fields using power law averaging. In these cases, the Winsorized mean treatment is applied to the gas curves for cases described by negative power law exponents (flow across incomplete layers). The accuracy of our analytical expressions for krjeff is demonstrated through extensive numerical tests, using low-variance and high-variance permeability realizations with a range of correlation structures. We also present integral expressions for geometric-mean and power law average krjeff for the systems considered, which enable derivation of closed-form series solutions for krjeff without generating permeability realizations.

Fluid mechanics in the perivascular space.

PubMed

Wang, Peng; Olbricht, William L

2011-04-07

Perivascular space (PVS) within the brain is an important pathway for interstitial fluid (ISF) and solute transport. Fluid flowing in the PVS can affect these transport processes and has significant impacts on physiology. In this paper, we carry out a theoretical analysis to investigate the fluid mechanics in the PVS. With certain assumptions and approximations, we are able to find an analytical solution to the problem. We discuss the physical meanings of the solution and particularly examine the consequences of the induced fluid flow in the context of convection-enhanced delivery (CED). We conclude that peristaltic motions of the blood vessel walls can facilitate fluid and solute transport in the PVS. Copyright © 2011 Elsevier Ltd. All rights reserved.

Shape determination and control for large space structures

NASA Technical Reports Server (NTRS)

Weeks, C. J.

1981-01-01

An integral operator approach is used to derive solutions to static shape determination and control problems associated with large space structures. Problem assumptions include a linear self-adjoint system model, observations and control forces at discrete points, and performance criteria for the comparison of estimates or control forms. Results are illustrated by simulations in the one dimensional case with a flexible beam model, and in the multidimensional case with a finite model of a large space antenna. Modal expansions for terms in the solution algorithms are presented, using modes from the static or associated dynamic mode. These expansions provide approximated solutions in the event that a used form analytical solution to the system boundary value problem is not available.

Online Solution of Two-Player Zero-Sum Games for Continuous-Time Nonlinear Systems With Completely Unknown Dynamics.

PubMed

Fu, Yue; Chai, Tianyou

2016-12-01

Regarding two-player zero-sum games of continuous-time nonlinear systems with completely unknown dynamics, this paper presents an online adaptive algorithm for learning the Nash equilibrium solution, i.e., the optimal policy pair. First, for known systems, the simultaneous policy updating algorithm (SPUA) is reviewed. A new analytical method to prove the convergence is presented. Then, based on the SPUA, without using a priori knowledge of any system dynamics, an online algorithm is proposed to simultaneously learn in real time either the minimal nonnegative solution of the Hamilton-Jacobi-Isaacs (HJI) equation or the generalized algebraic Riccati equation for linear systems as a special case, along with the optimal policy pair. The approximate solution to the HJI equation and the admissible policy pair is reexpressed by the approximation theorem. The unknown constants or weights of each are identified simultaneously by resorting to the recursive least square method. The convergence of the online algorithm to the optimal solutions is provided. A practical online algorithm is also developed. Simulation results illustrate the effectiveness of the proposed method.

An analytically solvable three-body break-up model problem in hyperspherical coordinates

NASA Astrophysics Data System (ADS)

Ancarani, L. U.; Gasaneo, G.; Mitnik, D. M.

2012-10-01

An analytically solvable S-wave model for three particles break-up processes is presented. The scattering process is represented by a non-homogeneous Coulombic Schrödinger equation where the driven term is given by a Coulomb-like interaction multiplied by the product of a continuum wave function and a bound state in the particles coordinates. The closed form solution is derived in hyperspherical coordinates leading to an analytic expression for the associated scattering transition amplitude. The proposed scattering model contains most of the difficulties encountered in real three-body scattering problem, e.g., non-separability in the electrons' spherical coordinates and Coulombic asymptotic behavior. Since the coordinates' coupling is completely different, the model provides an alternative test to that given by the Temkin-Poet model. The knowledge of the analytic solution provides an interesting benchmark to test numerical methods dealing with the double continuum, in particular in the asymptotic regions. An hyperspherical Sturmian approach recently developed for three-body collisional problems is used to reproduce to high accuracy the analytical results. In addition to this, we generalized the model generating an approximate wave function possessing the correct radial asymptotic behavior corresponding to an S-wave three-body Coulomb problem. The model allows us to explore the typical structure of the solution of a three-body driven equation, to identify three regions (the driven, the Coulombic and the asymptotic), and to analyze how far one has to go to extract the transition amplitude.

Exact analytical solution of shear-induced flexural vibration of functionally graded piezoelectric beam

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sharma, Pankaj, E-mail: psharma@rtu.ac.in; Parashar, Sandeep Kumar, E-mail: parashar2@yahoo.com

The priority of this paper is to obtain the exact analytical solution for free flexural vibration of FGPM beam actuated using the d{sub 15} effect. In piezoelectric actuators, the potential use of d{sub 15} effect has been of particular interest for engineering applications since shear piezoelectric coefficient d15 is much higher than the other piezoelectric coupling constants d{sub 31} and d{sub 33}. The applications of shear actuators are to induce and control the flexural vibrations of beams and plates. In this study, a modified Timoshenko beam theory is used where electric potential is assumed to vary sinusoidaly along the thicknessmore » direction. The material properties are assumed to be graded across the thickness in accordance with power law distribution. Hamilton's principle is employed to obtain the equations of motion along with the associated boundary conditions for FGPM beams. Exact analytical solution is derived thus obtained equations of motion. Results for clamped-clamped and clamped-free boundary conditions are presented. The presented result and method shell serve as benchmark for comparing the results obtained from the other approximate methods.« less

Role of partial miscibility on pressure buildup due to constant rate injection of CO2 into closed and open brine aquifers

NASA Astrophysics Data System (ADS)

Mathias, Simon A.; Gluyas, Jon G.; GonzáLez MartíNez de Miguel, Gerardo J.; Hosseini, Seyyed A.

2011-12-01

This work extends an existing analytical solution for pressure buildup because of CO2 injection in brine aquifers by incorporating effects associated with partial miscibility. These include evaporation of water into the CO2 rich phase and dissolution of CO2 into brine and salt precipitation. The resulting equations are closed-form, including the locations of the associated leading and trailing shock fronts. Derivation of the analytical solution involves making a number of simplifying assumptions including: vertical pressure equilibrium, negligible capillary pressure, and constant fluid properties. The analytical solution is compared to results from TOUGH2 and found to accurately approximate the extent of the dry-out zone around the well, the resulting permeability enhancement due to residual brine evaporation, the volumetric saturation of precipitated salt, and the vertically averaged pressure distribution in both space and time for the four scenarios studied. While brine evaporation is found to have a considerable effect on pressure, the effect of CO2 dissolution is found to be small. The resulting equations remain simple to evaluate in spreadsheet software and represent a significant improvement on current methods for estimating pressure-limited CO2 storage capacity.

Perfect fluidity of a dissipative system: Analytical solution for the Boltzmann equation in AdS 2 Ⓧ S 2

DOE PAGES

Noronha, Jorge; Denicol, Gabriel S.

2015-12-30

In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime AdS 2 Ⓧ S 2. We further derive explicit analytic expressions for the momentum dependence of the single-particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density doesmore » not match the equilibrium form. The nonequilibrium contribution to the entropy density is shown to be due to higher-order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Furthermore, in this system the slowly moving hydrodynamic degrees of freedom can exhibit true perfect fluidity while being totally decoupled from the fast moving, nonhydrodynamical microscopic degrees of freedom that lead to entropy production.« less

Tunneling-assisted transport of carriers through heterojunctions.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Wampler, William R.; Myers, Samuel M.; Modine, Normand A.

The formulation of carrier transport through heterojunctions by tunneling and thermionic emission is derived from first principles. The treatment of tunneling is discussed at three levels of approximation: numerical solution of the one-band envelope equation for an arbitrarily specified potential profile; the WKB approximation for an arbitrary potential; and, an analytic formulation assuming constant internal field. The effects of spatially varying carrier chemical potentials over tunneling distances are included. Illustrative computational results are presented. The described approach is used in exploratory physics models of irradiated heterojunction bipolar transistors within Sandia's QASPR program.

Numerical realization of the variational method for generating self-trapped beams.

PubMed

Duque, Erick I; Lopez-Aguayo, Servando; Malomed, Boris A

2018-03-19

We introduce a numerical variational method based on the Rayleigh-Ritz optimization principle for predicting two-dimensional self-trapped beams in nonlinear media. This technique overcomes the limitation of the traditional variational approximation in performing analytical Lagrangian integration and differentiation. Approximate soliton solutions of a generalized nonlinear Schrödinger equation are obtained, demonstrating robustness of the beams of various types (fundamental, vortices, multipoles, azimuthons) in the course of their propagation. The algorithm offers possibilities to produce more sophisticated soliton profiles in general nonlinear models.

Analysis of the impacts of horizontal translation and scaling on wavefront approximation coefficients with rectangular pupils for Chebyshev and Legendre polynomials.

PubMed

Sun, Wenqing; Chen, Lei; Tuya, Wulan; He, Yong; Zhu, Rihong

2013-12-01

Chebyshev and Legendre polynomials are frequently used in rectangular pupils for wavefront approximation. Ideally, the dataset completely fits with the polynomial basis, which provides the full-pupil approximation coefficients and the corresponding geometric aberrations. However, if there are horizontal translation and scaling, the terms in the original polynomials will become the linear combinations of the coefficients of the other terms. This paper introduces analytical expressions for two typical situations after translation and scaling. With a small translation, first-order Taylor expansion could be used to simplify the computation. Several representative terms could be selected as inputs to compute the coefficient changes before and after translation and scaling. Results show that the outcomes of the analytical solutions and the approximated values under discrete sampling are consistent. With the computation of a group of randomly generated coefficients, we contrasted the changes under different translation and scaling conditions. The larger ratios correlate the larger deviation from the approximated values to the original ones. Finally, we analyzed the peak-to-valley (PV) and root mean square (RMS) deviations from the uses of the first-order approximation and the direct expansion under different translation values. The results show that when the translation is less than 4%, the most deviated 5th term in the first-order 1D-Legendre expansion has a PV deviation less than 7% and an RMS deviation less than 2%. The analytical expressions and the computed results under discrete sampling given in this paper for the multiple typical function basis during translation and scaling in the rectangular areas could be applied in wavefront approximation and analysis.

Integration within the Felsenstein equation for improved Markov chain Monte Carlo methods in population genetics

PubMed Central

Hey, Jody; Nielsen, Rasmus

2007-01-01

In 1988, Felsenstein described a framework for assessing the likelihood of a genetic data set in which all of the possible genealogical histories of the data are considered, each in proportion to their probability. Although not analytically solvable, several approaches, including Markov chain Monte Carlo methods, have been developed to find approximate solutions. Here, we describe an approach in which Markov chain Monte Carlo simulations are used to integrate over the space of genealogies, whereas other parameters are integrated out analytically. The result is an approximation to the full joint posterior density of the model parameters. For many purposes, this function can be treated as a likelihood, thereby permitting likelihood-based analyses, including likelihood ratio tests of nested models. Several examples, including an application to the divergence of chimpanzee subspecies, are provided. PMID:17301231

Effects of Unsaturated Zones on Baseflow Recession: Analytical Solution and Application

NASA Astrophysics Data System (ADS)

Zhan, H.; Liang, X.; Zhang, Y. K.

2017-12-01

Unsaturated flow is an important process in baseflow recessions and its effect is rarely investigated. A mathematical model for a coupled unsaturated-saturated flow in a horizontally unconfined aquifer with time-dependent infiltrations is presented. Semi-analytical solutions for hydraulic heads and discharges are derived using Laplace transform and Cosine transform. The solutions are compared with solutions of the linearized Boussinesq equation (LB solution) and the linearized Laplace equation (LL solution), respectively. The result indicates that a larger dimensionless constitutive exponent κD of the unsaturated zone leads to a smaller discharge during the infiltration period and a larger discharge after the infiltration. The lateral discharge of the unsaturated zone is significant when κD≤1, and becomes negligible when κD≥100. For late times, the power index b of the recession curve-dQ/dt aQb, is 1 and independent of κD, where Q is the baseflow and a is a constant lumped aquifer parameter. For early times, b is approximately equal to 3 but it approaches infinity when t→1. The present solution is applied to synthetic and field cases. The present solution matched the synthetic data better than both the LL and LB solutions, with a minimum relative error of 16% for estimate of hydraulic conductivity. The present solution was applied to the observed streamflow discharge in Iowa, and the estimated values of the aquifer parameters were reasonable.

The propagation of the shock wave from a strong explosion in a plane-parallel stratified medium: the Kompaneets approximation

NASA Astrophysics Data System (ADS)

Olano, C. A.

2009-11-01

Context: Using certain simplifications, Kompaneets derived a partial differential equation that states the local geometrical and kinematical conditions that each surface element of a shock wave, created by a point blast in a stratified gaseous medium, must satisfy. Kompaneets could solve his equation analytically for the case of a wave propagating in an exponentially stratified medium, obtaining the form of the shock front at progressive evolutionary stages. Complete analytical solutions of the Kompaneets equation for shock wave motion in further plane-parallel stratified media were not found, except for radially stratified media. Aims: We aim to analytically solve the Kompaneets equation for the motion of a shock wave in different plane-parallel stratified media that can reflect a wide variety of astrophysical contexts. We were particularly interested in solving the Kompaneets equation for a strong explosion in the interstellar medium of the Galactic disk, in which, due to intense winds and explosions of stars, gigantic gaseous structures known as superbubbles and supershells are formed. Methods: Using the Kompaneets approximation, we derived a pair of equations that we call adapted Kompaneets equations, that govern the propagation of a shock wave in a stratified medium and that permit us to obtain solutions in parametric form. The solutions provided by the system of adapted Kompaneets equations are equivalent to those of the Kompaneets equation. We solved the adapted Kompaneets equations for shock wave propagation in a generic stratified medium by means of a power-series method. Results: Using the series solution for a shock wave in a generic medium, we obtained the series solutions for four specific media whose respective density distributions in the direction perpendicular to the stratification plane are of an exponential, power-law type (one with exponent k=-1 and the other with k =-2) and a quadratic hyperbolic-secant. From these series solutions, we deduced exact solutions for the four media in terms of elemental functions. The exact solution for shock wave propagation in a medium of quadratic hyperbolic-secant density distribution is very appropriate to describe the growth of superbubbles in the Galactic disk. Member of the Carrera del Investigador Científico del CONICET, Argentina.

Response to ``Comment on `Scalings for radiation from plasma bubbles' '' [Phys. Plasmas 18, 034701 (2011)

NASA Astrophysics Data System (ADS)

Thomas, A. G. R.

2011-03-01

In the preceding Comment, Corde, Stordeur, and Malka claim that the trapping threshold derived in my recent paper is incorrect. Their principal argument is that the elliptical orbits I used are not exact solutions of the equation of motion in the fields of the bubble. The original paper never claimed this—rather I claimed that the use of elliptical orbits was a reasonable approximation, which I based on observations from particle-in-cell simulations. Integration of the equation of motion for analytical expressions for idealized bubble fields (either analytically [I. Kostyukov, E. Nerush, A. Pukhov, and V. Seredov, Phys. Rev. Lett. 103, 175003 (2009)] or numerically [S. Corde, A. Stordeur, and V. Malka, "Comment on `Scalings for radiation from plasma bubbles,' " Phys. Plasmas 18, 034701 (2011)]) produces a trapping threshold wholly inconsistent with experiments and full particle-in-cell (PIC) simulations (e.g., requiring an estimated laser intensity of a0˜30 for ne˜1019 cm-3). The inconsistency in the particle trajectories between PIC and the numeric model used by the comment authors arises due to the fact that the analytical fields are only approximately true for "real" plasma bubbles, and lack certain key features of the field structure. Two possible methods of resolution to this inconsistency are either to find ever more complicated but accurate models for the bubble fields or to find approximate solutions to the equations of motion that capture the essential features of the self-consistent electron trajectories. The latter, heuristic approach used in my recent paper produced a threshold that is better matched to experimental observations. In this reply, I will also revisit the problem and examine the relationship between bubble radius and electron momentum at the point of trapping without reference to a particular trajectory.

An analytical model of iceberg drift

NASA Astrophysics Data System (ADS)

Eisenman, I.; Wagner, T. J. W.; Dell, R.

2017-12-01

Icebergs transport freshwater from glaciers and ice shelves, releasing the freshwater into the upper ocean thousands of kilometers from the source. This influences ocean circulation through its effect on seawater density. A standard empirical rule-of-thumb for estimating iceberg trajectories is that they drift at the ocean surface current velocity plus 2% of the atmospheric surface wind velocity. This relationship has been observed in empirical studies for decades, but it has never previously been physically derived or justified. In this presentation, we consider the momentum balance for an individual iceberg, which includes nonlinear drag terms. Applying a series of approximations, we derive an analytical solution for the iceberg velocity as a function of time. In order to validate the model, we force it with surface velocity and temperature data from an observational state estimate and compare the results with iceberg observations in both hemispheres. We show that the analytical solution reduces to the empirical 2% relationship in the asymptotic limit of small icebergs (or strong winds), which approximately applies for typical Arctic icebergs. We find that the 2% value arises due to a term involving the drag coefficients for water and air and the densities of the iceberg, ocean, and air. In the opposite limit of large icebergs (or weak winds), which approximately applies for typical Antarctic icebergs with horizontal length scales greater than about 12 km, we find that the 2% relationship is not applicable and that icebergs instead move with the ocean current, unaffected by the wind. The two asymptotic regimes can be understood by considering how iceberg size influences the relative importance of the wind and ocean current drag terms compared with the Coriolis and pressure gradient force terms in the iceberg momentum balance.

Electromagnetic field analysis and modeling of a relative position detection sensor for high speed maglev trains.

PubMed

Xue, Song; He, Ning; Long, Zhiqiang

2012-01-01

The long stator track for high speed maglev trains has a tooth-slot structure. The sensor obtains precise relative position information for the traction system by detecting the long stator tooth-slot structure based on nondestructive detection technology. The magnetic field modeling of the sensor is a typical three-dimensional (3-D) electromagnetic problem with complex boundary conditions, and is studied semi-analytically in this paper. A second-order vector potential (SOVP) is introduced to simplify the vector field problem to a scalar field one, the solution of which can be expressed in terms of series expansions according to Multipole Theory (MT) and the New Equivalent Source (NES) method. The coefficients of the expansions are determined by the least squares method based on the boundary conditions. Then, the solution is compared to the simulation result through Finite Element Analysis (FEA). The comparison results show that the semi-analytical solution agrees approximately with the numerical solution. Finally, based on electromagnetic modeling, a difference coil structure is designed to improve the sensitivity and accuracy of the sensor.

Electromagnetic Field Analysis and Modeling of a Relative Position Detection Sensor for High Speed Maglev Trains

PubMed Central

Xue, Song; He, Ning; Long, Zhiqiang

2012-01-01

The long stator track for high speed maglev trains has a tooth-slot structure. The sensor obtains precise relative position information for the traction system by detecting the long stator tooth-slot structure based on nondestructive detection technology. The magnetic field modeling of the sensor is a typical three-dimensional (3-D) electromagnetic problem with complex boundary conditions, and is studied semi-analytically in this paper. A second-order vector potential (SOVP) is introduced to simplify the vector field problem to a scalar field one, the solution of which can be expressed in terms of series expansions according to Multipole Theory (MT) and the New Equivalent Source (NES) method. The coefficients of the expansions are determined by the least squares method based on the boundary conditions. Then, the solution is compared to the simulation result through Finite Element Analysis (FEA). The comparison results show that the semi-analytical solution agrees approximately with the numerical solution. Finally, based on electromagnetic modeling, a difference coil structure is designed to improve the sensitivity and accuracy of the sensor. PMID:22778652

Deriving analytic solutions for compact binary inspirals without recourse to adiabatic approximations

NASA Astrophysics Data System (ADS)

Galley, Chad R.; Rothstein, Ira Z.

2017-05-01

We utilize the dynamical renormalization group formalism to calculate the real space trajectory of a compact binary inspiral for long times via a systematic resummation of secularly growing terms. This method generates closed form solutions without orbit averaging, and the accuracy can be systematically improved. The expansion parameter is v5ν Ω (t -t0) where t0 is the initial time, t is the time elapsed, and Ω and v are the angular orbital frequency and initial speed, respectively. ν is the binary's symmetric mass ratio. We demonstrate how to apply the renormalization group method to resum solutions beyond leading order in two ways. First, we calculate the second-order corrections of the leading radiation reaction force, which involves highly nontrivial checks of the formalism (i.e., its renormalizability). Second, we show how to systematically include post-Newtonian corrections to the radiation reaction force. By avoiding orbit averaging, we gain predictive power and eliminate ambiguities in the initial conditions. Finally, we discuss how this methodology can be used to find analytic solutions to the spin equations of motion that are valid over long times.

Infiltration into soils: Conceptual approaches and solutions

NASA Astrophysics Data System (ADS)

Assouline, Shmuel

2013-04-01

Infiltration is a key process in aspects of hydrology, agricultural and civil engineering, irrigation design, and soil and water conservation. It is complex, depending on soil and rainfall properties and initial and boundary conditions within the flow domain. During the last century, a great deal of effort has been invested to understand the physics of infiltration and to develop quantitative predictors of infiltration dynamics. Jean-Yves Parlange and Wilfried Brutsaert have made seminal contributions, especially in the area of infiltration theory and related analytical solutions to the flow equations. This review retraces the landmark discoveries and the evolution of the conceptual approaches and the mathematical solutions applied to the problem of infiltration into porous media, highlighting the pivotal contributions of Parlange and Brutsaert. A historical retrospective of physical models of infiltration is followed by the presentation of mathematical methods leading to analytical solutions of the flow equations. This review then addresses the time compression approximation developed to estimate infiltration at the transition between preponding and postponding conditions. Finally, the effects of special conditions, such as the presence of air and heterogeneity in soil properties, on infiltration are considered.

Generalized self-similar unsteady gas flows behind the strong shock wave front

NASA Astrophysics Data System (ADS)

Bogatko, V. I.; Potekhina, E. A.

2018-05-01

Two-dimensional (plane and axially symmetric) nonstationary gas flows behind the front of a strong shock wave are considered. All the gas parameters are functions of the ratio of Cartesian coordinates to some degree of time tn, where n is a self-similarity index. The problem is solved in Lagrangian variables. It is shown that the resulting system of partial differential equations is suitable for constructing an iterative process. ¢he "thin shock layer" method is used to construct an approximate analytical solution of the problem. The limit solution of the problem is constructed. A formula for determining the path traversed by a gas particle in the shock layer along the front of a shock wave is obtained. A system of equations for determining the first approximation corrections is constructed.

The lunar libration: comparisons between various models - a model fitted to LLR observations

NASA Astrophysics Data System (ADS)

Chapront, J.; Francou, G.

2005-09-01

We consider 4 libration models: 3 numerical models built by JPL (ephemerides for the libration in DE245, DE403 and DE405) and an analytical model improved with numerical complements fitted to recent LLR observations. The analytical solution uses 3 angular variables (ρ1, ρ2, τ) which represent the deviations with respect to Cassini's laws. After having referred the models to a unique reference frame, we study the differences between the models which depend on gravitational and tidal parameters of the Moon, as well as amplitudes and frequencies of the free librations. It appears that the differences vary widely depending of the above quantities. They correspond to a few meters displacement on the lunar surface, reminding that LLR distances are precise to the centimeter level. Taking advantage of the lunar libration theory built by Moons (1984) and improved by Chapront et al. (1999) we are able to establish 4 solutions and to represent their differences by Fourier series after a numerical substitution of the gravitational constants and free libration parameters. The results are confirmed by frequency analyses performed separately. Using DE245 as a basic reference ephemeris, we approximate the differences between the analytical and numerical models with Poisson series. The analytical solution - improved with numerical complements under the form of Poisson series - is valid over several centuries with an internal precision better than 5 centimeters.

The study of the Boltzmann equation of solid-gas two-phase flow with three-dimensional BGK model

NASA Astrophysics Data System (ADS)

Liu, Chang-jiang; Pang, Song; Xu, Qiang; He, Ling; Yang, Shao-peng; Qing, Yun-jie

2018-06-01

The motion of many solid-gas two-phase flows can be described by the Boltzmann equation. In order to simplify the Boltzmann equation, the convective-diffusion term is reserved and the collision term is replaced by the three-dimensional Bharnagar-Gross-Krook (BGK) model. Then the simplified Boltzmann equation is solved by homotopy perturbation method (HPM), and its approximate analytical solution is obtained. Through the analyzing, it is proved that the analytical solution satisfies all the constraint conditions, and its formation is in accord with the formation of the solution that is obtained by traditional Chapman-Enskog method, and the solving process of HPM is much more simple and convenient. This preliminarily shows the effectiveness and rapidness of HPM to solve the Boltzmann equation. The results obtained herein provide some theoretical basis for the further study of dynamic model of solid-gas two-phase flows, such as the sturzstrom of high-speed distant landslide caused by microseism and the sand storm caused by strong breeze.

Three-Dimensional Field Solutions for Multi-Pole Cylindrical Halbach Arrays in an Axial Orientation

NASA Technical Reports Server (NTRS)

Thompson, William K.

2006-01-01

This article presents three-dimensional B field solutions for the cylindrical Halbach array in an axial orientation. This arrangement has applications in the design of axial motors and passive axial magnetic bearings and couplers. The analytical model described here assumes ideal magnets with fixed and uniform magnetization. The field component functions are expressed as sums of 2-D definite integrals that are easily computed by a number of mathematical analysis software packages. The analysis is verified with sample calculations and the results are compared to equivalent results from traditional finite-element analysis (FEA). The field solutions are then approximated for use in flux linkage and induced EMF calculations in nearby stator windings by expressing the field variance with angular displacement as pure sinusoidal function whose amplitude depends on radial and axial position. The primary advantage of numerical implementation of the analytical approach presented in the article is that it lends itself more readily to parametric analysis and design tradeoffs than traditional FEA models.

An analytical solution for the squeeze film between a nondeformable sphere and groove

NASA Technical Reports Server (NTRS)

Allen, C. W.; Wilson, M. P.

1972-01-01

An analysis is presented to compute the film thickness, pressure and load relations between a rigid ball and rigid groove in normal approach when lubricated by a fluid with an exponential pressure-viscosity relationship. The geometry of the ball-groove system is reduced to the equivalent system of a paraboloid approaching a flat plate. Exact and approximate solutions are presented for the load and pressure relations. There is found to be a limiting load for a given geometry and lubricant regardless of the rate of approach.

Space proton transport in one dimension

NASA Technical Reports Server (NTRS)

Lamkin, S. L.; Khandelwal, G. S.; Shinn, J. L.; Wilson, J. W.

1994-01-01

An approximate evaluation procedure is derived for a second-order theory of coupled nucleon transport in one dimension. An analytical solution with a simplified interaction model is used to determine quadrature parameters to minimize truncation error. Effects of the improved method on transport solutions with the BRYNTRN data base are evaluated. Comparisons with Monte Carlo benchmarks are given. Using different shield materials, the computational procedure is used to study the physics of space protons. A transition effect occurs in tissue near the shield interface and is most important in shields of high atomic number.

Fluid displacement between two parallel plates: a non-empirical model displaying change of type from hyperbolic to elliptic equations

NASA Astrophysics Data System (ADS)

Shariati, M.; Talon, L.; Martin, J.; Rakotomalala, N.; Salin, D.; Yortsos, Y. C.

2004-11-01

We consider miscible displacement between parallel plates in the absence of diffusion, with a concentration-dependent viscosity. By selecting a piecewise viscosity function, this can also be considered as ‘three-fluid’ flow in the same geometry. Assuming symmetry across the gap and based on the lubrication (‘equilibrium’) approximation, a description in terms of two quasi-linear hyperbolic equations is obtained. We find that the system is hyperbolic and can be solved analytically, when the mobility profile is monotonic, or when the mobility of the middle phase is smaller than its neighbours. When the mobility of the middle phase is larger, a change of type is displayed, an elliptic region developing in the composition space. Numerical solutions of Riemann problems of the hyperbolic system spanning the elliptic region, with small diffusion added, show good agreement with the analytical outside, but an unstable behaviour inside the elliptic region. In these problems, the elliptic region arises precisely at the displacement front. Crossing the elliptic region requires the solution of essentially an eigenvalue problem of the full higher-dimensional model, obtained here using lattice BGK simulations. The hyperbolic-to-elliptic change-of-type reflects the failing of the lubrication approximation, underlying the quasi-linear hyperbolic formalism, to describe the problem uniformly. The obtained solution is analogous to non-classical shocks recently suggested in problems with change of type.

Planetary spectra for anisotropic scattering

NASA Technical Reports Server (NTRS)

Chamberlain, J. W.

1976-01-01

Some effects on planetary spectra that would be produced by departures from isotropic scattering are examined. The phase function is the simplest departure to handle analytically and the only phase function, other than the isotropic one, that can be incorporated into a Chandrasekhar first approximation. This approach has the advantage of illustrating effects resulting from anisotropies while retaining the simplicity that yields analytic solutions. The curve of growth is the sine qua non of planetary spectroscopy. The discussion emphasizes the difficulties and importance of ascertaining curves of growth as functions of observing geometry. A plea is made to observers to analyze their empirical curves of growth, whenever it seems feasible, in terms of coefficients of which are the leading terms in radiative-transfer analysis. An algebraic solution to the two sets of anisotropic H functions is developed which gives emergent intensities accurate to 0.3%.

Modulational Instability and Quantum Discrete Breather States of Cold Bosonic Atoms in a Zig-Zag Optical Lattice

NASA Astrophysics Data System (ADS)

Chang, Xia; Xie, Jiayu; Wu, Tianle; Tang, Bing

2018-07-01

A theoretical study on modulational instability and quantum discrete breather states in a system of cold bosonic atoms in zig-zag optical lattices is presented in this work. The time-dependent Hartree approximation is employed to deal with the multiple body problem. By means of a linear stability analysis, we analytically study the modulational instability, and estimate existence conditions of the bright stationary localized solutions for different values of the second-neighbor hopping constant. On the other hand, we get analytical bright stationary localized solutions, and analyze the influence of the second-neighbor hopping on their existence conditions. The predictions of the modulational instability analysis are shown to be reliable. Using these stationary localized single-boson wave functions, the quantum breather states corresponding to the system with different types of nonlinearities are constructed.

Quantifying non-Markovianity of continuous-variable Gaussian dynamical maps

DOE Office of Scientific and Technical Information (OSTI.GOV)

Vasile, Ruggero; Maniscalco, Sabrina; Paris, Matteo G. A.

2011-11-15

We introduce a non-Markovianity measure for continuous-variable open quantum systems based on the idea put forward in H.-P. Breuer et al.[Phys. Rev. Lett. 103, 210401 (2009);], that is, by quantifying the flow of information from the environment back to the open system. Instead of the trace distance we use here the fidelity to assess distinguishability of quantum states. We employ our measure to evaluate non-Markovianity of two paradigmatic Gaussian channels: the purely damping channel and the quantum Brownian motion channel with Ohmic environment. We consider different classes of Gaussian states and look for pairs of states maximizing the backflow ofmore » information. For coherent states we find simple analytical solutions, whereas for squeezed states we provide both exact numerical and approximate analytical solutions in the weak coupling limit.« less

Reemission spectra and inelastic processes at interaction of attosecond and shorter duration electromagnetic pulses with atoms

NASA Astrophysics Data System (ADS)

Makarov, D. N.; Matveev, V. I.

2017-01-01

Inelastic processes and the reemission of attosecond and shorter electromagnetic pulses by atoms have been considered within the analytical solution of the Schrödinger equation in the sudden perturbation approximation. A method of calculations with the exact inclusion of spatial inhomogeneity of the field of an ultrashort pulse and the momenta of photons in the reemission processes has been developed. The probabilities of inelastic processes and spectra of reemission of ultrashort electromagnetic pulses by one- and many-electron atoms have been calculated. The results have been presented in the form of analytical formulas.

A simplified analytical solution for thermal response of a one-dimensional, steady state transpiration cooling system in radiative and convective environment

NASA Technical Reports Server (NTRS)

Kubota, H.

1976-01-01

A simplified analytical method for calculation of thermal response within a transpiration-cooled porous heat shield material in an intense radiative-convective heating environment is presented. The essential assumptions of the radiative and convective transfer processes in the heat shield matrix are the two-temperature approximation and the specified radiative-convective heatings of the front surface. Sample calculations for porous silica with CO2 injection are presented for some typical parameters of mass injection rate, porosity, and material thickness. The effect of these parameters on the cooling system is discussed.

Theory and Circuit Model for Lossy Coaxial Transmission Line

DOE Office of Scientific and Technical Information (OSTI.GOV)

Genoni, T. C.; Anderson, C. N.; Clark, R. E.

2017-04-01

The theory of signal propagation in lossy coaxial transmission lines is revisited and new approximate analytic formulas for the line impedance and attenuation are derived. The accuracy of these formulas from DC to 100 GHz is demonstrated by comparison to numerical solutions of the exact field equations. Based on this analysis, a new circuit model is described which accurately reproduces the line response over the entire frequency range. Circuit model calculations are in excellent agreement with the numerical and analytic results, and with finite-difference-time-domain simulations which resolve the skindepths of the conducting walls.

Copper nanoparticles impinging on a curved channel with compliant walls and peristalsis

NASA Astrophysics Data System (ADS)

Akbar, Noreen Sher; Maraj, E. N.; Butt, Adil Wahid

2014-08-01

In the present article peristaltic transport of copper nanofluids in a curved channel with compliant walls is analytically studied. The mathematical analysis is carried out under the low Reynolds number and long wavelenght approximation. The exact solutions are computed for fluid velocity and temperature profile. The effect of meaningful parameters are shown graphically in the last section.

Gas gun dynamics

NASA Astrophysics Data System (ADS)

Denny, Mark

2013-09-01

The mechanics and thermodynamics of one- and two-stage gas guns are developed. Very high projectile muzzle speed can be obtained by the two-stage version. The physics of simple gas guns, such as air rifles, is accessible to undergraduates and the same level of presentation is used here to understand more complex designs. Numerical solutions to the equations of motion are shown, along with insightful analytic approximations.

Multipolar electromagnetic fields around neutron stars: general-relativistic vacuum solutions

NASA Astrophysics Data System (ADS)

Pétri, J.

2017-12-01

Magnetic fields inside and around neutron stars are at the heart of pulsar magnetospheric activity. Strong magnetic fields are responsible for quantum effects, an essential ingredient to produce leptonic pairs and the subsequent broad-band radiation. The variety of electromagnetic field topologies could lead to the observed diversity of neutron star classes. Thus, it is important to include multipolar components to a presumably dominant dipolar magnetic field. Exact analytical solutions for these multipoles in Newtonian gravity have been computed in recent literature. However, flat space-time is not adequate to describe physics in the immediate surroundings of neutron stars. We generalize the multipole expressions to the strong gravity regime by using a slowly rotating metric approximation such as the one expected around neutron stars. Approximate formulae for the electromagnetic field including frame dragging are computed from which we estimate the Poynting flux and the braking index. Corrections to leading order in compactness and spin parameter are presented. As far as spin-down luminosity is concerned, it is shown that frame dragging remains irrelevant. For high-order multipoles starting from the quadrupole, the electric part can radiate more efficiently than the magnetic part. Both analytical and numerical tools are employed.

Analytical and numerical construction of vertical periodic orbits about triangular libration points based on polynomial expansion relations among directions

NASA Astrophysics Data System (ADS)

Qian, Ying-Jing; Yang, Xiao-Dong; Zhai, Guan-Qiao; Zhang, Wei

2017-08-01

Innovated by the nonlinear modes concept in the vibrational dynamics, the vertical periodic orbits around the triangular libration points are revisited for the Circular Restricted Three-body Problem. The ζ -component motion is treated as the dominant motion and the ξ and η -component motions are treated as the slave motions. The slave motions are in nature related to the dominant motion through the approximate nonlinear polynomial expansions with respect to the ζ -position and ζ -velocity during the one of the periodic orbital motions. By employing the relations among the three directions, the three-dimensional system can be transferred into one-dimensional problem. Then the approximate three-dimensional vertical periodic solution can be analytically obtained by solving the dominant motion only on ζ -direction. To demonstrate the effectiveness of the proposed method, an accuracy study was carried out to validate the polynomial expansion (PE) method. As one of the applications, the invariant nonlinear relations in polynomial expansion form are used as constraints to obtain numerical solutions by differential correction. The nonlinear relations among the directions provide an alternative point of view to explore the overall dynamics of periodic orbits around libration points with general rules.

Analytical model of the optical vortex microscope.

PubMed

Płocinniczak, Łukasz; Popiołek-Masajada, Agnieszka; Masajada, Jan; Szatkowski, Mateusz

2016-04-20

This paper presents an analytical model of the optical vortex scanning microscope. In this microscope the Gaussian beam with an embedded optical vortex is focused into the sample plane. Additionally, the optical vortex can be moved inside the beam, which allows fine scanning of the sample. We provide an analytical solution of the whole path of the beam in the system (within paraxial approximation)-from the vortex lens to the observation plane situated on the CCD camera. The calculations are performed step by step from one optical element to the next. We show that at each step, the expression for light complex amplitude has the same form with only four coefficients modified. We also derive a simple expression for the vortex trajectory of small vortex displacements.

Analytic drain current model for III-V cylindrical nanowire transistors

NASA Astrophysics Data System (ADS)

Marin, E. G.; Ruiz, F. G.; Schmidt, V.; Godoy, A.; Riel, H.; Gámiz, F.

2015-07-01

An analytical model is proposed to determine the drain current of III-V cylindrical nanowires (NWs). The model uses the gradual channel approximation and takes into account the complete analytical solution of the Poisson and Schrödinger equations for the Γ-valley and for an arbitrary number of subbands. Fermi-Dirac statistics are considered to describe the 1D electron gas in the NWs, being the resulting recursive Fermi-Dirac integral of order -1/2 successfully integrated under reasonable assumptions. The model has been validated against numerical simulations showing excellent agreement for different semiconductor materials, diameters up to 40 nm, gate overdrive biases up to 0.7 V, and densities of interface states up to 1013eV-1cm-2 .

An Improved Analytical Model of the Local Interstellar Magnetic Field: The Extension to Compressibility

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kleimann, Jens; Fichtner, Horst; Röken, Christian, E-mail: jk@tp4.rub.de, E-mail: hf@tp4.rub.de, E-mail: christian.roeken@mathematik.uni-regensburg.de

A previously published analytical magnetohydrodynamic model for the local interstellar magnetic field in the vicinity of the heliopause (Röken et al. 2015) is extended from incompressible to compressible, yet predominantly subsonic flow, considering both isothermal and adiabatic equations of state. Exact expressions and suitable approximations for the density and the flow velocity are derived and discussed. In addition to the stationary induction equation, these expressions also satisfy the momentum balance equation along stream lines. The practical usefulness of the corresponding, still exact, analytical magnetic field solution is assessed by comparing it quantitatively to results from a fully self-consistent magnetohydrodynamic simulationmore » of the interstellar magnetic field draping around the heliopause.« less

An innovative hybrid 3D analytic-numerical model for air breathing parallel channel counter-flow PEM fuel cells.

PubMed

Tavčar, Gregor; Katrašnik, Tomaž

2014-01-01

The parallel straight channel PEM fuel cell model presented in this paper extends the innovative hybrid 3D analytic-numerical (HAN) approach previously published by the authors with capabilities to address ternary diffusion systems and counter-flow configurations. The model's core principle is modelling species transport by obtaining a 2D analytic solution for species concentration distribution in the plane perpendicular to the cannel gas-flow and coupling consecutive 2D solutions by means of a 1D numerical pipe-flow model. Electrochemical and other nonlinear phenomena are coupled to the species transport by a routine that uses derivative approximation with prediction-iteration. The latter is also the core of the counter-flow computation algorithm. A HAN model of a laboratory test fuel cell is presented and evaluated against a professional 3D CFD simulation tool showing very good agreement between results of the presented model and those of the CFD simulation. Furthermore, high accuracy results are achieved at moderate computational times, which is owed to the semi-analytic nature and to the efficient computational coupling of electrochemical kinetics and species transport.

Comparisons of characteristic timescales and approximate models for Brownian magnetic nanoparticle rotations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Reeves, Daniel B., E-mail: dbr@Dartmouth.edu; Weaver, John B.

2015-06-21

Magnetic nanoparticles are promising tools for a host of therapeutic and diagnostic medical applications. The dynamics of rotating magnetic nanoparticles in applied magnetic fields depend strongly on the type and strength of the field applied. There are two possible rotation mechanisms and the decision for the dominant mechanism is often made by comparing the equilibrium relaxation times. This is a problem when particles are driven with high-amplitude fields because they are not necessarily at equilibrium at all. Instead, it is more appropriate to consider the “characteristic timescales” that arise in various applied fields. Approximate forms for the characteristic time ofmore » Brownian particle rotations do exist and we show agreement between several analytical and phenomenological-fit models to simulated data from a stochastic Langevin equation approach. We also compare several approximate models with solutions of the Fokker-Planck equation to determine their range of validity for general fields and relaxation times. The effective field model is an excellent approximation, while the linear response solution is only useful for very low fields and frequencies for realistic Brownian particle rotations.« less

A multi-species reactive transport model to estimate biogeochemical rates based on single-well push-pull test data

NASA Astrophysics Data System (ADS)

Phanikumar, Mantha S.; McGuire, Jennifer T.

2010-08-01

Push-pull tests are a popular technique to investigate various aquifer properties and microbial reaction kinetics in situ. Most previous studies have interpreted push-pull test data using approximate analytical solutions to estimate (generally first-order) reaction rate coefficients. Though useful, these analytical solutions may not be able to describe important complexities in rate data. This paper reports the development of a multi-species, radial coordinate numerical model (PPTEST) that includes the effects of sorption, reaction lag time and arbitrary reaction order kinetics to estimate rates in the presence of mixing interfaces such as those created between injected "push" water and native aquifer water. The model has the ability to describe an arbitrary number of species and user-defined reaction rate expressions including Monod/Michelis-Menten kinetics. The FORTRAN code uses a finite-difference numerical model based on the advection-dispersion-reaction equation and was developed to describe the radial flow and transport during a push-pull test. The accuracy of the numerical solutions was assessed by comparing numerical results with analytical solutions and field data available in the literature. The model described the observed breakthrough data for tracers (chloride and iodide-131) and reactive components (sulfate and strontium-85) well and was found to be useful for testing hypotheses related to the complex set of processes operating near mixing interfaces.

An Operator Method for Field Moments from the Extended Parabolic Wave Equation and Analytical Solutions of the First and Second Moments for Atmospheric Electromagnetic Wave Propagation

NASA Technical Reports Server (NTRS)

Manning, Robert M.

2004-01-01

The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and solutions are found that transcend those which incorporate the full paraxial approximation at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov approximation (i.e., the delta-correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial approximation is accurate even at millimeter wavelengths. The comprehensive operator solution also allows one to obtain expressions for the longitudinal (generalized) second order moment. This is also considered and the solution for the atmospheric case is obtained and discussed. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.

Solute Migration from the Aquifer Matrix into a Solution Conduit and the Reverse.

PubMed

Li, Guangquan; Field, Malcolm S

2016-09-01

A solution conduit has a permeable wall allowing for water exchange and solute transfer between the conduit and its surrounding aquifer matrix. In this paper, we use Laplace Transform to solve a one-dimensional equation constructed using the Euler approach to describe advective transport of solute in a conduit, a production-value problem. Both nonuniform cross-section of the conduit and nonuniform seepage at the conduit wall are considered in the solution. Physical analysis using the Lagrangian approach and a lumping method is performed to verify the solution. Two-way transfer between conduit water and matrix water is also investigated by using the solution for the production-value problem as a first-order approximation. The approximate solution agrees well with the exact solution if dimensionless travel time in the conduit is an order of magnitude smaller than unity. Our analytical solution is based on the assumption that the spatial and/or temporal heterogeneity in the wall solute flux is the dominant factor in the spreading of spring-breakthrough curves, and conduit dispersion is only a secondary mechanism. Such an approach can lead to the better understanding of water exchange and solute transfer between conduits and aquifer matrix. Euler and Lagrangian approaches are used to solve transport in conduit. Two-way transfer between conduit and matrix is investigated. The solution is applicable to transport in conduit of persisting solute from matrix. © 2016, National Ground Water Association.

A numerical scheme to solve unstable boundary value problems

NASA Technical Reports Server (NTRS)

Kalnay-Rivas, E.

1977-01-01

The considered scheme makes it possible to determine an unstable steady state solution in cases in which, because of lack of symmetry, such a solution cannot be obtained analytically, and other time integration or relaxation schemes, because of instability, fail to converge. The iterative solution of a single complex equation is discussed and a nonlinear system of equations is considered. Described applications of the scheme are related to a steady state solution with shear instability, an unstable nonlinear Ekman boundary layer, and the steady state solution of a baroclinic atmosphere with asymmetric forcing. The scheme makes use of forward and backward time integrations of the original spatial differential operators and of an approximation of the adjoint operators. Only two computations of the time derivative per iteration are required.

Analytical solutions for combined close-contact and natural convection melting in horizontal cylindrical heat storage capsule

DOE Office of Scientific and Technical Information (OSTI.GOV)

Saitoh, T.S.; Hoshi, A.

1998-07-01

Melting and solidification of a phase change material (PCM) in a capsule is of practical importance in latent heat thermal energy storage (LHTES) systems which are considered to be very promising to reduce a peak demand of electricity in the summer season and carbon dioxide (CO{sub 2}) emissions. Two melting modes are involved in melting of capsules. One is close-contact melting between the solid bulk and the capsule wall, and another is natural convection melting in the liquid region. Close-contact melting processes for a single enclosure have been solved using several numerical methods (e.g., Saitoh and Kato (1994)). In additionmore » close-contact melting heat transfer characteristics including melt flow in the liquid film under inner wall temperature distribution were analyzed and simple approximate equations were already presented by Saitoh and Hoshi (1997). The effects of Stefan number and variable temperature profile etc. were clarified in detail. And the melting velocity of the solid bulk under various conditions was also studied theoretically. In addition the effects of variable inner wall temperature on molten mass fraction were investigated. The present paper reports analytical solutions for combined close-contact and natural convection melting in horizontal cylindrical capsule. Moreover, natural convection melting in the liquid region were analyzed in this report. The upper interface shape of the solid bulk is approximated by a circular arc throughout the melting process. For the sake of verification, close-contact melting heat-transfer characteristics including natural convection in the liquid region were studied experimentally. Apparent shift of upper solid-liquid interface is good agreement with the experiment. The present simple approximate solutions will be useful to facilitate designing of the practical capsule bed LHTES systems.« less

A new mathematical solution for predicting char activation reactions

USGS Publications Warehouse

Rafsanjani, H.H.; Jamshidi, E.; Rostam-Abadi, M.

2002-01-01

The differential conservation equations that describe typical gas-solid reactions, such as activation of coal chars, yield a set of coupled second-order partial differential equations. The solution of these coupled equations by exact analytical methods is impossible. In addition, an approximate or exact solution only provides predictions for either reaction- or diffusion-controlling cases. A new mathematical solution, the quantize method (QM), was applied to predict the gasification rates of coal char when both chemical reaction and diffusion through the porous char are present. Carbon conversion rates predicted by the QM were in closer agreement with the experimental data than those predicted by the random pore model and the simple particle model. ?? 2002 Elsevier Science Ltd. All rights reserved.

Ergodicity of the Stochastic Nosé-Hoover Heat Bath

NASA Astrophysics Data System (ADS)

Wei Chung Lo,; Baowen Li,

2010-07-01

We numerically study the ergodicity of the stochastic Nosé-Hoover heat bath whose formalism is based on the Markovian approximation for the Nosé-Hoover equation [J. Phys. Soc. Jpn. 77 (2008) 103001]. The approximation leads to a Langevin-like equation driven by a fluctuating dissipative force and multiplicative Gaussian white noise. The steady state solution of the associated Fokker-Planck equation is the canonical distribution. We investigate the dynamics of this method for the case of (i) free particle, (ii) nonlinear oscillators and (iii) lattice chains. We derive the Fokker-Planck equation for the free particle and present approximate analytical solution for the stationary distribution in the context of the Markovian approximation. Numerical simulation results for nonlinear oscillators show that this method results in a Gaussian distribution for the particles velocity. We also employ the method as heat baths to study nonequilibrium heat flow in one-dimensional Fermi-Pasta-Ulam (FPU-β) and Frenkel-Kontorova (FK) lattices. The establishment of well-defined temperature profiles are observed only when the lattice size is large. Our results provide numerical justification for such Markovian approximation for classical single- and many-body systems.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chhiber, R; Usmanov, AV; Matthaeus, WH

Simple estimates of the number of Coulomb collisions experienced by the interplanetary plasma to the point of observation, i.e., the “collisional age”, can be usefully employed in the study of non-thermal features of the solar wind. Usually these estimates are based on local plasma properties at the point of observation. Here we improve the method of estimation of the collisional age by employing solutions obtained from global three-dimensional magnetohydrodynamics simulations. This enables evaluation of the complete analytical expression for the collisional age without using approximations. The improved estimation of the collisional timescale is compared with turbulence and expansion timescales tomore » assess the relative importance of collisions. The collisional age computed using the approximate formula employed in previous work is compared with the improved simulation-based calculations to examine the validity of the simplified formula. We also develop an analytical expression for the evaluation of the collisional age and we find good agreement between the numerical and analytical results. Finally, we briefly discuss the implications for an improved estimation of collisionality along spacecraft trajectories, including Solar Probe Plus.« less

Shape sensitivity analysis of flutter response of a laminated wing

NASA Technical Reports Server (NTRS)

Bergen, Fred D.; Kapania, Rakesh K.

1988-01-01

A method is presented for calculating the shape sensitivity of a wing aeroelastic response with respect to changes in geometric shape. Yates' modified strip method is used in conjunction with Giles' equivalent plate analysis to predict the flutter speed, frequency, and reduced frequency of the wing. Three methods are used to calculate the sensitivity of the eigenvalue. The first method is purely a finite difference calculation of the eigenvalue derivative directly from the solution of the flutter problem corresponding to the two different values of the shape parameters. The second method uses an analytic expression for the eigenvalue sensitivities of a general complex matrix, where the derivatives of the aerodynamic, mass, and stiffness matrices are computed using a finite difference approximation. The third method also uses an analytic expression for the eigenvalue sensitivities, but the aerodynamic matrix is computed analytically. All three methods are found to be in good agreement with each other. The sensitivities of the eigenvalues were used to predict the flutter speed, frequency, and reduced frequency. These approximations were found to be in good agreement with those obtained using a complete reanalysis.

Analytical Debye-Huckel model for electrostatic potentials around dissolved DNA.

PubMed

Wagner, K; Keyes, E; Kephart, T W; Edwards, G

1997-07-01

We present an analytical, Green-function-based model for the electric potential of DNA in solution, treating the surrounding solvent with the Debye-Huckel approximation. The partial charge of each atom is accounted for by modeling DNA as linear distributions of atoms on concentric cylindrical surfaces. The condensed ions of the solvent are treated with the Debye-Huckel approximation. The resultant leading term of the potential is that of a continuous shielded line charge, and the higher order terms account for the helical structure. Within several angstroms of the surface there is sufficient information in the electric potential to distinguish features and symmetries of DNA. Plots of the potential and equipotential surfaces, dominated by the phosphate charges, reflect the structural differences between the A, B, and Z conformations and, to a smaller extent, the difference between base sequences. As the distances from the helices increase, the magnitudes of the potentials decrease. However, the bases and sugars account for a larger fraction of the double helix potential with increasing distance. We have found that when the solvent is treated with the Debye-Huckel approximation, the potential decays more rapidly in every direction from the surface than it did in the concentric dielectric cylinder approximation.

Analytic derivation of an approximate SU(3) symmetry inside the symmetry triangle of the interacting boson approximation model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bonatsos, Dennis; Karampagia, S.; Casten, R. F.

2011-05-15

Using a contraction of the SU(3) algebra to the algebra of the rigid rotator in the large-boson-number limit of the interacting boson approximation (IBA) model, a line is found inside the symmetry triangle of the IBA, along which the SU(3) symmetry is preserved. The line extends from the SU(3) vertex to near the critical line of the first-order shape/phase transition separating the spherical and prolate deformed phases, and it lies within the Alhassid-Whelan arc of regularity, the unique valley of regularity connecting the SU(3) and U(5) vertices in the midst of chaotic regions. In addition to providing an explanation formore » the existence of the arc of regularity, the present line represents an example of an analytically determined approximate symmetry in the interior of the symmetry triangle of the IBA. The method is applicable to algebraic models possessing subalgebras amenable to contraction. This condition is equivalent to algebras in which the equilibrium ground state and its rotational band become energetically isolated from intrinsic excitations, as typified by deformed solutions to the IBA for large numbers of valence nucleons.« less

Analytical Modeling of Groundwater Seepages to St. Lucie Estuary

NASA Astrophysics Data System (ADS)

Lee, J.; Yeh, G.; Hu, G.

2008-12-01

In this paper, six analytical models describing hydraulic interaction of stream-aquifer systems were applied to St Lucie Estuary (SLE) River Estuaries. These are analytical solutions for: (1) flow from a finite aquifer to a canal, (2) flow from an infinite aquifer to a canal, (3) the linearized Laplace system in a seepage surface, (4) wave propagation in the aquifer, (5) potential flow through stratified unconfined aquifers, and (6) flow through stratified confined aquifers. Input data for analytical solutions were obtained from monitoring wells and river stages at seepage-meter sites. Four transects in the study area are available: Club Med, Harbour Ridge, Lutz/MacMillan, and Pendarvis Cove located in the St. Lucie River. The analytical models were first calibrated with seepage meter measurements and then used to estimate of groundwater discharges into St. Lucie River. From this process, analytical relationships between the seepage rate and river stages and/or groundwater tables were established to predict the seasonal and monthly variation in groundwater seepage into SLE. It was found the seepage rate estimations by analytical models agreed well with measured data for some cases but only fair for some other cases. This is not unexpected because analytical solutions have some inherently simplified assumptions, which may be more valid for some cases than the others. From analytical calculations, it is possible to predict approximate seepage rates in the study domain when the assumptions underlying these analytical models are valid. The finite and infinite aquifer models and the linearized Laplace method are good for sites Pendarvis Cove and Lutz/MacMillian, but fair for the other two sites. The wave propagation model gave very good agreement in phase but only fairly agreement in magnitude for all four sites. The stratified unconfined and confined aquifer models gave similarly good agreements with measurements at three sites but poorly at the Club Med site. None of the analytical models presented here can fit the data at this site. To give better estimates at all sites numerical modeling that couple river hydraulics and groundwater flow involving less simplifications of and assumptions for the system may have to be adapted.

Diagnostics of seeded RF plasmas: An experimental study related to the gaseous core reactor

NASA Technical Reports Server (NTRS)

Thompson, S. D.; Clement, J. D.; Williams, J. R.

1974-01-01

Measurements of the temperature profiles in an RF argon plasma were made over magnetic field intensities ranging from 20 amp turns/cm to 80 amp turns/cm. The results were compared with a one-dimensional numerical treatment of the governing equations and with an approximate closed form analytical solution that neglected radiation losses. The average measured temperatures in the plasma compared well with the numerical treatment, though the experimental profile showed less of an off center temperature peak than predicted by theory. This may be a result of the complex turbulent flow pattern present in the experimental torch and not modeled in the numerical treatment. The radiation term cannot be neglected for argon at the power levels investigated. The closed form analytical approximation that neglected radiation led to temperature predictions on the order of 1000 K to 2000 K higher than measured or predicted by the numerical treatment which considered radiation losses.

An Analytic Approximation to Very High Specific Impulse and Specific Power Interplanetary Space Mission Analysis

NASA Technical Reports Server (NTRS)

Williams, Craig Hamilton

1995-01-01

A simple, analytic approximation is derived to calculate trip time and performance for propulsion systems of very high specific impulse (50,000 to 200,000 seconds) and very high specific power (10 to 1000 kW/kg) for human interplanetary space missions. The approach assumed field-free space, constant thrust/constant specific power, and near straight line (radial) trajectories between the planets. Closed form, one dimensional equations of motion for two-burn rendezvous and four-burn round trip missions are derived as a function of specific impulse, specific power, and propellant mass ratio. The equations are coupled to an optimizing parameter that maximizes performance and minimizes trip time. Data generated for hypothetical one-way and round trip human missions to Jupiter were found to be within 1% and 6% accuracy of integrated solutions respectively, verifying that for these systems, credible analysis does not require computationally intensive numerical techniques.

Iron line spectroscopy with Einstein-dilaton-Gauss-Bonnet black holes

NASA Astrophysics Data System (ADS)

Nampalliwar, Sourabh; Bambi, Cosimo; Kokkotas, Kostas D.; Konoplya, Roman A.

2018-06-01

Einstein-dilaton-Gauss-Bonnet gravity is a well-motivated alternative theory of gravity that emerges naturally from string theory. While black hole solutions have been known in this theory in numerical form for a while, an approximate analytical metric was obtained recently by some of us, which allows for faster and more detailed analysis. Here we test the accuracy of the analytical metric in the context of X-ray reflection spectroscopy. We analyze innermost stable circular orbits (ISCO) and relativistically broadened iron lines and find that both the ISCO and iron lines are determined sufficiently accurately up to the limit of the approximation. We also find that, though the ISCO increases by about 7% as dilaton coupling increases from zero to extremal values, the redshift at ISCO changes by less than 1%. Consequently, the shape of the iron line is much less sensitive to the dilaton charge than expected.

Front dynamics and entanglement in the XXZ chain with a gradient

NASA Astrophysics Data System (ADS)

Eisler, Viktor; Bauernfeind, Daniel

2017-11-01

We consider the XXZ spin chain with a magnetic field gradient and study the profiles of the magnetization as well as the entanglement entropy. For a slowly varying field, it is shown that, by means of a local density approximation, the ground-state magnetization profile can be obtained with standard Bethe ansatz techniques. Furthermore, it is argued that the low-energy description of the theory is given by a Luttinger liquid with slowly varying parameters. This allows us to obtain a very good approximation of the entanglement profile using a recently introduced technique of conformal field theory in curved spacetime. Finally, the front dynamics is also studied after the gradient field has been switched off, following arguments of generalized hydrodynamics for integrable systems. While for the XX chain the hydrodynamic solution can be found analytically, the XXZ case appears to be more complicated and the magnetization profiles are recovered only around the edge of the front via an approximate numerical solution.

The Dynamics of Current Carriers In Standing Alfven Waves

NASA Astrophysics Data System (ADS)

Wright, A. N.; Allan, W.; Ruderman, M. S.; Elphic, R. C.

The acceleration of current carriers in an Alfvén wave current system is considered. The model incorporates a dipole magnetic field geometry, and we present an analyt- ical solution of the two-fluid equations by successive approximations. The leading solution corresponds to the familiar single-fluid toroidal oscillations. The next order describes the nonlinear dynamics of electrons responsible for carrying a few µAm-2 field aligned current into the ionosphere. The solution shows how most of the elec- tron acceleration in the magnetosphere occurs within 1 RE of the ionosphere, and that a parallel electric field of the order of 1 mVm-1 is reponsible for energising the electrons to 1 keV. The limitations of the electron fluid approximation are considered, and a qualitative solution including electron beams and a modified E is developed in accord with observations. We find that the electron acceleration can be nonlinear, (ve )ve > ve , as a result of our nonuniform equilibrium field geometry even when ve is less than the Alfvén speed. Our calculation also elucidates the processes through which E is generated and supported.

A hybrid perturbation-Galerkin technique for partial differential equations

NASA Technical Reports Server (NTRS)

Geer, James F.; Anderson, Carl M.

1990-01-01

A two-step hybrid perturbation-Galerkin technique for improving the usefulness of perturbation solutions to partial differential equations which contain a parameter is presented and discussed. In the first step of the method, the leading terms in the asymptotic expansion(s) of the solution about one or more values of the perturbation parameter are obtained using standard perturbation methods. In the second step, the perturbation functions obtained in the first step are used as trial functions in a Bubnov-Galerkin approximation. This semi-analytical, semi-numerical hybrid technique appears to overcome some of the drawbacks of the perturbation and Galerkin methods when they are applied by themselves, while combining some of the good features of each. The technique is illustrated first by a simple example. It is then applied to the problem of determining the flow of a slightly compressible fluid past a circular cylinder and to the problem of determining the shape of a free surface due to a sink above the surface. Solutions obtained by the hybrid method are compared with other approximate solutions, and its possible application to certain problems associated with domain decomposition is discussed.

Application of an Extended Parabolic Equation to the Calculation of the Mean Field and the Transverse and Longitudinal Mutual Coherence Functions Within Atmospheric Turbulence

NASA Technical Reports Server (NTRS)

Manning, Robert M.

2005-01-01

Solutions are derived for the generalized mutual coherence function (MCF), i.e., the second order moment, of a random wave field propagating through a random medium within the context of the extended parabolic equation. Here, "generalized" connotes the consideration of both the transverse as well as the longitudinal second order moments (with respect to the direction of propagation). Such solutions will afford a comparison between the results of the parabolic equation within the pararaxial approximation and those of the wide-angle extended theory. To this end, a statistical operator method is developed which gives a general equation for an arbitrary spatial statistical moment of the wave field. The generality of the operator method allows one to obtain an expression for the second order field moment in the direction longitudinal to the direction of propagation. Analytical solutions to these equations are derived for the Kolmogorov and Tatarskii spectra of atmospheric permittivity fluctuations within the Markov approximation.

A reduced-order model from high-dimensional frictional hysteresis

PubMed Central

Biswas, Saurabh; Chatterjee, Anindya

2014-01-01

Hysteresis in material behaviour includes both signum nonlinearities as well as high dimensionality. Available models for component-level hysteretic behaviour are empirical. Here, we derive a low-order model for rate-independent hysteresis from a high-dimensional massless frictional system. The original system, being given in terms of signs of velocities, is first solved incrementally using a linear complementarity problem formulation. From this numerical solution, to develop a reduced-order model, basis vectors are chosen using the singular value decomposition. The slip direction in generalized coordinates is identified as the minimizer of a dissipation-related function. That function includes terms for frictional dissipation through signum nonlinearities at many friction sites. Luckily, it allows a convenient analytical approximation. Upon solution of the approximated minimization problem, the slip direction is found. A final evolution equation for a few states is then obtained that gives a good match with the full solution. The model obtained here may lead to new insights into hysteresis as well as better empirical modelling thereof. PMID:24910522

A simple closed-form solution for assessing concentration uncertainty

NASA Astrophysics Data System (ADS)

de Barros, F. P. J.; Fiori, Aldo; Bellin, Alberto

2011-12-01

We propose closed-form approximate solutions for the moments of a nonreactive tracer that can be used in applications, such as risk analysis. This is in line with the tenet that analytical solutions provide useful information, with minimum cost, during initial site characterization efforts and can serve as a preliminary screening tool when used with prior knowledge. We show that with the help of a few assumptions, the first-order solutions of the concentration moments proposed by Fiori and Dagan (2000) can be further simplified to assume a form similar to well-known deterministic solutions, therefore facilitating their use in applications. A highly anisotropic formation is assumed, and we neglect the transverse components of the two-particle correlation trajectory. The proposed solution compares well with the work of Fiori and Dagan while presenting the same simplicity of use of existing solutions for homogeneous porous media.

Limiting cases of the small-angle scattering approximation solutions for the propagation of laser beams in anisotropic scattering media

NASA Technical Reports Server (NTRS)

Box, M. A.; Deepak, A.

1981-01-01

The propagation of photons in a medium with strongly anisotropic scattering is a problem with a considerable history. Like the propagation of electrons in metal foils, it may be solved in the small-angle scattering approximation by the use of Fourier-transform techniques. In certain limiting cases, one may even obtain analytic expressions. This paper presents some of these results in a model-independent form and also illustrates them by the use of four different phase-function models. Sample calculations are provided for comparison purposes

Analytic descriptions of stochastic bistable systems under force ramp

DOE PAGES

Friddle, Raymond W.

2016-05-13

Solving the two-state master equation with time-dependent rates, the ubiquitous driven bistable system, is a long-standing problem that does not permit a complete solution for all driving rates. We show an accurate approximation to this problem by considering the system in the control parameter regime. Moreover, the results are immediately applicable to a diverse range of bistable systems including single-molecule mechanics.

Low-frequency vibrations of a cylindrical shell rotating on rollers

NASA Astrophysics Data System (ADS)

Filippov, S. B.

2018-05-01

Small free low-frequency vibrations of a rotating closed cylindrical shell which is in a contact with rigid cylindrical rollers are considered. Assumptions of semi-momentless shell theory are used. By means of the expansion of solutions in truncated Fourier series in circumference coordinate the system of the algebraic equations for the approximate calculation of the vibration frequencies and the mode shapes is obtained. The algorithm for the evaluation of frequencies and vibration modes based on analytical solution is developed. In particular, the lowest frequencies of thin cylindrical shell, representing greatest interest for applications, were found. Approximate results are compared with results of numerical calculations carried out by the Finite Elements Analysis. It is shown that the semi-momentless theory can be used for the evaluation of the low frequencies of a cylindrical shell rotating on rollers.

Transport coefficients in ultrarelativistic kinetic theory

NASA Astrophysics Data System (ADS)

Ambruş, Victor E.

2018-02-01

A spatially periodic longitudinal wave is considered in relativistic dissipative hydrodynamics. At sufficiently small wave amplitudes, an analytic solution is obtained in the linearized limit of the macroscopic conservation equations within the first- and second-order relativistic hydrodynamics formulations. A kinetic solver is used to obtain the numerical solution of the relativistic Boltzmann equation for massless particles in the Anderson-Witting approximation for the collision term. It is found that, at small values of the Anderson-Witting relaxation time τ , the transport coefficients emerging from the relativistic Boltzmann equation agree with those predicted through the Chapman-Enskog procedure, while the relaxation times of the heat flux and shear pressure are equal to τ . These claims are further strengthened by considering a moment-type approximation based on orthogonal polynomials under which the Chapman-Enskog results for the transport coefficients are exactly recovered.

On the dynamics of approximating schemes for dissipative nonlinear equations

NASA Technical Reports Server (NTRS)

Jones, Donald A.

1993-01-01

Since one can rarely write down the analytical solutions to nonlinear dissipative partial differential equations (PDE's), it is important to understand whether, and in what sense, the behavior of approximating schemes to these equations reflects the true dynamics of the original equations. Further, because standard error estimates between approximations of the true solutions coming from spectral methods - finite difference or finite element schemes, for example - and the exact solutions grow exponentially in time, this analysis provides little value in understanding the infinite time behavior of a given approximating scheme. The notion of the global attractor has been useful in quantifying the infinite time behavior of dissipative PDEs, such as the Navier-Stokes equations. Loosely speaking, the global attractor is all that remains of a sufficiently large bounded set in phase space mapped infinitely forward in time under the evolution of the PDE. Though the attractor has been shown to have some nice properties - it is compact, connected, and finite dimensional, for example - it is in general quite complicated. Nevertheless, the global attractor gives a way to understand how the infinite time behavior of approximating schemes such as the ones coming from a finite difference, finite element, or spectral method relates to that of the original PDE. Indeed, one can often show that such approximations also have a global attractor. We therefore only need to understand how the structure of the attractor for the PDE behaves under approximation. This is by no means a trivial task. Several interesting results have been obtained in this direction. However, we will not go into the details. We mention here that approximations generally lose information about the system no matter how accurate they are. There are examples that show certain parts of the attractor may be lost by arbitrary small perturbations of the original equations.

An Efficient Algorithm for Perturbed Orbit Integration Combining Analytical Continuation and Modified Chebyshev Picard Iteration

NASA Astrophysics Data System (ADS)

Elgohary, T.; Kim, D.; Turner, J.; Junkins, J.

2014-09-01

Several methods exist for integrating the motion in high order gravity fields. Some recent methods use an approximate starting orbit, and an efficient method is needed for generating warm starts that account for specific low order gravity approximations. By introducing two scalar Lagrange-like invariants and employing Leibniz product rule, the perturbed motion is integrated by a novel recursive formulation. The Lagrange-like invariants allow exact arbitrary order time derivatives. Restricting attention to the perturbations due to the zonal harmonics J2 through J6, we illustrate an idea. The recursively generated vector-valued time derivatives for the trajectory are used to develop a continuation series-based solution for propagating position and velocity. Numerical comparisons indicate performance improvements of ~ 70X over existing explicit Runge-Kutta methods while maintaining mm accuracy for the orbit predictions. The Modified Chebyshev Picard Iteration (MCPI) is an iterative path approximation method to solve nonlinear ordinary differential equations. The MCPI utilizes Picard iteration with orthogonal Chebyshev polynomial basis functions to recursively update the states. The key advantages of the MCPI are as follows: 1) Large segments of a trajectory can be approximated by evaluating the forcing function at multiple nodes along the current approximation during each iteration. 2) It can readily handle general gravity perturbations as well as non-conservative forces. 3) Parallel applications are possible. The Picard sequence converges to the solution over large time intervals when the forces are continuous and differentiable. According to the accuracy of the starting solutions, however, the MCPI may require significant number of iterations and function evaluations compared to other integrators. In this work, we provide an efficient methodology to establish good starting solutions from the continuation series method; this warm start improves the performance of the MCPI significantly and will likely be useful for other applications where efficiently computed approximate orbit solutions are needed.

Amine-capped ZnS-Mn2+ nanocrystals for fluorescence detection of trace TNT explosive.

PubMed

Tu, Renyong; Liu, Bianhua; Wang, Zhenyang; Gao, Daming; Wang, Feng; Fang, Qunling; Zhang, Zhongping

2008-05-01

Mn2+-doped ZnS nanocrystals with an amine-capping layer have been synthesized and used for the fluorescence detection of ultratrace 2,4,6-trinitrotoluene (TNT) by quenching the strong orange Mn2+ photoluminescence. The organic amine-capped nanocrystals can bind TNT species from solution and atmosphere by the acid-base pairing interaction between electron-rich amino ligands and electron-deficient aromatic rings. The resultant TNT anions bound onto the amino monolayer can efficiently quench the Mn2+ photoluminescence through the electron transfer from the conductive band of ZnS to the lowest unoccupied molecular orbital (LUMO) of TNT anions. The amino ligands provide an amplified response to the binding events of nitroaromatic compounds by the 2- to approximately 5-fold increase in quenching constants. Moreover, a large difference in quenching efficiency was observed for different types of nitroaromatic analytes, dependent on the affinity of nitro analytes to the amino monolayer and their electron-accepting abilities. The amine-capped nanocrystals can sensitively detect down to 1 nM TNT in solution or several parts-per-billion of TNT vapor in atmosphere. The ion-doped nanocrystal sensors reported here show a remarkable air/solution stability, high quantum yield, and strong analyte affinity and, therefore, are well-suited for detecting the ultratrace TNT and distinguishing different nitro compounds.

Collisionless kinetic theory of oblique tearing instabilities

DOE PAGES

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

2018-02-15

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. In this paper, we find that this stabilization is associated with the density-gradient-driven diamagnetic drift. Themore » analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. Finally, a simple analytic estimate for the stability criterion is provided.« less

Collisionless kinetic theory of oblique tearing instabilities

DOE Office of Scientific and Technical Information (OSTI.GOV)

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. In this paper, we find that this stabilization is associated with the density-gradient-driven diamagnetic drift. Themore » analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. Finally, a simple analytic estimate for the stability criterion is provided.« less

Collisionless kinetic theory of oblique tearing instabilities

NASA Astrophysics Data System (ADS)

Baalrud, S. D.; Bhattacharjee, A.; Daughton, W.

2018-02-01

The linear dispersion relation for collisionless kinetic tearing instabilities is calculated for the Harris equilibrium. In contrast to the conventional 2D geometry, which considers only modes at the center of the current sheet, modes can span the current sheet in 3D. Modes at each resonant surface have a unique angle with respect to the guide field direction. Both kinetic simulations and numerical eigenmode solutions of the linearized Vlasov-Maxwell equations have recently revealed that standard analytic theories vastly overestimate the growth rate of oblique modes. We find that this stabilization is associated with the density-gradient-driven diamagnetic drift. The analytic theories miss this drift stabilization because the inner tearing layer broadens at oblique angles sufficiently far that the assumption of scale separation between the inner and outer regions of boundary-layer theory breaks down. The dispersion relation obtained by numerically solving a single second order differential equation is found to approximately capture the drift stabilization predicted by solutions of the full integro-differential eigenvalue problem. A simple analytic estimate for the stability criterion is provided.

Analytical study of magnetohydrodynamic propulsion stability

NASA Astrophysics Data System (ADS)

Abdollahzadeh Jamalabadi, M. Y.

2014-09-01

In this paper an analytical solution for the stability of the fully developed flow drive in a magneto-hydro-dynamic pump with pulsating transverse Eletro-magnetic fields is presented. To do this, a theoretical model of the flow is developed and the analytical results are obtained for both the cylindrical and Cartesian configurations that are proper to use in the propulsion of marine vessels. The governing parabolic momentum PDEs are transformed into an ordinary differential equation using approximate velocity distribution. The numerical results are obtained and asymptotic analyses are built to discover the mathematical behavior of the solutions. The maximum velocity in a magneto-hydro-dynamic pump versus time for various values of the Stuart number, electro-magnetic interaction number, Reynolds number, aspect ratio, as well as the magnetic and electrical angular frequency and the shift of the phase angle is presented. Results show that for a high Stuart number there is a frequency limit for stability of the fluid flow in a certain direction of the flow. This stability frequency is dependent on the geometric parameters of a channel.

Sound Emission of Rotor Induced Deformations of Generator Casings

NASA Technical Reports Server (NTRS)

Polifke, W.; Mueller, B.; Yee, H. C.; Mansour, Nagi (Technical Monitor)

2001-01-01

The casing of large electrical generators can be deformed slightly by the rotor's magnetic field. The sound emission produced by these periodic deformations, which could possibly exceed guaranteed noise emission limits, is analysed analytically and numerically. From the deformation of the casing, the normal velocity of the generator's surface is computed. Taking into account the corresponding symmetry, an analytical solution for the acoustic pressure outside the generator is round in terms of the Hankel function of second order. The normal velocity or the generator surface provides the required boundary condition for the acoustic pressure and determines the magnitude of pressure oscillations. For the numerical simulation, the nonlinear 2D Euler equations are formulated In a perturbation form for low Mach number Computational Aeroacoustics (CAA). The spatial derivatives are discretized by the classical sixth-order central interior scheme and a third-order boundary scheme. Spurious high frequency oscillations are damped by a characteristic-based artificial compression method (ACM) filter. The time derivatives are approximated by the classical 4th-order Runge-Kutta method. The numerical results are In excellent agreement with the analytical solution.

Electromagnetic jets from stars and black holes

NASA Astrophysics Data System (ADS)

Gralla, Samuel E.; Lupsasca, Alexandru; Rodriguez, Maria J.

2016-02-01

We present analytic force-free solutions modeling rotating stars and black holes immersed in the magnetic field of a thin disk that terminates at an inner radius. The solutions are exact in flat spacetime and approximate in Kerr spacetime. The compact object produces a conical jet whose properties carry information about its nature. For example, the jet from a star is surrounded by a current sheet, while that of a black hole is smooth. We compute an effective resistance in each case and compare to the canonical values used in circuit models of energy extraction. These solutions illustrate all of the basic features of the Blandford-Znajek process for energy extraction and jet formation in a clean setting.

Complete characterization of the spasing (L-L) curve of a three-level quantum coherence enhanced spaser for design optimization

NASA Astrophysics Data System (ADS)

Kumarapperuma, Lakshitha; Premaratne, Malin; Jha, Pankaj K.; Stockman, Mark I.; Agrawal, Govind P.

2018-05-01

We demonstrate that it is possible to derive an approximate analytical expression to characterize the spasing (L-L) curve of a coherently enhanced spaser with 3-level gain-medium chromophores. The utility of this solution stems from the fact that it enables optimization of the large parameter space associated with spaser designing, a functionality not offered by the methods currently available in the literature. This is vital for the advancement of spaser technology towards the level of device realization. Owing to the compact nature of the analytical expressions, our solution also facilitates the grouping and identification of key processes responsible for the spasing action, whilst providing significant physical insights. Furthermore, we show that our expression generates results within 0.1% error compared to numerically obtained results for pumping rates higher than the spasing threshold, thereby drastically reducing the computational cost associated with spaser designing.

Peristalsis of nonconstant viscosity Jeffrey fluid with nanoparticles

NASA Astrophysics Data System (ADS)

Alvi, N.; Latif, T.; Hussain, Q.; Asghar, S.

Mixed convective peristaltic activity of variable viscosity nanofluids is addressed. Unlike the conventional consideration of constant viscosity; the viscosity is taken as temperature dependent. Constitutive relations for linear viscoelastic Jeffrey fluid are employed and uniform magnetic field is applied in the transverse direction. For nanofluids, the formulation is completed in presence of Brownian motion, thermophoresis, viscous dissipation and Joule heating. Consideration of temperature dependence of viscosity is not a choice but the realistic requirement of the wall temperature and the heat generated due to the viscous dissipation. Well established large wavelength and small Reynolds number approximations are invoked. Non-linear coupled system is analytically solved for the convergent series solutions identifying the interval of convergence explicitly. A comparative study between analytical and numerical solution is made for certainty. Influence of the parameters undertaken for the description of the problem is pointed out and its physics explained.

Nonequilibrium mechanisms of weak electrolyte electrification under the action of constant voltage

NASA Astrophysics Data System (ADS)

Stishkov, Yu. K.; Chirkov, V. A.

2016-07-01

The formation of space charge in weak electrolytes, specifically in liquid dielectrics, has been considered. An analytical solution is given to a simplified set of Nernst-Planck equations that describe the formation of nonequilibrium recombination layers in weak electrolytes. This approximate analytical solution is compared with computer simulation data for a complete set of Poisson-Nernst-Planck equations. It has been shown that the current passage in weak electrolytes can be described by a single dimensionless parameter that equals the length of a near-electrode recombination layer divided by the width of the interelectrode gap. The formation mechanism and the structure of charged nonequilibrium near-electrode layers in the nonstationary regime have been analyzed for different injection-to-conduction current ratios. It has been found that almost all charge structures encountered in weak dielectrics can be accounted for by the nonequilibrium dissociation-recombination mechanism of space charge formation.

Numerical analysis of two-fluid tearing mode instability in a finite aspect ratio cylinder

NASA Astrophysics Data System (ADS)

Ito, Atsushi; Ramos, Jesús J.

2018-01-01

The two-fluid resistive tearing mode instability in a periodic plasma cylinder of finite aspect ratio is investigated numerically for parameters such that the cylindrical aspect ratio and two-fluid effects are of order unity, hence the real and imaginary parts of the mode eigenfunctions and growth rate are comparable. Considering a force-free equilibrium, numerical solutions of the complete eigenmode equations for general aspect ratios and ion skin depths are compared and found to be in very good agreement with the corresponding analytic solutions derived by means of the boundary layer theory [A. Ito and J. J. Ramos, Phys. Plasmas 24, 072102 (2017)]. Scaling laws for the growth rate and the real frequency of the mode are derived from the analytic dispersion relation by using Taylor expansions and Padé approximations. The cylindrical finite aspect ratio effect is inferred from the scaling law for the real frequency of the mode.

The Hubbard Dimer: A Complete DFT Solution to a Many-Body Problem

NASA Astrophysics Data System (ADS)

Smith, Justin; Carrascal, Diego; Ferrer, Jaime; Burke, Kieron

2015-03-01

In this work we explain the relationship between density functional theory and strongly correlated models using the simplest possible example, the two-site asymmetric Hubbard model. We discuss the connection between the lattice and real-space and how this is a simple model for stretched H2. We can solve this elementary example analytically, and with that we can illuminate the underlying logic and aims of DFT. While the many-body solution is analytic, the density functional is given only implicitly. We overcome this difficulty by creating a highly accurate parameterization of the exact function. We use this parameterization to perform benchmark calculations of correlation kinetic energy, the adiabatic connection, etc. We also test Hartree-Fock and the Bethe Ansatz Local Density Approximation. We also discuss and illustrate the derivative discontinuity in the exchange-correlation energy and the infamous gap problem in DFT. DGE-1321846, DE-FG02-08ER46496.

Characterization of gold nanoparticles with different hydrophilic coatings via capillary electrophoresis and Taylor dispersion analysis. Part I: determination of the zeta potential employing a modified analytic approximation.

PubMed

Pyell, Ute; Jalil, Alaa H; Pfeiffer, Christian; Pelaz, Beatriz; Parak, Wolfgang J

2015-07-15

Taking gold nanoparticles with different hydrophilic coatings as an example, it is investigated whether capillary electrophoresis in combination with Taylor dispersion analysis allows for the precise determination of mean electrophoretic mobilities, electrophoretic mobility distributions, and zeta potentials in a matrix of exactly known composition and the calibration-free determination of number-weighted mean hydrodynamic radii. Our experimental data confirm that the calculation of the zeta potential for colloidal nanoparticles with ζ>25 mV requires to take the relaxation effect into account. Because of the requirement to avoid particle-wall interactions, a solution of disodiumtetraborate decahydrate (borax) in deionized water had been selected as suitable electrolyte. Measurements of the electrophoretic mobility at different ionic strength and application of the analytic approximation developed by Ohshima show that in the present case of a buffered solution with a weak electrolyte co-ion and a strong electrolyte counterion, the effective ionic drag coefficient should be approximated with the ionic drag coefficient of the counterion. The obtained results are in good agreement with theoretical expectations regarding the dependence of the zeta potential and the electrokinetic surface charge density on the ionic strength. We also show that Taylor dispersion analysis (besides estimation of the number-weighted mean hydrodynamic radius) provides additional information on the type and width of the number-weighted particle distribution. Copyright © 2015 Elsevier Inc. All rights reserved.

Modeling rainfall infiltration on hillslopes using Flux-concentration relation and time compression approximation

NASA Astrophysics Data System (ADS)

Wang, Jie; Chen, Li; Yu, Zhongbo

2018-02-01

Rainfall infiltration on hillslopes is an important issue in hydrology, which is related to many environmental problems, such as flood, soil erosion, and nutrient and contaminant transport. This study aimed to improve the quantification of infiltration on hillslopes under both steady and unsteady rainfalls. Starting from Darcy's law, an analytical integral infiltrability equation was derived for hillslope infiltration by use of the flux-concentration relation. Based on this equation, a simple scaling relation linking the infiltration times on hillslopes and horizontal planes was obtained which is applicable for both small and large times and can be used to simplify the solution procedure of hillslope infiltration. The infiltrability equation also improved the estimation of ponding time for infiltration under rainfall conditions. For infiltration after ponding, the time compression approximation (TCA) was applied together with the infiltrability equation. To improve the computational efficiency, the analytical integral infiltrability equation was approximated with a two-term power-like function by nonlinear regression. Procedures of applying this approach to both steady and unsteady rainfall conditions were proposed. To evaluate the performance of the new approach, it was compared with the Green-Ampt model for sloping surfaces by Chen and Young (2006) and Richards' equation. The proposed model outperformed the sloping Green-Ampt, and both ponding time and infiltration predictions agreed well with the solutions of Richards' equation for various soil textures, slope angles, initial water contents, and rainfall intensities for both steady and unsteady rainfalls.

A non-grey analytical model for irradiated atmospheres. II. Analytical vs. numerical solutions

NASA Astrophysics Data System (ADS)

Parmentier, Vivien; Guillot, Tristan; Fortney, Jonathan J.; Marley, Mark S.

2015-02-01

Context. The recent discovery and characterization of the diversity of the atmospheres of exoplanets and brown dwarfs calls for the development of fast and accurate analytical models. Aims: We wish to assess the goodness of the different approximations used to solve the radiative transfer problem in irradiated atmospheres analytically, and we aim to provide a useful tool for a fast computation of analytical temperature profiles that remains correct over a wide range of atmospheric characteristics. Methods: We quantify the accuracy of the analytical solution derived in paper I for an irradiated, non-grey atmosphere by comparing it to a state-of-the-art radiative transfer model. Then, using a grid of numerical models, we calibrate the different coefficients of our analytical model for irradiated solar-composition atmospheres of giant exoplanets and brown dwarfs. Results: We show that the so-called Eddington approximation used to solve the angular dependency of the radiation field leads to relative errors of up to ~5% on the temperature profile. For grey or semi-grey atmospheres (i.e., when the visible and thermal opacities, respectively, can be considered independent of wavelength), we show that the presence of a convective zone has a limited effect on the radiative atmosphere above it and leads to modifications of the radiative temperature profile of approximately ~2%. However, for realistic non-grey planetary atmospheres, the presence of a convective zone that extends to optical depths smaller than unity can lead to changes in the radiative temperature profile on the order of 20% or more. When the convective zone is located at deeper levels (such as for strongly irradiated hot Jupiters), its effect on the radiative atmosphere is again on the same order (~2%) as in the semi-grey case. We show that the temperature inversion induced by a strong absorber in the optical, such as TiO or VO is mainly due to non-grey thermal effects reducing the ability of the upper atmosphere to cool down rather than an enhanced absorption of the stellar light as previously thought. Finally, we provide a functional form for the coefficients of our analytical model for solar-composition giant exoplanets and brown dwarfs. This leads to fully analytical pressure-temperature profiles for irradiated atmospheres with a relative accuracy better than 10% for gravities between 2.5 m s-2 and 250 m s-2 and effective temperatures between 100 K and 3000 K. This is a great improvement over the commonly used Eddington boundary condition. A FORTRAN implementation of the analytical model is only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (ftp://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/574/A35 or at http://www.oca.eu/parmentier/nongrey.Appendix A is available in electronic form at http://www.aanda.org

A combined analytical and numerical analysis of the flow-acoustic coupling in a cavity-pipe system

NASA Astrophysics Data System (ADS)

Langthjem, Mikael A.; Nakano, Masami

2018-05-01

The generation of sound by flow through a closed, cylindrical cavity (expansion chamber) accommodated with a long tailpipe is investigated analytically and numerically. The sound generation is due to self-sustained flow oscillations in the cavity. These oscillations may, in turn, generate standing (resonant) acoustic waves in the tailpipe. The main interest of the paper is in the interaction between these two sound sources. An analytical, approximate solution of the acoustic part of the problem is obtained via the method of matched asymptotic expansions. The sound-generating flow is represented by a discrete vortex method, based on axisymmetric vortex rings. It is demonstrated through numerical examples that inclusion of acoustic feedback from the tailpipe is essential for a good representation of the sound characteristics.

Convective fluid flows in a horizontal channel with evaporation: analytical and experimental investigations

NASA Astrophysics Data System (ADS)

Lyulin, Y. V.; Rezanova, E. V.

2017-11-01

Heat- and mass transfer processes in a two-layer system of the liquid and gas are studied with respect to evaporation at interface. The stationary convective flows of two immiscible viscous incompressible fluids filling an infinite channel and being under action of the transverse gravitation field are studied analytically. Mathematical modeling of the flows is carried out with the help of the Navier-Stokes equations in Boussinesq approximation. The Dufour and Soret effects are taken into consideration in the gas-vapor phase. In the two-dimensional case the exact solutions of special type are constructed under condition of a given specific gas flow rate. Comparison of the analytical results with results of the physical experiments with the “liquid-gas” system like “ethanol-air” are presented.

Development of models of the magnetorheological fluid damper

NASA Astrophysics Data System (ADS)

Kazakov, Yu. B.; Morozov, N. A.; Nesterov, S. A.

2017-06-01

The algorithm for analytical calculation of a power characteristic of magnetorheological (MR) dampers taking into account the rheological properties of MR fluid is considered. The nonlinear magnetorheological characteristics are represented by piecewise linear approximation to MR fluid areas with different viscosities. The extended calculated power characteristics of a MR damper are received and they coincide with actual results. The finite element model of a MR damper is developed; it allows carrying out the analysis of a MR damper taking into account the mutual influence of electromagnetic, hydrodynamic and thermal fields. The results of finite element simulation coincide with analytical solutions that allows using them for design development of a MR damper.

An Approximate Solution to the Plastic Indentation of Circular Sandwich Panels

NASA Astrophysics Data System (ADS)

Xie, Z.

2018-05-01

The plastic indentation response of circular sandwich panels loaded by the flat end of a cylinder is investigated employing a velocity field model. Using the principles of virtual velocities and minimum work, an expression for the indenter load in relation to the indenter displacement and displacement field of the deformed face sheet is derived. The analytical solutions obtained are in good agreement with those found by simulations using the ABAQUS code. The radial tensile strain of the deformed face sheet and the ratio of energy absorption rate of the core to that of the face sheet are discussed.

On the singular perturbations for fractional differential equation.

PubMed

Atangana, Abdon

2014-01-01

The goal of this paper is to examine the possible extension of the singular perturbation differential equation to the concept of fractional order derivative. To achieve this, we presented a review of the concept of fractional calculus. We make use of the Laplace transform operator to derive exact solution of singular perturbation fractional linear differential equations. We make use of the methodology of three analytical methods to present exact and approximate solution of the singular perturbation fractional, nonlinear, nonhomogeneous differential equation. These methods are including the regular perturbation method, the new development of the variational iteration method, and the homotopy decomposition method.

Contact problem on indentation of an elastic half-plane with an inhomogeneous coating by a flat punch in the presence of tangential stresses on a surface

NASA Astrophysics Data System (ADS)

Volkov, Sergei S.; Vasiliev, Andrey S.; Aizikovich, Sergei M.; Sadyrin, Evgeniy V.

2018-05-01

Indentation of an elastic half-space with functionally graded coating by a rigid flat punch is studied. The half-plane is additionally subjected to distributed tangential stresses. Tangential stresses are represented in a form of Fourier series. The problem is reduced to the solution of two dual integral equations over even and odd functions describing distribution of unknown normal contact stresses. The solutions of these dual integral equations are constructed by the bilateral asymptotic method. Approximated analytical expressions for contact normal stresses are provided.

Exact solution for the layered convection of a viscous incompressible fluid at specified temperature gradients and tangential forces on the free boundary

NASA Astrophysics Data System (ADS)

Burmasheva, N. V.; Prosviryakov, E. Yu.

2017-12-01

A new exact analytical solution of a system of thermal convection equations in the Boussinesq approximation describing layered flows in an incompressible viscous fluid is obtained. A fluid flow in an infinite layer is considered. Convection in the fluid is induced by tangential stresses specified on the upper non-deformable boundary. At the fixed lower boundary, the no-slip condition is satisfied. Temperature corrections are given on the both boundaries of the fluid layer. The possibility of physical field stratification is investigated.

Transport toward a well in highly heterogeneous aquifer

NASA Astrophysics Data System (ADS)

Di Dato, Mariaines; de Barros, Felipe P. J.; Bellin, Alberto; Fiori, Aldo

2017-04-01

Solute transport toward a well is a challenging subject in subsurface hydrology since the complexity of the mathematical model is tremendously increased by the non-uniformity of the mean flow and heterogeneity of the formation. Up to date, analytical solutions for such flow configurations are limited to low heterogeneous conditions. On the other hand, numerical simulations in 3D highly heterogeneous formations are computationally expensive and plagued by numerical errors. In this work we propose an analytical solution for the Breakthrough Curve (BTC) at the well for an instantaneous linear injection across the aquifer's thickness for any degree of heterogeneity of the porous medium. Our solution makes use of the Multi Indicator Model-Self Consistent Approximation (MIMSCA), by which the aquifer is conceptualized as an ensemble of blocks of constant hydraulic conductivity K randomly drawn from a lognormal distribution. In order to apply MIMSCA, we assume the flow as locally uniform, given that K is uniform within the block. With this approximation, the travel time to the well is equal to the superposition of the time spent by the solute particle within each block. We emphasize that, despite the approximations introduced, the model is able to reproduce the laboratory experiment of [1] without the need to fit any transport parameters. In this work, we present results for two different injection modes: a resident injection (e.g., residual DNAPL) and a flux proportional injection (e.g., leakage from a passive well). The proposed methodology allows to quantify the BTC at the well as a function of few parameters such as the injection mode and the statistical structure of the aquifer (geometric mean, variance and integral scale of the hydraulic conductivity field). Results illustrate that the release condition has a strong impact on the shape of the BTC. Furthermore, the difference between different injection modes increases with the heterogeneity of the K-field. The importance of the both injection mode and heterogeneity degree are also elucidated on the early and late solute arrival times at the well. Finally, we show how travel times become ergodic only for very thick aquifers, even in case of mild heterogeneity. We emphasize that the present framework has a practical validity, giving an affordable, although approximated, first estimation of mass arrival at an extraction well. References [1] Fernàndez-Garcia, D., T. H. Illangasekare, and H. Rajaram (2004), Conservative and sorptive forced-gradient and uniform flow tracer tests in a three-dimensional laboratory test aquifer, Water Resour. Res., 40, W10103, doi:10.1029/2004WR003112.

Analytical model for the radio-frequency sheath

NASA Astrophysics Data System (ADS)

Czarnetzki, Uwe

2013-12-01

A simple analytical model for the planar radio-frequency (rf) sheath in capacitive discharges is developed that is based on the assumptions of a step profile for the electron front, charge exchange collisions with constant cross sections, negligible ionization within the sheath, and negligible ion dynamics. The continuity, momentum conservation, and Poisson equations are combined in a single integro-differential equation for the square of the ion drift velocity, the so called sheath equation. Starting from the kinetic Boltzmann equation, special attention is paid to the derivation and the validity of the approximate fluid equation for momentum balance. The integrals in the sheath equation appear in the screening function which considers the relative contribution of the temporal mean of the electron density to the space charge in the sheath. It is shown that the screening function is quite insensitive to variations of the effective sheath parameters. The two parameters defining the solution are the ratios of the maximum sheath extension to the ion mean free path and the Debye length, respectively. A simple general analytic expression for the screening function is introduced. By means of this expression approximate analytical solutions are obtained for the collisionless as well as the highly collisional case that compare well with the exact numerical solution. A simple transition formula allows application to all degrees of collisionality. In addition, the solutions are used to calculate all static and dynamic quantities of the sheath, e.g., the ion density, fields, and currents. Further, the rf Child-Langmuir laws for the collisionless as well as the collisional case are derived. An essential part of the model is the a priori knowledge of the wave form of the sheath voltage. This wave form is derived on the basis of a cubic charge-voltage relation for individual sheaths, considering both sheaths and the self-consistent self-bias in a discharge with arbitrary symmetry. The externally applied rf voltage is assumed to be sinusoidal, although the model can be extended to arbitrary wave forms, e.g., for dual-frequency discharges. The model calculates explicitly the cubic correction parameter in the charge-voltage relation for the case of highly asymmetric discharges. It is shown that the cubic correction is generally moderate but more pronounced in the collisionless case. The analytical results are compared to experimental data from the literature obtained by laser electric field measurements of the mean and dynamic fields in the capacitive sheath for various gases and pressures. Very good agreement is found throughout.

Analytical model for the radio-frequency sheath.

PubMed

Czarnetzki, Uwe

2013-12-01

A simple analytical model for the planar radio-frequency (rf) sheath in capacitive discharges is developed that is based on the assumptions of a step profile for the electron front, charge exchange collisions with constant cross sections, negligible ionization within the sheath, and negligible ion dynamics. The continuity, momentum conservation, and Poisson equations are combined in a single integro-differential equation for the square of the ion drift velocity, the so called sheath equation. Starting from the kinetic Boltzmann equation, special attention is paid to the derivation and the validity of the approximate fluid equation for momentum balance. The integrals in the sheath equation appear in the screening function which considers the relative contribution of the temporal mean of the electron density to the space charge in the sheath. It is shown that the screening function is quite insensitive to variations of the effective sheath parameters. The two parameters defining the solution are the ratios of the maximum sheath extension to the ion mean free path and the Debye length, respectively. A simple general analytic expression for the screening function is introduced. By means of this expression approximate analytical solutions are obtained for the collisionless as well as the highly collisional case that compare well with the exact numerical solution. A simple transition formula allows application to all degrees of collisionality. In addition, the solutions are used to calculate all static and dynamic quantities of the sheath, e.g., the ion density, fields, and currents. Further, the rf Child-Langmuir laws for the collisionless as well as the collisional case are derived. An essential part of the model is the a priori knowledge of the wave form of the sheath voltage. This wave form is derived on the basis of a cubic charge-voltage relation for individual sheaths, considering both sheaths and the self-consistent self-bias in a discharge with arbitrary symmetry. The externally applied rf voltage is assumed to be sinusoidal, although the model can be extended to arbitrary wave forms, e.g., for dual-frequency discharges. The model calculates explicitly the cubic correction parameter in the charge-voltage relation for the case of highly asymmetric discharges. It is shown that the cubic correction is generally moderate but more pronounced in the collisionless case. The analytical results are compared to experimental data from the literature obtained by laser electric field measurements of the mean and dynamic fields in the capacitive sheath for various gases and pressures. Very good agreement is found throughout.

Analytical model of a corona discharge from a conical electrode under saturation

NASA Astrophysics Data System (ADS)

Boltachev, G. Sh.; Zubarev, N. M.

2012-11-01

Exact partial solutions are found for the electric field distribution in the outer region of a stationary unipolar corona discharge from an ideal conical needle in the space-charge-limited current mode with allowance for the electric field dependence of the ion mobility. It is assumed that only the very tip of the cone is responsible for the discharge, i.e., that the ionization zone is a point. The solutions are obtained by joining the spherically symmetric potential distribution in the drift space and the self-similar potential distribution in the space-charge-free region. Such solutions are outside the framework of the conventional Deutsch approximation, according to which the space charge insignificantly influences the shape of equipotential surfaces and electric lines of force. The dependence is derived of the corona discharge saturation current on the apex angle of the conical electrode and applied potential difference. A simple analytical model is suggested that describes drift in the point-plane electrode geometry under saturation as a superposition of two exact solutions for the field potential. In terms of this model, the angular distribution of the current density over the massive plane electrode is derived, which agrees well with Warburg's empirical law.

Injection and swirl driven flowfields in solid and liquid rocket motors

NASA Astrophysics Data System (ADS)

Vyas, Anand B.

In this work, we seek approximate analytical solutions to describe the bulk flow motion in certain types of solid and liquid rocket motors. In the case of an idealized solid rocket motor, a cylindrical double base propellant grain with steady regression rate is considered. The well known inviscid profile determined by Culick is extended here to include the effects of viscosity and steady grain regression. The approximate analytical solution for the cold flow is obtained from similarity principles, perturbation methods and the method of variation of parameters. The velocity, vorticity, pressure gradient and the shear stress distributions are determined and interpreted for different rates of wall regression and injection Reynolds number. The liquid propellant rocket engine considered here is based on a novel design that gives rise to a cyclonic flow. The resulting bidirectional motion is triggered by the tangential injection of an oxidizer just upstream of the chamber nozzle. Velocity, vorticity and pressure gradient distributions are determined for the bulk gas dynamics using a non-reactive inviscid model. Viscous corrections are then incorporated to explain the formation of a forced vortex near the core. Our results compare favorably with numerical simulations and experimental measurements obtained by other researchers. They also indicate that the bidirectional vortex in a cylindrical chamber is a physical solution of the Euler equations. In closing, we investigate the possibility of multi-directional flow behavior as predicted by Euler's equation and as reported recently in laboratory experiments.

Force-Free Magnetic Fields Calculated from Automated Tracing of Coronal Loops with AIA/SDO

NASA Astrophysics Data System (ADS)

Aschwanden, M. J.

2013-12-01

One of the most realistic magnetic field models of the solar corona is a nonlinear force-free field (NLFFF) solution. There exist about a dozen numeric codes that compute NLFFF solutions based on extrapolations of photospheric vector magnetograph data. However, since the photosphere and lower chromosphere is not force-free, a suitable correction has to be applied to the lower boundary condition. Despite of such "pre-processing" corrections, the resulting theoretical magnetic field lines deviate substantially from observed coronal loop geometries. - Here we developed an alternative method that fits an analytical NLFFF approximation to the observed geometry of coronal loops. The 2D coordinates of the geometry of coronal loop structures observed with AIA/SDO are traced with the "Oriented Coronal CUrved Loop Tracing" (OCCULT-2) code, an automated pattern recognition algorithm that has demonstrated the fidelity in loop tracing matching visual perception. A potential magnetic field solution is then derived from a line-of-sight magnetogram observed with HMI/SDO, and an analytical NLFFF approximation is then forward-fitted to the twisted geometry of coronal loops. We demonstrate the performance of this magnetic field modeling method for a number of solar active regions, before and after major flares observed with SDO. The difference of the NLFFF and the potential field energies allows us then to compute the free magnetic energy, which is an upper limit of the energy that is released during a solar flare.

On the Accuracy of Double Scattering Approximation for Atmospheric Polarization Computations

NASA Technical Reports Server (NTRS)

Korkin, Sergey V.; Lyapustin, Alexei I.; Marshak, Alexander L.

2011-01-01

Interpretation of multi-angle spectro-polarimetric data in remote sensing of atmospheric aerosols require fast and accurate methods of solving the vector radiative transfer equation (VRTE). The single and double scattering approximations could provide an analytical framework for the inversion algorithms and are relatively fast, however accuracy assessments of these approximations for the aerosol atmospheres in the atmospheric window channels have been missing. This paper provides such analysis for a vertically homogeneous aerosol atmosphere with weak and strong asymmetry of scattering. In both cases, the double scattering approximation gives a high accuracy result (relative error approximately 0.2%) only for the low optical path - 10(sup -2) As the error rapidly grows with optical thickness, a full VRTE solution is required for the practical remote sensing analysis. It is shown that the scattering anisotropy is not important at low optical thicknesses neither for reflected nor for transmitted polarization components of radiation.

Polynomial solutions of the Monge-Ampère equation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Aminov, Yu A

2014-11-30

The question of the existence of polynomial solutions to the Monge-Ampère equation z{sub xx}z{sub yy}−z{sub xy}{sup 2}=f(x,y) is considered in the case when f(x,y) is a polynomial. It is proved that if f is a polynomial of the second degree, which is positive for all values of its arguments and has a positive squared part, then no polynomial solution exists. On the other hand, a solution which is not polynomial but is analytic in the whole of the x, y-plane is produced. Necessary and sufficient conditions for the existence of polynomial solutions of degree up to 4 are found and methods for the construction ofmore » such solutions are indicated. An approximation theorem is proved. Bibliography: 10 titles.« less

Flow and Transport in Highly Heterogeneous Porous Formations: Numerical Experiments Performed Using the Analytic Element Method

NASA Astrophysics Data System (ADS)

Jankovic, I.

2002-05-01

Flow and transport in porous formations are analyzed using numerical simulations. Hydraulic conductivity is treated as a spatial random function characterized by a probability density function and a two-point covariance function. Simulations are performed for a multi-indicator conductivity structure developed by Gedeon Dagan (personal communication). This conductivity structure contains inhomogeneities (inclusions) of elliptical and ellipsoidal geometry that are embedded in a homogeneous background. By varying the distribution of sizes and conductivities of inclusions, any probability density function and two-point covariance may be reproduced. The multi-indicator structure is selected since it yields simple approximate transport solutions (Aldo Fiori, personal communication) and accurate numerical solutions (based on the Analytic Element Method). The dispersion is examined for two conceptual models. Both models are based on the multi-indicator conductivity structure. The first model is designed to examine dispersion in aquifers with continuously varying conductivity. The inclusions in this model cover as much area/volume of the porous formation as possible. The second model is designed for aquifers that contain clay/sand/gravel lenses embedded in otherwise homogeneous background. The dispersion in both aquifer types is simulated numerically. Simulation results are compared to those obtained using simple approximate solutions. In order to infer transport statistics that are representative of an infinite domain using the numerical experiments, the inclusions are placed in a domain that was shaped as a large ellipse (2D) and a large spheroid (3D) that were submerged in an unbounded homogeneous medium. On a large scale, the large body of inclusions behaves like a single large inhomogeneity. The analytic solution for a uniform flow past the single inhomogeneity of such geometry yields uniform velocity inside the domain. The velocity differs from that at infinity and can be used to infer the effective conductivity of the medium. As many as 100,000 inhomogeneities are placed inside the domain for 2D simulations. Simulations in 3D were limited to 50,000 inclusions. A large number of simulations was conducted on a massively parallel supercomputer cluster at the Center for Computational Research, University at Buffalo. Simulations range from mildly heterogeneous formations to highly heterogeneous formations (variance of the logarithm of conductivity equal to 10) and from sparsely populated systems to systems where inhomogeneities cover 95% of the volume. Particles are released and tracked inside the core of constant mean velocity. Following the particle tracking, various medium, flow, and transport statistics are computed. These include: spatial moments of particle positions, probability density function of hydraulic conductivity and each component of velocity, their two-point covariance function in the direction of flow and normal to it, covariance of Lagrangean velocities, and probability density function of travel times to various break-through locations. Following the analytic nature of the flow solution, all the results are presented in dimensionless forms. For example, the dispersion coefficients are made dimensionless with respect to the mean velocity and size of inhomogeneities. Detailed results will be presented and compared to well known first-order results and the results that are based on simple approximate transport solutions of Aldo Fiori.

Explosive attractor solutions to a universal cubic delay equation

NASA Astrophysics Data System (ADS)

Sanz-Orozco, D.; Berk, H. L.

2017-05-01

New explosive attractor solutions have been found in a universal cubic delay equation that has been studied in both the plasma and the fluid mechanics literature. Through computational simulations and analytic approximations, it is found that the oscillatory component of the explosive mode amplitude solutions are described through multi-frequency Fourier expansions with respect to a pseudo-time variable. The spectral dependence of these solutions as a function of a system parameter, ϕ , is studied. The mode amplitude that is described in the explosive regime has two main features: a well-known envelope ( t 0 - t ) - 5 / 2 , with t0 the blow-up time of the amplitude, and a spectrum of discrete oscillations with ever-increasing frequencies, which may give experimental information about the properties of a system's equilibrium.

Shock compression modeling of metallic single crystals: comparison of finite difference, steady wave, and analytical solutions

DOE PAGES

Lloyd, Jeffrey T.; Clayton, John D.; Austin, Ryan A.; ...

2015-07-10

Background: The shock response of metallic single crystals can be captured using a micro-mechanical description of the thermoelastic-viscoplastic material response; however, using a such a description within the context of traditional numerical methods may introduce a physical artifacts. Advantages and disadvantages of complex material descriptions, in particular the viscoplastic response, must be framed within approximations introduced by numerical methods. Methods: Three methods of modeling the shock response of metallic single crystals are summarized: finite difference simulations, steady wave simulations, and algebraic solutions of the Rankine-Hugoniot jump conditions. For the former two numerical techniques, a dislocation density based framework describes themore » rate- and temperature-dependent shear strength on each slip system. For the latter analytical technique, a simple (two-parameter) rate- and temperature-independent linear hardening description is necessarily invoked to enable simultaneous solution of the governing equations. For all models, the same nonlinear thermoelastic energy potential incorporating elastic constants of up to order 3 is applied. Results: Solutions are compared for plate impact of highly symmetric orientations (all three methods) and low symmetry orientations (numerical methods only) of aluminum single crystals shocked to 5 GPa (weak shock regime) and 25 GPa (overdriven regime). Conclusions: For weak shocks, results of the two numerical methods are very similar, regardless of crystallographic orientation. For strong shocks, artificial viscosity affects the finite difference solution, and effects of transverse waves for the lower symmetry orientations not captured by the steady wave method become important. The analytical solution, which can only be applied to highly symmetric orientations, provides reasonable accuracy with regards to prediction of most variables in the final shocked state but, by construction, does not provide insight into the shock structure afforded by the numerical methods.« less

Unsteady Flow Simulation: A Numerical Challenge

DTIC Science & Technology

2003-03-01

drive to convergence the numerical unsteady term. The time marching procedure is based on the approximate implicit Newton method for systems of non...computed through analytical derivatives of S. The linear system stemming from equation (3) is solved at each integration step by the same iterative method...significant reduction of memory usage, thanks to the reduced dimensions of the linear system matrix during the implicit marching of the solution. The

Spatiotemporal Airy Ince-Gaussian wave packets in strongly nonlocal nonlinear media.

PubMed

Peng, Xi; Zhuang, Jingli; Peng, Yulian; Li, DongDong; Zhang, Liping; Chen, Xingyu; Zhao, Fang; Deng, Dongmei

2018-03-08

The self-accelerating Airy Ince-Gaussian (AiIG) and Airy helical Ince-Gaussian (AihIG) wave packets in strongly nonlocal nonlinear media (SNNM) are obtained by solving the strongly nonlocal nonlinear Schrödinger equation. For the first time, the propagation properties of three dimensional localized AiIG and AihIG breathers and solitons in the SNNM are demonstrated, these spatiotemporal wave packets maintain the self-accelerating and approximately non-dispersion properties in temporal dimension, periodically oscillating (breather state) or steady (soliton state) in spatial dimension. In particular, their numerical experiments of spatial intensity distribution, numerical simulations of spatiotemporal distribution, as well as the transverse energy flow and the angular momentum in SNNM are presented. Typical examples of the obtained solutions are based on the ratio between the input power and the critical power, the ellipticity and the strong nonlocality parameter. The comparisons of analytical solutions with numerical simulations and numerical experiments of the AiIG and AihIG optical solitons show that the numerical results agree well with the analytical solutions in the case of strong nonlocality.

Providing the physical basis of SCS curve number method and its proportionality relationship from Richards' equation

NASA Astrophysics Data System (ADS)

Hooshyar, M.; Wang, D.

2016-12-01

The empirical proportionality relationship, which indicates that the ratio of cumulative surface runoff and infiltration to their corresponding potentials are equal, is the basis of the extensively used Soil Conservation Service Curve Number (SCS-CN) method. The objective of this paper is to provide the physical basis of the SCS-CN method and its proportionality hypothesis from the infiltration excess runoff generation perspective. To achieve this purpose, an analytical solution of Richards' equation is derived for ponded infiltration in shallow water table environment under the following boundary conditions: 1) the soil is saturated at the land surface; and 2) there is a no-flux boundary which moves downward. The solution is established based on the assumptions of negligible gravitational effect, constant soil water diffusivity, and hydrostatic soil moisture profile between the no-flux boundary and water table. Based on the derived analytical solution, the proportionality hypothesis is a reasonable approximation for rainfall partitioning at the early stage of ponded infiltration in areas with a shallow water table for coarse textured soils.

Study of microvascular non-Newtonian blood flow modulated by electroosmosis.

PubMed

Tripathi, Dharmendra; Yadav, Ashu; Anwar Bég, O; Kumar, Rakesh

2018-05-01

An analytical study of microvascular non-Newtonian blood flow is conducted incorporating the electro-osmosis phenomenon. Blood is considered as a Bingham rheological aqueous ionic solution. An externally applied static axial electrical field is imposed on the system. The Poisson-Boltzmann equation for electrical potential distribution is implemented to accommodate the electrical double layer in the microvascular regime. With long wavelength, lubrication and Debye-Hückel approximations, the boundary value problem is rendered non-dimensional. Analytical solutions are derived for the axial velocity, volumetric flow rate, pressure gradient, volumetric flow rate, averaged volumetric flow rate along one time period, pressure rise along one wavelength and stream function. A plug swidth is featured in the solutions. Via symbolic software (Mathematica), graphical plots are generated for the influence of Bingham plug flow width parameter, electrical Debye length and Helmholtz-Smoluchowski velocity (maximum electro-osmotic velocity) on the key hydrodynamic variables. This study reveals that blood flow rate accelerates with decreasing the plug width (i.e. viscoplastic nature of fluids) and also with increasing the Debye length parameter. Copyright © 2018 Elsevier Inc. All rights reserved.

Optimal guidance law development for an advanced launch system

NASA Technical Reports Server (NTRS)

Calise, Anthony J.; Leung, Martin S. K.

1995-01-01

The objective of this research effort was to develop a real-time guidance approach for launch vehicles ascent to orbit injection. Various analytical approaches combined with a variety of model order and model complexity reduction have been investigated. Singular perturbation methods were first attempted and found to be unsatisfactory. The second approach based on regular perturbation analysis was subsequently investigated. It also fails because the aerodynamic effects (ignored in the zero order solution) are too large to be treated as perturbations. Therefore, the study demonstrates that perturbation methods alone (both regular and singular perturbations) are inadequate for use in developing a guidance algorithm for the atmospheric flight phase of a launch vehicle. During a second phase of the research effort, a hybrid analytic/numerical approach was developed and evaluated. The approach combines the numerical methods of collocation and the analytical method of regular perturbations. The concept of choosing intelligent interpolating functions is also introduced. Regular perturbation analysis allows the use of a crude representation for the collocation solution, and intelligent interpolating functions further reduce the number of elements without sacrificing the approximation accuracy. As a result, the combined method forms a powerful tool for solving real-time optimal control problems. Details of the approach are illustrated in a fourth order nonlinear example. The hybrid approach is then applied to the launch vehicle problem. The collocation solution is derived from a bilinear tangent steering law, and results in a guidance solution for the entire flight regime that includes both atmospheric and exoatmospheric flight phases.

A discussion on validity of the diffusion theory by Monte Carlo method

NASA Astrophysics Data System (ADS)

Peng, Dong-qing; Li, Hui; Xie, Shusen

2008-12-01

Diffusion theory was widely used as a basis of the experiments and methods in determining the optical properties of biological tissues. A simple analytical solution could be obtained easily from the diffusion equation after a series of approximations. Thus, a misinterpret of analytical solution would be made: while the effective attenuation coefficient of several semi-infinite bio-tissues were the same, the distribution of light fluence in the tissues would be the same. In order to assess the validity of knowledge above, depth resolved internal fluence of several semi-infinite biological tissues which have the same effective attenuation coefficient were simulated with wide collimated beam in the paper by using Monte Carlo method in different condition. Also, the influence of bio-tissue refractive index on the distribution of light fluence was discussed in detail. Our results showed that, when the refractive index of several bio-tissues which had the same effective attenuation coefficient were the same, the depth resolved internal fluence would be the same; otherwise, the depth resolved internal fluence would be not the same. The change of refractive index of tissue would have affection on the light depth distribution in tissue. Therefore, the refractive index is an important optical property of tissue, and should be taken in account while using the diffusion approximation theory.

Phase slips in superconducting weak links

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kimmel, Gregory; Glatz, Andreas; Aranson, Igor S.

2017-01-01

Superconducting vortices and phase slips are primary mechanisms of dissipation in superconducting, superfluid, and cold-atom systems. While the dynamics of vortices is fairly well described, phase slips occurring in quasi-one- dimensional superconducting wires still elude understanding. The main reason is that phase slips are strongly nonlinear time-dependent phenomena that cannot be cast in terms of small perturbations of the superconducting state. Here we study phase slips occurring in superconducting weak links. Thanks to partial suppression of superconductivity in weak links, we employ a weakly nonlinear approximation for dynamic phase slips. This approximation is not valid for homogeneous superconducting wires andmore » slabs. Using the numerical solution of the time-dependent Ginzburg-Landau equation and bifurcation analysis of stationary solutions, we show that the onset of phase slips occurs via an infinite period bifurcation, which is manifested in a specific voltage-current dependence. Our analytical results are in good agreement with simulations.« less

Analysis and Sizing for Transient Thermal Heating of Insulated Aerospace Vehicle Structures

NASA Technical Reports Server (NTRS)

Blosser, Max L.

2012-01-01

An analytical solution was derived for the transient response of an insulated structure subjected to a simplified heat pulse. The solution is solely a function of two nondimensional parameters. Simpler functions of these two parameters were developed to approximate the maximum structural temperature over a wide range of parameter values. Techniques were developed to choose constant, effective thermal properties to represent the relevant temperature and pressure-dependent properties for the insulator and structure. A technique was also developed to map a time-varying surface temperature history to an equivalent square heat pulse. Equations were also developed for the minimum mass required to maintain the inner, unheated surface below a specified temperature. In the course of the derivation, two figures of merit were identified. Required insulation masses calculated using the approximate equation were shown to typically agree with finite element results within 10%-20% over the relevant range of parameters studied.

Shadows, signals, and stability in Einsteinian cubic gravity

NASA Astrophysics Data System (ADS)

Hennigar, Robie A.; Jahani Poshteh, Mohammad Bagher; Mann, Robert B.

2018-03-01

We conduct a preliminary investigation into the phenomenological implications of Einsteinian cubic gravity (ECG), a four-dimensional theory of gravity cubic in curvature of interest for its unique formulation and properties. We find an analytic approximation for a spherically symmetric black hole solution to this theory using a continued fraction ansatz. This approximate solution is valid everywhere outside of the horizon and we use it to study the orbit of massive test bodies near a black hole, specifically computing the innermost stable circular orbit. We compute constraints on the ECG coupling parameter imposed by Shapiro time delay. We then compute the shadow of an ECG black hole and find it to be larger than its Einsteinian counterpart in general relativity for the same value of the mass. Applying our results to Sgr A*, we find that departures from general relativity are small but in principle distinguishable.

A modular finite-element model (MODFE) for areal and axisymmetric ground-water-flow problems, Part 2: Derivation of finite-element equations and comparisons with analytical solutions

USGS Publications Warehouse

Cooley, Richard L.

1992-01-01

MODFE, a modular finite-element model for simulating steady- or unsteady-state, area1 or axisymmetric flow of ground water in a heterogeneous anisotropic aquifer is documented in a three-part series of reports. In this report, part 2, the finite-element equations are derived by minimizing a functional of the difference between the true and approximate hydraulic head, which produces equations that are equivalent to those obtained by either classical variational or Galerkin techniques. Spatial finite elements are triangular with linear basis functions, and temporal finite elements are one dimensional with linear basis functions. Physical processes that can be represented by the model include (1) confined flow, unconfined flow (using the Dupuit approximation), or a combination of both; (2) leakage through either rigid or elastic confining units; (3) specified recharge or discharge at points, along lines, or areally; (4) flow across specified-flow, specified-head, or head-dependent boundaries; (5) decrease of aquifer thickness to zero under extreme water-table decline and increase of aquifer thickness from zero as the water table rises; and (6) head-dependent fluxes from springs, drainage wells, leakage across riverbeds or confining units combined with aquifer dewatering, and evapotranspiration. The matrix equations produced by the finite-element method are solved by the direct symmetric-Doolittle method or the iterative modified incomplete-Cholesky conjugate-gradient method. The direct method can be efficient for small- to medium-sized problems (less than about 500 nodes), and the iterative method is generally more efficient for larger-sized problems. Comparison of finite-element solutions with analytical solutions for five example problems demonstrates that the finite-element model can yield accurate solutions to ground-water flow problems.

General relativistic razor-thin disks with magnetically polarized matter

NASA Astrophysics Data System (ADS)

Navarro-Noguera, Anamaría; Lora-Clavijo, F. D.; González, Guillermo A.

2018-06-01

The origin of magnetic fields in the universe still remains unknown and constitutes one of the most intriguing questions in astronomy and astrophysics. Their significance is enormous since they have a strong influence on many astrophysical phenomena. In regards of this motivation, theoretical models of galactic disks with sources of magnetic field may contribute to understand the physics behind them. Inspired by this, we present a new family of analytical models for thin disks composed by magnetized material. The solutions are axially symmetric, conformastatic and are obtained by solving the Einstein-Maxwell Field Equations for continuum media without the test field approximation, and assuming that the sources are razor-thin disk of magnetically polarized matter. We find analytical expressions for the surface energy density, the pressure, the polarization vector, the electromagnetic fields, the mass and the rotational velocity for circular orbits, for two particular solutions. In each case, the energy-momentum tensor agrees with the energy conditions and also the convergence of the mass for all the solutions is proved. Since the solutions are well-behaved, they may be used to model astrophysical thin disks, and also may contribute as initial data in numerical simulations. In addition, the process to obtain the solutions is described in detail, which may be used as a guide to find solutions with magnetized material in General Relativity.

The Relationship Between Partial Contaminant Source Zone Remediation and Groundwater Plume Attenuation

NASA Astrophysics Data System (ADS)

Falta, R. W.

2004-05-01

Analytical solutions are developed that relate changes in the contaminant mass in a source area to the behavior of biologically reactive dissolved contaminant groundwater plumes. Based on data from field experiments, laboratory experiments, numerical streamtube models, and numerical multiphase flow models, the chemical discharge from a source region is assumed to be a nonlinear power function of the fraction of contaminant mass removed from the source zone. This function can approximately represent source zone mass discharge behavior over a wide range of site conditions ranging from simple homogeneous systems, to complex heterogeneous systems. A mass balance on the source zone with advective transport and first order decay leads to a nonlinear differential equation that is solved analytically to provide a prediction of the time-dependent contaminant mass discharge leaving the source zone. The solution for source zone mass discharge is coupled semi-analytically with a modified version of the Domenico (1987) analytical solution for three-dimensional reactive advective and dispersive transport in groundwater. The semi-analytical model then employs the BIOCHLOR (Aziz et al., 2000; Sun et al., 1999) transformations to model sequential first order parent-daughter biological decay reactions of chlorinated ethenes and ethanes in the groundwater plume. The resulting semi-analytic model thus allows for transient simulation of complex source zone behavior that is fully coupled to a dissolved contaminant plume undergoing sequential biological reactions. Analyses of several realistic scenarios show that substantial changes in the ground water plume can result from the partial removal of contaminant mass from the source zone. These results, however, are sensitive to the nature of the source mass reduction-source discharge reduction curve, and to the rates of degradation of the primary contaminant and its daughter products in the ground water plume. Aziz, C.E., C.J. Newell, J.R. Gonzales, P. Haas, T.P. Clement, and Y. Sun, 2000, BIOCHLOR Natural Attenuation Decision Support System User's Manual Version 1.0, US EPA Report EPA/600/R-00/008 Domenico, P.A., 1987, An analytical model for multidimensional transport of a decaying contaminant species, J. Hydrol., 91: 49-58. Sun, Y., J.N. Petersen, T.P. Clement, and R.S. Skeen, 1999, A new analytical solution for multi-species transport equations with serial and parallel reactions, Water Resour. Res., 35(1): 185-190.

Global Properties of Fully Convective Accretion Disks from Local Simulations

NASA Astrophysics Data System (ADS)

Bodo, G.; Cattaneo, F.; Mignone, A.; Ponzo, F.; Rossi, P.

2015-08-01

We present an approach to deriving global properties of accretion disks from the knowledge of local solutions derived from numerical simulations based on the shearing box approximation. The approach consists of a two-step procedure. First, a local solution valid for all values of the disk height is constructed by piecing together an interior solution obtained numerically with an analytical exterior radiative solution. The matching is obtained by assuming hydrostatic balance and radiative equilibrium. Although in principle the procedure can be carried out in general, it simplifies considerably when the interior solution is fully convective. In these cases, the construction is analogous to the derivation of the Hayashi tracks for protostars. The second step consists of piecing together the local solutions at different radii to obtain a global solution. Here we use the symmetry of the solutions with respect to the defining dimensionless numbers—in a way similar to the use of homology relations in stellar structure theory—to obtain the scaling properties of the various disk quantities with radius.

Analytical Debye-Huckel model for electrostatic potentials around dissolved DNA.

PubMed Central

Wagner, K; Keyes, E; Kephart, T W; Edwards, G

1997-01-01

We present an analytical, Green-function-based model for the electric potential of DNA in solution, treating the surrounding solvent with the Debye-Huckel approximation. The partial charge of each atom is accounted for by modeling DNA as linear distributions of atoms on concentric cylindrical surfaces. The condensed ions of the solvent are treated with the Debye-Huckel approximation. The resultant leading term of the potential is that of a continuous shielded line charge, and the higher order terms account for the helical structure. Within several angstroms of the surface there is sufficient information in the electric potential to distinguish features and symmetries of DNA. Plots of the potential and equipotential surfaces, dominated by the phosphate charges, reflect the structural differences between the A, B, and Z conformations and, to a smaller extent, the difference between base sequences. As the distances from the helices increase, the magnitudes of the potentials decrease. However, the bases and sugars account for a larger fraction of the double helix potential with increasing distance. We have found that when the solvent is treated with the Debye-Huckel approximation, the potential decays more rapidly in every direction from the surface than it did in the concentric dielectric cylinder approximation. Images FIGURE 1 FIGURE 2 FIGURE 3 FIGURE 4 FIGURE 5 FIGURE 7 PMID:9199767

Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity.

PubMed

Zeng, Jianhua; Malomed, Boris A

2017-05-01

Recent studies have demonstrated that defocusing cubic nonlinearity with local strength growing from the center to the periphery faster than r^{D}, in space of dimension D with radial coordinate r, supports a vast variety of robust bright solitons. In the framework of the same model, but with a weaker spatial-growth rate ∼r^{α} with α≤D, we test here the possibility to create stable localized continuous waves (LCWs) in one-dimensional (1D) and 2D geometries, localized dark solitons (LDSs) in one dimension, and localized dark vortices (LDVs) in two dimensions, which are all realized as loosely confined states with a divergent norm. Asymptotic tails of the solutions, which determine the divergence of the norm, are constructed in a universal analytical form by means of the Thomas-Fermi approximation (TFA). Global approximations for the LCWs, LDSs, and LDVs are constructed on the basis of interpolations between analytical approximations available far from (TFA) and close to the center. In particular, the interpolations for the 1D LDS, as well as for the 2D LDVs, are based on a deformed-tanh expression, which is suggested by the usual 1D dark-soliton solution. The analytical interpolations produce very accurate results, in comparison with numerical findings, for the 1D and 2D LCWs, 1D LDSs, and 2D LDVs with vorticity S=1. In addition to the 1D fundamental LDSs with the single notch and 2D vortices with S=1, higher-order LDSs with multiple notches are found too, as well as double LDVs, with S=2. Stability regions for the modes under consideration are identified by means of systematic simulations, the LCWs being completely stable in one and two dimensions, as they are ground states in the corresponding settings. Basic evolution scenarios are identified for those vortices that are unstable. The settings considered in this work may be implemented in nonlinear optics and in Bose-Einstein condensates.

Localized dark solitons and vortices in defocusing media with spatially inhomogeneous nonlinearity

NASA Astrophysics Data System (ADS)

Zeng, Jianhua; Malomed, Boris A.

2017-05-01

Recent studies have demonstrated that defocusing cubic nonlinearity with local strength growing from the center to the periphery faster than rD, in space of dimension D with radial coordinate r , supports a vast variety of robust bright solitons. In the framework of the same model, but with a weaker spatial-growth rate ˜rα with α ≤D , we test here the possibility to create stable localized continuous waves (LCWs) in one-dimensional (1D) and 2D geometries, localized dark solitons (LDSs) in one dimension, and localized dark vortices (LDVs) in two dimensions, which are all realized as loosely confined states with a divergent norm. Asymptotic tails of the solutions, which determine the divergence of the norm, are constructed in a universal analytical form by means of the Thomas-Fermi approximation (TFA). Global approximations for the LCWs, LDSs, and LDVs are constructed on the basis of interpolations between analytical approximations available far from (TFA) and close to the center. In particular, the interpolations for the 1D LDS, as well as for the 2D LDVs, are based on a deformed-tanh expression, which is suggested by the usual 1D dark-soliton solution. The analytical interpolations produce very accurate results, in comparison with numerical findings, for the 1D and 2D LCWs, 1D LDSs, and 2D LDVs with vorticity S =1 . In addition to the 1D fundamental LDSs with the single notch and 2D vortices with S =1 , higher-order LDSs with multiple notches are found too, as well as double LDVs, with S =2 . Stability regions for the modes under consideration are identified by means of systematic simulations, the LCWs being completely stable in one and two dimensions, as they are ground states in the corresponding settings. Basic evolution scenarios are identified for those vortices that are unstable. The settings considered in this work may be implemented in nonlinear optics and in Bose-Einstein condensates.

The Surface Density Distribution in the Solar Nebula

NASA Technical Reports Server (NTRS)

Davis, Sanford S.

2004-01-01

The commonly used minimum mass power law representation of the pre-solar nebula is reanalyzed using a new cumulative-mass-model. This model predicts a smoother surface density approximation compared with methods based on direct computation of surface density. The density is quantified using two independent analytical formulations. First, a best-fit transcendental function is applied directly to the basic planetary data. Next a solution to the time-dependent disk evolution equation is parametrically adapted to the solar nebula data. The latter model is shown to be a good approximation to the finite-size early Solar Nebula, and by extension to other extra solar protoplanetary disks.

Higher order approximation to the Hill problem dynamics about the libration points

NASA Astrophysics Data System (ADS)

Lara, Martin; Pérez, Iván L.; López, Rosario

2018-06-01

An analytical solution to the Hill problem Hamiltonian expanded about the libration points has been obtained by means of perturbation techniques. In order to compute the higher orders of the perturbation solution that are needed to capture all the relevant periodic orbits originated from the libration points within a reasonable accuracy, the normalization is approached in complex variables. The validity of the solution extends to energy values considerably far away from that of the libration points and, therefore, can be used in the computation of Halo orbits as an alternative to the classical Lindstedt-Poincaré approach. Furthermore, the theory correctly predicts the existence of the two-lane bridge of periodic orbits linking the families of planar and vertical Lyapunov orbits.

Neural Network and Regression Methods Demonstrated in the Design Optimization of a Subsonic Aircraft

NASA Technical Reports Server (NTRS)

Hopkins, Dale A.; Lavelle, Thomas M.; Patnaik, Surya

2003-01-01

The neural network and regression methods of NASA Glenn Research Center s COMETBOARDS design optimization testbed were used to generate approximate analysis and design models for a subsonic aircraft operating at Mach 0.85 cruise speed. The analytical model is defined by nine design variables: wing aspect ratio, engine thrust, wing area, sweep angle, chord-thickness ratio, turbine temperature, pressure ratio, bypass ratio, fan pressure; and eight response parameters: weight, landing velocity, takeoff and landing field lengths, approach thrust, overall efficiency, and compressor pressure and temperature. The variables were adjusted to optimally balance the engines to the airframe. The solution strategy included a sensitivity model and the soft analysis model. Researchers generated the sensitivity model by training the approximators to predict an optimum design. The trained neural network predicted all response variables, within 5-percent error. This was reduced to 1 percent by the regression method. The soft analysis model was developed to replace aircraft analysis as the reanalyzer in design optimization. Soft models have been generated for a neural network method, a regression method, and a hybrid method obtained by combining the approximators. The performance of the models is graphed for aircraft weight versus thrust as well as for wing area and turbine temperature. The regression method followed the analytical solution with little error. The neural network exhibited 5-percent maximum error over all parameters. Performance of the hybrid method was intermediate in comparison to the individual approximators. Error in the response variable is smaller than that shown in the figure because of a distortion scale factor. The overall performance of the approximators was considered to be satisfactory because aircraft analysis with NASA Langley Research Center s FLOPS (Flight Optimization System) code is a synthesis of diverse disciplines: weight estimation, aerodynamic analysis, engine cycle analysis, propulsion data interpolation, mission performance, airfield length for landing and takeoff, noise footprint, and others.

Verification and Validation of a Coordinate Transformation Method in Axisymmetric Transient Magnetics.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ashcraft, C. Chace; Niederhaus, John Henry; Robinson, Allen C.

We present a verification and validation analysis of a coordinate-transformation-based numerical solution method for the two-dimensional axisymmetric magnetic diffusion equation, implemented in the finite-element simulation code ALEGRA. The transformation, suggested by Melissen and Simkin, yields an equation set perfectly suited for linear finite elements and for problems with large jumps in material conductivity near the axis. The verification analysis examines transient magnetic diffusion in a rod or wire in a very low conductivity background by first deriving an approximate analytic solution using perturbation theory. This approach for generating a reference solution is shown to be not fully satisfactory. A specializedmore » approach for manufacturing an exact solution is then used to demonstrate second-order convergence under spatial refinement and tem- poral refinement. For this new implementation, a significant improvement relative to previously available formulations is observed. Benefits in accuracy for computed current density and Joule heating are also demonstrated. The validation analysis examines the circuit-driven explosion of a copper wire using resistive magnetohydrodynamics modeling, in comparison to experimental tests. The new implementation matches the accuracy of the existing formulation, with both formulations capturing the experimental burst time and action to within approximately 2%.« less

The dynamics of current carriers in standing Alfvén waves: Parallel electric fields in the auroral acceleration region

NASA Astrophysics Data System (ADS)

Wright, Andrew N.; Allan, W.; Ruderman, Michael S.; Elphic, R. C.

2002-07-01

The acceleration of current carriers in an Alfvén wave current system is considered. The model incorporates a dipole magnetic field geometry, and we present an analytical solution of the two-fluid equations by successive approximations. The leading solution corresponds to the familiar single-fluid toroidal oscillations. The next order describes the nonlinear dynamics of electrons responsible for carrying a few μAm-2 field aligned current into the ionosphere. The solution shows how most of the electron acceleration in the magnetosphere occurs within 1 RE of the ionosphere, and that a parallel electric field of the order of 1 mVm-1 is responsible for energising the electrons to 1 keV. The limitations of the electron fluid approximation are considered, and a qualitative solution including electron beams and a modified E∥ is developed in accord with observations. We find that the electron acceleration can be nonlinear, (ve∥∇∥)ve∥ > ωve∥, as a result of our nonuniform equilibrium field geometry even when ve∥ is less than the Alfvén speed. Our calculation also elucidates the processes through which E∥ is generated and supported.

Analytical and Numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations

NASA Astrophysics Data System (ADS)

Gómez-Aguilar, J. F.

2018-03-01

In this paper, we analyze an alcoholism model which involves the impact of Twitter via Liouville-Caputo and Atangana-Baleanu-Caputo fractional derivatives with constant- and variable-order. Two fractional mathematical models are considered, with and without delay. Special solutions using an iterative scheme via Laplace and Sumudu transform were obtained. We studied the uniqueness and existence of the solutions employing the fixed point postulate. The generalized model with variable-order was solved numerically via the Adams method and the Adams-Bashforth-Moulton scheme. Stability and convergence of the numerical solutions were presented in details. Numerical examples of the approximate solutions are provided to show that the numerical methods are computationally efficient. Therefore, by including both the fractional derivatives and finite time delays in the alcoholism model studied, we believe that we have established a more complete and more realistic indicator of alcoholism model and affect the spread of the drinking.

3DRISM-HI-D2MSA: an improved analytic theory to compute solvent structure around hydrophobic solutes with proper treatment of solute–solvent electrostatic interactions

NASA Astrophysics Data System (ADS)

Cao, Siqin; Zhu, Lizhe; Huang, Xuhui

2018-04-01

The 3D reference interaction site model (3DRISM) is a powerful tool to study the thermodynamic and structural properties of liquids. However, for hydrophobic solutes, the inhomogeneity of the solvent density around them poses a great challenge to the 3DRISM theory. To address this issue, we have previously introduced the hydrophobic-induced density inhomogeneity theory (HI) for purely hydrophobic solutes. To further consider the complex hydrophobic solutes containing partial charges, here we propose the D2MSA closure to incorporate the short-range and long-range interactions with the D2 closure and the mean spherical approximation, respectively. We demonstrate that our new theory can compute the solvent distributions around real hydrophobic solutes in water and complex organic solvents that agree well with the explicit solvent molecular dynamics simulations.

Semi-analytical modelling of positive corona discharge in air

NASA Astrophysics Data System (ADS)

Pontiga, Francisco; Yanallah, Khelifa; Chen, Junhong

2013-09-01

Semianalytical approximate solutions of the spatial distribution of electric field and electron and ion densities have been obtained by solving Poisson's equations and the continuity equations for the charged species along the Laplacian field lines. The need to iterate for the correct value of space charge on the corona electrode has been eliminated by using the corona current distribution over the grounded plane derived by Deutsch, which predicts a cos m θ law similar to Warburg's law. Based on the results of the approximated model, a parametric study of the influence of gas pressure, the corona wire radius, and the inter-electrode wire-plate separation has been carried out. Also, the approximate solutions of the electron number density has been combined with a simplified plasma chemistry model in order to compute the ozone density generated by the corona discharge in the presence of a gas flow. This work was supported by the Consejeria de Innovacion, Ciencia y Empresa (Junta de Andalucia) and by the Ministerio de Ciencia e Innovacion, Spain, within the European Regional Development Fund contracts FQM-4983 and FIS2011-25161.

A Semianalytical Model for Pumping Tests in Finite Heterogeneous Confined Aquifers With Arbitrarily Shaped Boundary

NASA Astrophysics Data System (ADS)

Wang, Lei; Dai, Cheng; Xue, Liang

2018-04-01

This study presents a Laplace-transform-based boundary element method to model the groundwater flow in a heterogeneous confined finite aquifer with arbitrarily shaped boundaries. The boundary condition can be Dirichlet, Neumann or Robin-type. The derived solution is analytical since it is obtained through the Green's function method within the domain. However, the numerical approximation is required on the boundaries, which essentially renders it a semi-analytical solution. The proposed method can provide a general framework to derive solutions for zoned heterogeneous confined aquifers with arbitrarily shaped boundary. The requirement of the boundary element method presented here is that the Green function must exist for a specific PDE equation. In this study, the linear equations for the two-zone and three-zone confined aquifers with arbitrarily shaped boundary is established in Laplace space, and the solution can be obtained by using any linear solver. Stehfest inversion algorithm can be used to transform it back into time domain to obtain the transient solution. The presented solution is validated in the two-zone cases by reducing the arbitrarily shaped boundaries to circular ones and comparing it with the solution in Lin et al. (2016, https://doi.org/10.1016/j.jhydrol.2016.07.028). The effect of boundary shape and well location on dimensionless drawdown in two-zone aquifers is investigated. Finally the drawdown distribution in three-zone aquifers with arbitrarily shaped boundary for constant-rate tests (CRT) and flow rate distribution for constant-head tests (CHT) are analyzed.

Self-induced transparency of an extremely short pulse

NASA Technical Reports Server (NTRS)

Lee, C. T.

1973-01-01

An extremely short pulse propagation in a resonant medium is properly described by a closed form steady-state analytic solution. The usual slowly varying envelope approximation (SVEA) is not made. Instead, different assumptions with respect to pulse speed and pulse duration are used, and any possible nonresonant loss is ignored. This study indicates that the results obtained by the SVEA approach are much better than they have been intuitively expected to be.

Traffic signal synchronization.

PubMed

Huang, Ding-wei; Huang, Wei-neng

2003-05-01

The benefits of traffic signal synchronization are examined within the cellular automata approach. The microsimulations of traffic flow are obtained with different settings of signal period T and time delay delta. Both numerical results and analytical approximations are presented. For undersaturated traffic, the green-light wave solutions can be realized. For saturated traffic, the correlation among the traffic signals has no effect on the throughput. For oversaturated traffic, the benefits of synchronization are manifest only when stochastic noise is suppressed.

Mathematical modeling of damage in unidirectional composites

NASA Technical Reports Server (NTRS)

Goree, J. G.; Dharani, L. R.; Jones, W. F.

1981-01-01

A review of some approximate analytical models for damaged, fiber reinforced composite materials is presented. Using the classical shear lag stress displacement assumption, solutions are presented for a unidirectional laminate containing a notch, a rectangular cut-out, and a circular hole. The models account for longitudinal matrix yielding and splitting as well as transverse matrix yielding and fiber breakage. The constraining influence of a cover sheet on the unidirectional laminate is also modeled.