Sample records for exact averaged equations

  1. Coarse-grained forms for equations describing the microscopic motion of particles in a fluid.

    PubMed

    Das, Shankar P; Yoshimori, Akira

    2013-10-01

    Exact equations of motion for the microscopically defined collective density ρ(x,t) and the momentum density ĝ(x,t) of a fluid have been obtained in the past starting from the corresponding Langevin equations representing the dynamics of the fluid particles. In the present work we average these exact equations of microscopic dynamics over the local equilibrium distribution to obtain stochastic partial differential equations for the coarse-grained densities with smooth spatial and temporal dependence. In particular, we consider Dean's exact balance equation for the microscopic density of a system of interacting Brownian particles to obtain the basic equation of the dynamic density functional theory with noise. Our analysis demonstrates that on thermal averaging the dependence of the exact equations on the bare interaction potential is converted to dependence on the corresponding thermodynamic direct correlation functions in the coarse-grained equations.

  2. When is quasi-linear theory exact. [particle acceleration

    NASA Technical Reports Server (NTRS)

    Jones, F. C.; Birmingham, T. J.

    1975-01-01

    We use the cumulant expansion technique of Kubo (1962, 1963) to derive an integrodifferential equation for the average one-particle distribution function for particles being accelerated by electric and magnetic fluctuations of a general nature. For a very restricted class of fluctuations, the equation for this function degenerates exactly to a differential equation of Fokker-Planck type. Quasi-linear theory, including the adiabatic assumption, is an exact theory only for this limited class of fluctuations.

  3. On the stability of dust orbits in mean-motion resonances perturbed by from an interstellar wind

    NASA Astrophysics Data System (ADS)

    Pástor, Pavol

    2014-09-01

    Circumstellar dust particles can be captured in a mean-motion resonance (MMR) with a planet and simultaneously be affected by non-gravitational effects. It is possible to describe the secular variations of a particle orbit in the MMR analytically using averaged resonant equations. We derive the averaged resonant equations from the equations of motion in near-canonical form. The secular variations of the particle orbit depending on the orientation of the orbit in space are taken into account. The averaged resonant equations can be derived/confirmed also from Lagrange's planetary equations. We apply the derived theory to the case when the non-gravitational effects are the Poynting-Robertson effect, the radial stellar wind, and an interstellar wind. The analytical and numerical results obtained are in excellent agreement. We found that the types of orbits correspond to libration centers of the conservative problem. The averaged resonant equations can lead to a system of equations which holds for stationary points in a subset of resonant variables. Using this system we show analytically that for the considered non-gravitational effects, all stationary points should correspond to orbits which are stationary in interplanetary space after an averaging over a synodic period. In an exact resonance, the stationary orbits are stable. The stability is achieved by a periodic repetition of the evolution during the synodic period. Numerical solutions of this system show that there are no stationary orbits for either the exact or non-exact resonances.

  4. On the structure of the master equation for a two-level system coupled to a thermal bath

    NASA Astrophysics Data System (ADS)

    de Vega, Inés

    2015-04-01

    We derive a master equation from the exact stochastic Liouville-von-Neumann (SLN) equation (Stockburger and Grabert 2002 Phys. Rev. Lett. 88 170407). The latter depends on two correlated noises and describes exactly the dynamics of an oscillator (which can be either harmonic or present an anharmonicity) coupled to an environment at thermal equilibrium. The newly derived master equation is obtained by performing analytically the average over different noise trajectories. It is found to have a complex hierarchical structure that might be helpful to explain the convergence problems occurring when performing numerically the stochastic average of trajectories given by the SLN equation (Koch et al 2008 Phys. Rev. Lett. 100 230402, Koch 2010 PhD thesis Fakultät Mathematik und Naturwissenschaften der Technischen Universitat Dresden).

  5. Quasi-linear theory via the cumulant expansion approach

    NASA Technical Reports Server (NTRS)

    Jones, F. C.; Birmingham, T. J.

    1974-01-01

    The cumulant expansion technique of Kubo was used to derive an intergro-differential equation for f , the average one particle distribution function for particles being accelerated by electric and magnetic fluctuations of a general nature. For a very restricted class of fluctuations, the f equation degenerates exactly to a differential equation of Fokker-Planck type. Quasi-linear theory, including the adiabatic assumption, is an exact theory for this limited class of fluctuations. For more physically realistic fluctuations, however, quasi-linear theory is at best approximate.

  6. Spectra of turbulently advected scalars that have small Schmidt number

    NASA Astrophysics Data System (ADS)

    Hill, Reginald J.

    2017-09-01

    Exact statistical equations are derived for turbulent advection of a passive scalar having diffusivity much larger than the kinematic viscosity, i.e., small Schmidt number. The equations contain all terms needed for precise direct numerical simulation (DNS) quantification. In the appropriate limit, the equations reduce to the classical theory for which the scalar spectrum is proportional to the energy spectrum multiplied by k-4, which, in turn, results in the inertial-diffusive range power law, k-17 /3. The classical theory was derived for the case of isotropic velocity and scalar fields. The exact equations are simplified for less restrictive cases: (1) locally isotropic scalar fluctuations at dissipation scales with no restriction on symmetry of the velocity field, (2) isotropic velocity field with averaging over all wave-vector directions with no restriction on the symmetry of the scalar, motivated by that average being used for DNS, and (3) isotropic velocity field with axisymmetric scalar fluctuations, motivated by the mean-scalar-gradient-source case. The equations are applied to recently published DNSs of passive scalars for the cases of a freely decaying scalar and a mean-scalar-gradient source. New terms in the exact equations are estimated for those cases and are found to be significant; those terms cause the deviations from the classical theory found by the DNS studies. A new formula for the mean-scalar-gradient case explains the variation of the scalar spectra for the DNS of the smallest Schmidt-number cases. Expansion in Legendre polynomials reveals the effect of axisymmetry. Inertial-diffusive-range formulas for both the zero- and second-order Legendre contributions are given. Exact statistical equations reveal what must be quantified using DNS to determine what causes deviations from asymptotic relationships.

  7. BHR equations re-derived with immiscible particle effects

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

    Schwarzkopf, John Dennis; Horwitz, Jeremy A.

    2015-05-01

    Compressible and variable density turbulent flows with dispersed phase effects are found in many applications ranging from combustion to cloud formation. These types of flows are among the most challenging to simulate. While the exact equations governing a system of particles and fluid are known, computational resources limit the scale and detail that can be simulated in this type of problem. Therefore, a common method is to simulate averaged versions of the flow equations, which still capture salient physics and is relatively less computationally expensive. Besnard developed such a model for variable density miscible turbulence, where ensemble-averaging was applied tomore » the flow equations to yield a set of filtered equations. Besnard further derived transport equations for the Reynolds stresses, the turbulent mass flux, and the density-specific volume covariance, to help close the filtered momentum and continuity equations. We re-derive the exact BHR closure equations which include integral terms owing to immiscible effects. Physical interpretations of the additional terms are proposed along with simple models. The goal of this work is to extend the BHR model to allow for the simulation of turbulent flows where an immiscible dispersed phase is non-trivially coupled with the carrier phase.« less

  8. Electromagnetic energy momentum in dispersive media

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

    Philbin, T. G.

    2011-01-15

    The standard derivations of electromagnetic energy and momentum in media take Maxwell's equations as the starting point. It is well known that for dispersive media this approach does not directly yield exact expressions for the energy and momentum densities. Although Maxwell's equations fully describe electromagnetic fields, the general approach to conserved quantities in field theory is not based on the field equations, but rather on the action. Here an action principle for macroscopic electromagnetism in dispersive, lossless media is used to derive the exact conserved energy-momentum tensor. The time-averaged energy density reduces to Brillouin's simple formula when the fields aremore » monochromatic. The time-averaged momentum density for monochromatic fields corresponds to the familiar Minkowski expression DxB, but for general fields in dispersive media the momentum density does not have the Minkowski value. The results are unaffected by the debate over momentum balance in light-matter interactions.« less

  9. Conservative-variable average states for equilibrium gas multi-dimensional fluxes

    NASA Technical Reports Server (NTRS)

    Iannelli, G. S.

    1992-01-01

    Modern split component evaluations of the flux vector Jacobians are thoroughly analyzed for equilibrium-gas average-state determinations. It is shown that all such derivations satisfy a fundamental eigenvalue consistency theorem. A conservative-variable average state is then developed for arbitrary equilibrium-gas equations of state and curvilinear-coordinate fluxes. Original expressions for eigenvalues, sound speed, Mach number, and eigenvectors are then determined for a general average Jacobian, and it is shown that the average eigenvalues, Mach number, and eigenvectors may not coincide with their classical pointwise counterparts. A general equilibrium-gas equation of state is then discussed for conservative-variable computational fluid dynamics (CFD) Euler formulations. The associated derivations lead to unique compatibility relations that constrain the pressure Jacobian derivatives. Thereafter, alternative forms for the pressure variation and average sound speed are developed in terms of two average pressure Jacobian derivatives. Significantly, no additional degree of freedom exists in the determination of these two average partial derivatives of pressure. Therefore, they are simultaneously computed exactly without any auxiliary relation, hence without any geometric solution projection or arbitrary scale factors. Several alternative formulations are then compared and key differences highlighted with emphasis on the determination of the pressure variation and average sound speed. The relevant underlying assumptions are identified, including some subtle approximations that are inherently employed in published average-state procedures. Finally, a representative test case is discussed for which an intrinsically exact average state is determined. This exact state is then compared with the predictions of recent methods, and their inherent approximations are appropriately quantified.

  10. Multispecies reaction-diffusion systems

    NASA Astrophysics Data System (ADS)

    Aghamohammadi, A.; Fatollahi, A. H.; Khorrami, M.; Shariati, A.

    2000-10-01

    Multispecies reaction-diffusion systems, for which the time evolution equations of correlation functions become a closed set, are considered. A formal solution for the average densities is found. Some special interactions and the exact time dependence of the average densities in these cases are also studied. For the general case, the large-time behavior of the average densities has also been obtained.

  11. Ensemble Averaged Probability Density Function (APDF) for Compressible Turbulent Reacting Flows

    NASA Technical Reports Server (NTRS)

    Shih, Tsan-Hsing; Liu, Nan-Suey

    2012-01-01

    In this paper, we present a concept of the averaged probability density function (APDF) for studying compressible turbulent reacting flows. The APDF is defined as an ensemble average of the fine grained probability density function (FG-PDF) with a mass density weighting. It can be used to exactly deduce the mass density weighted, ensemble averaged turbulent mean variables. The transport equation for APDF can be derived in two ways. One is the traditional way that starts from the transport equation of FG-PDF, in which the compressible Navier- Stokes equations are embedded. The resulting transport equation of APDF is then in a traditional form that contains conditional means of all terms from the right hand side of the Navier-Stokes equations except for the chemical reaction term. These conditional means are new unknown quantities that need to be modeled. Another way of deriving the transport equation of APDF is to start directly from the ensemble averaged Navier-Stokes equations. The resulting transport equation of APDF derived from this approach appears in a closed form without any need for additional modeling. The methodology of ensemble averaging presented in this paper can be extended to other averaging procedures: for example, the Reynolds time averaging for statistically steady flow and the Reynolds spatial averaging for statistically homogeneous flow. It can also be extended to a time or spatial filtering procedure to construct the filtered density function (FDF) for the large eddy simulation (LES) of compressible turbulent reacting flows.

  12. Evaluation of MOSTAS computer code for predicting dynamic loads in two bladed wind turbines

    NASA Technical Reports Server (NTRS)

    Kaza, K. R. V.; Janetzke, D. C.; Sullivan, T. L.

    1979-01-01

    Calculated dynamic blade loads were compared with measured loads over a range of yaw stiffnesses of the DOE/NASA Mod-O wind turbine to evaluate the performance of two versions of the MOSTAS computer code. The first version uses a time-averaged coefficient approximation in conjunction with a multi-blade coordinate transformation for two bladed rotors to solve the equations of motion by standard eigenanalysis. The second version accounts for periodic coefficients while solving the equations by a time history integration. A hypothetical three-degree of freedom dynamic model was investigated. The exact equations of motion of this model were solved using the Floquet-Lipunov method. The equations with time-averaged coefficients were solved by standard eigenanalysis.

  13. Expansion and growth of structure observables in a macroscopic gravity averaged universe

    NASA Astrophysics Data System (ADS)

    Wijenayake, Tharake; Ishak, Mustapha

    2015-03-01

    We investigate the effect of averaging inhomogeneities on expansion and large-scale structure growth observables using the exact and covariant framework of macroscopic gravity (MG). It is well known that applying the Einstein's equations and spatial averaging do not commute and lead to the averaging problem and backreaction terms. For the MG formalism applied to the Friedman-Lemaitre-Robertson-Walker (FLRW) metric, the extra term can be encapsulated as an averaging density parameter denoted ΩA . An exact isotropic cosmological solution of MG for the flat FLRW metric is already known in the literature; we derive here an anisotropic exact solution. Using the isotropic solution, we compare the expansion history to current available data of distances to supernovae, baryon acoustic oscillations, cosmic microwave background last scattering surface data, and Hubble constant measurements, and find -0.05 ≤ΩA≤0.07 (at the 95% confidence level). For the flat metric case this reduces to -0.03 ≤ΩA≤0.05 . The positive part of the intervals can be rejected if a mathematical (and physical) prior is taken into account. We also find that the inclusion of this term in the fits can shift the values of the usual cosmological parameters by a few to several percents. Next, we derive an equation for the growth rate of large-scale structure in MG that includes a term due to the averaging and assess its effect on the evolution of the growth compared to that of the Lambda cold dark matter (Λ CDM ) concordance model. We find that an ΩA term of an amplitude range of [-0.04 ,-0.02 ] lead to a relative deviation of the growth from that of the Λ CDM of up to 2%-4% at late times. Thus, the shift in the growth could be of comparable amplitude to that caused by similar changes in cosmological parameters like the dark energy density parameter or its equation of state. The effect could also be comparable in amplitude to some systematic effects considered for future surveys. This indicates that the averaging term and its possible effect need to be tightly constrained in future precision cosmological studies.

  14. Wide localized solutions of the parity-time-symmetric nonautonomous nonlinear Schrödinger equation

    NASA Astrophysics Data System (ADS)

    Meza, L. E. Arroyo; Dutra, A. de Souza; Hott, M. B.; Roy, P.

    2015-01-01

    By using canonical transformations we obtain localized (in space) exact solutions of the nonlinear Schrödinger equation (NLSE) with cubic and quintic space and time modulated nonlinearities and in the presence of time-dependent and inhomogeneous external potentials and amplification or absorption (source or drain) coefficients. We obtain a class of wide localized exact solutions of NLSE in the presence of a number of non-Hermitian parity-time (PT )-symmetric external potentials, which are constituted by a mixing of external potentials and source or drain terms. The exact solutions found here can be applied to theoretical studies of ultrashort pulse propagation in optical fibers with focusing and defocusing nonlinearities. We show that, even in the presence of gain or loss terms, stable solutions can be found and that the PT symmetry is an important feature to guarantee the conservation of the average energy of the system.

  15. Gyro-Landau fluid models for toroidal geometry

    NASA Astrophysics Data System (ADS)

    Waltz, R. E.; Dominguez, R. R.; Hammett, G. W.

    1992-10-01

    Gyro-Landau fluid model equations provide first-order time advancement for a limited number of moments of the gyrokinetic equation, while approximately preserving the effects of the gyroradius averaging and Landau damping. This paper extends the work of Hammett and Perkins [Phys. Rev. Lett. 64, 3019 (1990)] for electrostatic motion parallel to the magnetic field and E×B motion to include the gyroaveraging linearly and the curvature drift motion. The equations are tested by comparing the ion-temperature-gradient mode linear growth rates for the model equations with those of the exact gyrokinetic theory over a full range of parameters.

  16. Rectification of thermal fluctuations in ideal gases.

    PubMed

    Meurs, P; Van den Broeck, C; Garcia, A

    2004-11-01

    We calculate the systematic average speed of the adiabatic piston and a thermal Brownian motor, introduced by C. Van den Broeck, R, Kawai and P. Meurs [Phys. Rev. Lett. 93, 090601 (2004)], by an expansion of the Boltzmann equation and compare with the exact numerical solution.

  17. Weighed scalar averaging in LTB dust models: part II. A formalism of exact perturbations

    NASA Astrophysics Data System (ADS)

    Sussman, Roberto A.

    2013-03-01

    We examine the exact perturbations that arise from the q-average formalism that was applied in the preceding article (part I) to Lemaître-Tolman-Bondi (LTB) models. By introducing an initial value parametrization, we show that all LTB scalars that take an FLRW ‘look-alike’ form (frequently used in the literature dealing with LTB models) follow as q-averages of covariant scalars that are common to FLRW models. These q-scalars determine for every averaging domain a unique FLRW background state through Darmois matching conditions at the domain boundary, though the definition of this background does not require an actual matching with an FLRW region (Swiss cheese-type models). Local perturbations describe the deviation from the FLRW background state through the local gradients of covariant scalars at the boundary of every comoving domain, while non-local perturbations do so in terms of the intuitive notion of a ‘contrast’ of local scalars with respect to FLRW reference values that emerge from q-averages assigned to the whole domain or the whole time slice in the asymptotic limit. We derive fluid flow evolution equations that completely determine the dynamics of the models in terms of the q-scalars and both types of perturbations. A rigorous formalism of exact spherical nonlinear perturbations is defined over the FLRW background state associated with the q-scalars, recovering the standard results of linear perturbation theory in the appropriate limit. We examine the notion of the amplitude and illustrate the differences between local and non-local perturbations by qualitative diagrams and through an example of a cosmic density void that follows from the numeric solution of the evolution equations.

  18. A unified stochastic formulation of dissipative quantum dynamics. I. Generalized hierarchical equations

    NASA Astrophysics Data System (ADS)

    Hsieh, Chang-Yu; Cao, Jianshu

    2018-01-01

    We extend a standard stochastic theory to study open quantum systems coupled to a generic quantum environment. We exemplify the general framework by studying a two-level quantum system coupled bilinearly to the three fundamental classes of non-interacting particles: bosons, fermions, and spins. In this unified stochastic approach, the generalized stochastic Liouville equation (SLE) formally captures the exact quantum dissipations when noise variables with appropriate statistics for different bath models are applied. Anharmonic effects of a non-Gaussian bath are precisely encoded in the bath multi-time correlation functions that noise variables have to satisfy. Starting from the SLE, we devise a family of generalized hierarchical equations by averaging out the noise variables and expand bath multi-time correlation functions in a complete basis of orthonormal functions. The general hierarchical equations constitute systems of linear equations that provide numerically exact simulations of quantum dynamics. For bosonic bath models, our general hierarchical equation of motion reduces exactly to an extended version of hierarchical equation of motion which allows efficient simulation for arbitrary spectral densities and temperature regimes. Similar efficiency and flexibility can be achieved for the fermionic bath models within our formalism. The spin bath models can be simulated with two complementary approaches in the present formalism. (I) They can be viewed as an example of non-Gaussian bath models and be directly handled with the general hierarchical equation approach given their multi-time correlation functions. (II) Alternatively, each bath spin can be first mapped onto a pair of fermions and be treated as fermionic environments within the present formalism.

  19. Quantum corrections of the truncated Wigner approximation applied to an exciton transport model.

    PubMed

    Ivanov, Anton; Breuer, Heinz-Peter

    2017-04-01

    We modify the path integral representation of exciton transport in open quantum systems such that an exact description of the quantum fluctuations around the classical evolution of the system is possible. As a consequence, the time evolution of the system observables is obtained by calculating the average of a stochastic difference equation which is weighted with a product of pseudoprobability density functions. From the exact equation of motion one can clearly identify the terms that are also present if we apply the truncated Wigner approximation. This description of the problem is used as a basis for the derivation of a new approximation, whose validity goes beyond the truncated Wigner approximation. To demonstrate this we apply the formalism to a donor-acceptor transport model.

  20. New Exact Solutions of Relativistic Hydrodynamics for Longitudinally Expanding Fireballs

    NASA Astrophysics Data System (ADS)

    Csörgő, Tamás; Kasza, Gábor; Csanád, Máté; Jiang, Zefang

    2018-06-01

    We present new, exact, finite solutions of relativistic hydrodynamics for longitudinally expanding fireballs for arbitrary constant value of the speed of sound. These new solutions generalize earlier, longitudinally finite, exact solutions, from an unrealistic to a reasonable equation of state, characterized by a temperature independent (average) value of the speed of sound. Observables like the rapidity density and the pseudorapidity density are evaluated analytically, resulting in simple and easy to fit formulae that can be matched to the high energy proton-proton and heavy ion collision data at RHIC and LHC. In the longitudinally boost-invariant limit, these new solutions approach the Hwa-Bjorken solution and the corresponding rapidity distributions approach a rapidity plateaux.

  1. Exact anisotropic viscous fluid solutions of Einstein's equations

    NASA Astrophysics Data System (ADS)

    Goenner, H. F. M.; Kowalewski, F.

    1989-05-01

    A method for obtaining anisotropic, rotationless viscous fluid matter solutions of Bianchi type I and Segré type [1, 111] with the barotropic equation of state is presented. Solutions for which the anisotropy decreases exponentially or with a power law as well as solutions with average Hubble parameterH ˜t -1 are discussed. Also, a class of solutions with constant anisotropy and Bianchi type VIh is found. The dominant energy condition holds and the transport coefficients show the right sign.

  2. Higher order solution of the Euler equations on unstructured grids using quadratic reconstruction

    NASA Technical Reports Server (NTRS)

    Barth, Timothy J.; Frederickson, Paul O.

    1990-01-01

    High order accurate finite-volume schemes for solving the Euler equations of gasdynamics are developed. Central to the development of these methods are the construction of a k-exact reconstruction operator given cell-averaged quantities and the use of high order flux quadrature formulas. General polygonal control volumes (with curved boundary edges) are considered. The formulations presented make no explicit assumption as to complexity or convexity of control volumes. Numerical examples are presented for Ringleb flow to validate the methodology.

  3. Universal shocks in the Wishart random-matrix ensemble.

    PubMed

    Blaizot, Jean-Paul; Nowak, Maciej A; Warchoł, Piotr

    2013-05-01

    We show that the derivative of the logarithm of the average characteristic polynomial of a diffusing Wishart matrix obeys an exact partial differential equation valid for an arbitrary value of N, the size of the matrix. In the large N limit, this equation generalizes the simple inviscid Burgers equation that has been obtained earlier for Hermitian or unitary matrices. The solution, through the method of characteristics, presents singularities that we relate to the precursors of shock formation in the Burgers equation. The finite N effects appear as a viscosity term in the Burgers equation. Using a scaling analysis of the complete equation for the characteristic polynomial, in the vicinity of the shocks, we recover in a simple way the universal Bessel oscillations (so-called hard-edge singularities) familiar in random-matrix theory.

  4. Exact solutions for (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and coupled Klein-Gordon equations.

    PubMed

    Khan, Kamruzzaman; Akbar, M Ali; Islam, S M Rayhanul

    2014-01-01

    In this work, recently developed modified simple equation (MSE) method is applied to find exact traveling wave solutions of nonlinear evolution equations (NLEEs). To do so, we consider the (1 + 1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony (DMBBM) equation and coupled Klein-Gordon (cKG) equations. Two classes of explicit exact solutions-hyperbolic and trigonometric solutions of the associated equations are characterized with some free parameters. Then these exact solutions correspond to solitary waves for particular values of the parameters. 02.30.Jr; 02.70.Wz; 05.45.Yv; 94.05.Fg.

  5. Exact combinatorial approach to finite coagulating systems

    NASA Astrophysics Data System (ADS)

    Fronczak, Agata; Chmiel, Anna; Fronczak, Piotr

    2018-02-01

    This paper outlines an exact combinatorial approach to finite coagulating systems. In this approach, cluster sizes and time are discrete and the binary aggregation alone governs the time evolution of the systems. By considering the growth histories of all possible clusters, an exact expression is derived for the probability of a coagulating system with an arbitrary kernel being found in a given cluster configuration when monodisperse initial conditions are applied. Then this probability is used to calculate the time-dependent distribution for the number of clusters of a given size, the average number of such clusters, and that average's standard deviation. The correctness of our general expressions is proved based on the (analytical and numerical) results obtained for systems with the constant kernel. In addition, the results obtained are compared with the results arising from the solutions to the mean-field Smoluchowski coagulation equation, indicating its weak points. The paper closes with a brief discussion on the extensibility to other systems of the approach presented herein, emphasizing the issue of arbitrary initial conditions.

  6. Study of coupled nonlinear partial differential equations for finding exact analytical solutions.

    PubMed

    Khan, Kamruzzaman; Akbar, M Ali; Koppelaar, H

    2015-07-01

    Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G'/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd-Sokolov-Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics.

  7. Similarity solutions of reaction–diffusion equation with space- and time-dependent diffusion and reaction terms

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

    Ho, C.-L.; Lee, C.-C., E-mail: chieh.no27@gmail.com

    2016-01-15

    We consider solvability of the generalized reaction–diffusion equation with both space- and time-dependent diffusion and reaction terms by means of the similarity method. By introducing the similarity variable, the reaction–diffusion equation is reduced to an ordinary differential equation. Matching the resulting ordinary differential equation with known exactly solvable equations, one can obtain corresponding exactly solvable reaction–diffusion systems. Several representative examples of exactly solvable reaction–diffusion equations are presented.

  8. Exact solutions to the Mo-Papas and Landau-Lifshitz equations

    NASA Astrophysics Data System (ADS)

    Rivera, R.; Villarroel, D.

    2002-10-01

    Two exact solutions of the Mo-Papas and Landau-Lifshitz equations for a point charge in classical electrodynamics are presented here. Both equations admit as an exact solution the motion of a charge rotating with constant speed in a circular orbit. These equations also admit as an exact solution the motion of two identical charges rotating with constant speed at the opposite ends of a diameter. These exact solutions allow one to obtain, starting from the equation of motion, a definite formula for the rate of radiation. In both cases the rate of radiation can also be obtained, with independence of the equation of motion, from the well known fields of a point charge, that is, from the Maxwell equations. The rate of radiation obtained from the Mo-Papas equation in the one-charge case coincides with the rate of radiation that comes from the Maxwell equations; but in the two-charge case the results do not coincide. On the other hand, the rate of radiation obtained from the Landau-Lifshitz equation differs from the one that follows from the Maxwell equations in both the one-charge and two-charge cases. This last result does not support a recent statement by Rohrlich in favor of considering the Landau-Lifshitz equation as the correct and exact equation of motion for a point charge in classical electrodynamics.

  9. Study of coupled nonlinear partial differential equations for finding exact analytical solutions

    PubMed Central

    Khan, Kamruzzaman; Akbar, M. Ali; Koppelaar, H.

    2015-01-01

    Exact solutions of nonlinear partial differential equations (NPDEs) are obtained via the enhanced (G′/G)-expansion method. The method is subsequently applied to find exact solutions of the Drinfel'd–Sokolov–Wilson (DSW) equation and the (2+1)-dimensional Painlevé integrable Burgers (PIB) equation. The efficiency of this method for finding these exact solutions is demonstrated. The method is effective and applicable for many other NPDEs in mathematical physics. PMID:26587256

  10. Microscopic derivation of particle-based coarse-grained dynamics: Exact expression for memory function

    NASA Astrophysics Data System (ADS)

    Izvekov, Sergei

    2017-03-01

    We consider the generalized Langevin equations of motion describing exactly the particle-based coarse-grained dynamics in the classical microscopic ensemble that were derived recently within the Mori-Zwanzig formalism based on new projection operators [S. Izvekov, J. Chem. Phys. 138(13), 134106 (2013)]. The fundamental difference between the new family of projection operators and the standard Zwanzig projection operator used in the past to derive the coarse-grained equations of motion is that the new operators average out the explicit irrelevant trajectories leading to the possibility of solving the projected dynamics exactly. We clarify the definition of the projection operators and revisit the formalism to compute the projected dynamics exactly for the microscopic system in equilibrium. The resulting expression for the projected force is in the form of a "generalized additive fluctuating force" describing the departure of the generalized microscopic force associated with the coarse-grained coordinate from its projection. Starting with this key expression, we formulate a new exact formula for the memory function in terms of microscopic and coarse-grained conservative forces. We conclude by studying two independent limiting cases of practical importance: the Markov limit (vanishing correlations of projected force) and the limit of weak dependence of the memory function on the particle momenta. We present computationally affordable expressions which can be efficiently evaluated from standard molecular dynamics simulations.

  11. Exact solution of a linear molecular motor model driven by two-step fluctuations and subject to protein friction.

    PubMed

    Fogedby, Hans C; Metzler, Ralf; Svane, Axel

    2004-08-01

    We investigate by analytical means the stochastic equations of motion of a linear molecular motor model based on the concept of protein friction. Solving the coupled Langevin equations originally proposed by Mogilner et al. [Phys. Lett. A 237, 297 (1998)], and averaging over both the two-step internal conformational fluctuations and the thermal noise, we present explicit, analytical expressions for the average motion and the velocity-force relationship. Our results allow for a direct interpretation of details of this motor model which are not readily accessible from numerical solutions. In particular, we find that the model is able to predict physiologically reasonable values for the load-free motor velocity and the motor mobility.

  12. A Well-Balanced Path-Integral f-Wave Method for Hyperbolic Problems with Source Terms

    PubMed Central

    2014-01-01

    Systems of hyperbolic partial differential equations with source terms (balance laws) arise in many applications where it is important to compute accurate time-dependent solutions modeling small perturbations of equilibrium solutions in which the source terms balance the hyperbolic part. The f-wave version of the wave-propagation algorithm is one approach, but requires the use of a particular averaged value of the source terms at each cell interface in order to be “well balanced” and exactly maintain steady states. A general approach to choosing this average is developed using the theory of path conservative methods. A scalar advection equation with a decay or growth term is introduced as a model problem for numerical experiments. PMID:24563581

  13. FAST TRACK COMMUNICATION Time-dependent exact solutions of the nonlinear Kompaneets equation

    NASA Astrophysics Data System (ADS)

    Ibragimov, N. H.

    2010-12-01

    Time-dependent exact solutions of the Kompaneets photon diffusion equation are obtained for several approximations of this equation. One of the approximations describes the case when the induced scattering is dominant. In this case, the Kompaneets equation has an additional symmetry which is used for constructing some exact solutions as group invariant solutions.

  14. Multidimensional discrete compactons in nonlinear Schrödinger lattices with strong nonlinearity management

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

    D'Ambroise, J.; Salerno, M.; Kevrekidis, P. G.

    The existence of multidimensional lattice compactons in the discrete nonlinear Schrödinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. By averaging over the period of the fast modulations, an effective averaged dynamical equation arises with coupling constants involving Bessel functions of the first and zeroth kinds. We show that these terms allow one to solve, at this averaged level, for exact discrete compacton solution configurations in the corresponding stationary equation. We focus on seven types of compacton solutions. Single-site and vortex solutions are found to be always stable in the parametric regimes we examined.more » We also found that other solutions such as double-site in- and out-of-phase, four-site symmetric and antisymmetric, and a five-site compacton solution are found to have regions of stability and instability in two-dimensional parametric planes, involving variations of the strength of the coupling and of the nonlinearity. We also explore the time evolution of the solutions and compare the dynamics according to the averaged equations with those of the original dynamical system. Finally, the possible observation of compactons in Bose-Einstein condensates loaded in a deep two-dimensional optical lattice with interactions modulated periodically in time is also discussed.« less

  15. Multidimensional discrete compactons in nonlinear Schrödinger lattices with strong nonlinearity management

    DOE PAGES

    D'Ambroise, J.; Salerno, M.; Kevrekidis, P. G.; ...

    2015-11-19

    The existence of multidimensional lattice compactons in the discrete nonlinear Schrödinger equation in the presence of fast periodic time modulations of the nonlinearity is demonstrated. By averaging over the period of the fast modulations, an effective averaged dynamical equation arises with coupling constants involving Bessel functions of the first and zeroth kinds. We show that these terms allow one to solve, at this averaged level, for exact discrete compacton solution configurations in the corresponding stationary equation. We focus on seven types of compacton solutions. Single-site and vortex solutions are found to be always stable in the parametric regimes we examined.more » We also found that other solutions such as double-site in- and out-of-phase, four-site symmetric and antisymmetric, and a five-site compacton solution are found to have regions of stability and instability in two-dimensional parametric planes, involving variations of the strength of the coupling and of the nonlinearity. We also explore the time evolution of the solutions and compare the dynamics according to the averaged equations with those of the original dynamical system. Finally, the possible observation of compactons in Bose-Einstein condensates loaded in a deep two-dimensional optical lattice with interactions modulated periodically in time is also discussed.« less

  16. On symmetries, conservation laws and exact solutions of the nonlinear Schrödinger-Hirota equation

    NASA Astrophysics Data System (ADS)

    Akbulut, Arzu; Taşcan, Filiz

    2018-04-01

    In this paper, conservation laws and exact solution are found for nonlinear Schrödinger-Hirota equation. Conservation theorem is used for finding conservation laws. We get modified conservation laws for given equation. Modified simple equation method is used to obtain the exact solutions of the nonlinear Schrödinger-Hirota equation. It is shown that the suggested method provides a powerful mathematical instrument for solving nonlinear equations in mathematical physics and engineering.

  17. A multi-dimensional nonlinearly implicit, electromagnetic Vlasov-Darwin particle-in-cell (PIC) algorithm

    NASA Astrophysics Data System (ADS)

    Chen, Guangye; Chacón, Luis; CoCoMans Team

    2014-10-01

    For decades, the Vlasov-Darwin model has been recognized to be attractive for PIC simulations (to avoid radiative noise issues) in non-radiative electromagnetic regimes. However, the Darwin model results in elliptic field equations that renders explicit time integration unconditionally unstable. Improving on linearly implicit schemes, fully implicit PIC algorithms for both electrostatic and electromagnetic regimes, with exact discrete energy and charge conservation properties, have been recently developed in 1D. This study builds on these recent algorithms to develop an implicit, orbit-averaged, time-space-centered finite difference scheme for the particle-field equations in multiple dimensions. The algorithm conserves energy, charge, and canonical-momentum exactly, even with grid packing. A simple fluid preconditioner allows efficient use of large timesteps, O (√{mi/me}c/veT) larger than the explicit CFL. We demonstrate the accuracy and efficiency properties of the of the algorithm with various numerical experiments in 2D3V.

  18. A simple extension of Roe's scheme for real gases

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

    Arabi, Sina, E-mail: sina.arabi@polymtl.ca; Trépanier, Jean-Yves; Camarero, Ricardo

    The purpose of this paper is to develop a highly accurate numerical algorithm to model real gas flows in local thermodynamic equilibrium (LTE). The Euler equations are solved using a finite volume method based on Roe's flux difference splitting scheme including real gas effects. A novel algorithm is proposed to calculate the Jacobian matrix which satisfies the flux difference splitting exactly in the average state for a general equation of state. This algorithm increases the robustness and accuracy of the method, especially around the contact discontinuities and shock waves where the gas properties jump appreciably. The results are compared withmore » an exact solution of the Riemann problem for the shock tube which considers the real gas effects. In addition, the method is applied to a blunt cone to illustrate the capability of the proposed extension in solving two dimensional flows.« less

  19. Given a one-step numerical scheme, on which ordinary differential equations is it exact?

    NASA Astrophysics Data System (ADS)

    Villatoro, Francisco R.

    2009-01-01

    A necessary condition for a (non-autonomous) ordinary differential equation to be exactly solved by a one-step, finite difference method is that the principal term of its local truncation error be null. A procedure to determine some ordinary differential equations exactly solved by a given numerical scheme is developed. Examples of differential equations exactly solved by the explicit Euler, implicit Euler, trapezoidal rule, second-order Taylor, third-order Taylor, van Niekerk's second-order rational, and van Niekerk's third-order rational methods are presented.

  20. A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrödinger equation

    NASA Astrophysics Data System (ADS)

    Ghanbari, Behzad; Inc, Mustafa

    2018-04-01

    The present paper suggests a novel technique to acquire exact solutions of nonlinear partial differential equations. The main idea of the method is to generalize the exponential rational function method. In order to examine the ability of the method, we consider the resonant nonlinear Schrödinger equation (R-NLSE). Many variants of exact soliton solutions for the equation are derived by the proposed method. Physical interpretations of some obtained solutions is also included. One can easily conclude that the new proposed method is very efficient and finds the exact solutions of the equation in a relatively easy way.

  1. Study of analytical method to seek for exact solutions of variant Boussinesq equations.

    PubMed

    Khan, Kamruzzaman; Akbar, M Ali

    2014-01-01

    In this paper, we have been acquired the soliton solutions of the Variant Boussinesq equations. Primarily, we have used the enhanced (G'/G)-expansion method to find exact solutions of Variant Boussinesq equations. Then, we attain some exact solutions including soliton solutions, hyperbolic and trigonometric function solutions of this equation. 35 K99; 35P05; 35P99.

  2. Nonhyperbolic reflection moveout for horizontal transverse isotropy

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

    Al-Dajani, A.F.; Tsvankin, I.

    1998-09-01

    The transversely isotropic model with a horizontal axis of symmetry (HTI) has been used extensively in studies of shear-wave splitting to describe fractured formations with a single system of parallel vertical penny-shaped cracks. Here, the authors present an analytic description of long-spread reflection moveout in horizontally layered HTI media with arbitrary strength of anisotropy. To account for nonhyperbolic moveout, the authors have derived an exact expression for the azimuthally dependent quartic term of the Taylor series traveltime expansion valid for any pure mode in an HTI layer. The quartic moveout coefficient and the NMO velocity are then substituted into themore » nonhyperbolic moveout equation of Tsvankin and Thomsen, originally designed for vertical transverse isotropy (VTI). In multilayered HTI media, the NMO velocity and the quartic moveout coefficient reflect the influence of layering as well as azimuthal anisotropy. The authors show that the conventional Dix equation for NMO velocity remains entirely valid for any azimuth in HTI media if the group-velocity vectors (rays) for data in a common-midpoint (CMP) gather do not deviate from the vertical incidence plane. Although this condition is not exactly satisfied in the presence of azimuthal velocity variations, rms averaging of the interval NMO velocities represents a good approximation for models with moderate azimuthal anisotropy. Furthermore, the quartic moveout coefficient for multilayered HTI media can also be calculated with acceptable accuracy using the known averaging equations for vertical transverse isotropy. This allows one to extend the nonhyperbolic moveout equation to horizontally stratified media composed of any combination of isotropic, VTI, and HTI layers.« less

  3. Corrigenda of 'explicit wave-averaged primitive equations using a generalized Lagrangian Mean'

    NASA Astrophysics Data System (ADS)

    Ardhuin, F.; Rascle, N.; Belibassakis, K. A.

    2017-05-01

    Ardhuin et al. (2008) gave a second-order approximation in the wave slope of the exact Generalized Lagrangian Mean (GLM) equations derived by Andrews and McIntyre (1978), and also performed a coordinate transformation, going from GLM to a 'GLMz' set of equations. That latter step removed the wandering of the GLM mean sea level away from the Eulerian-mean sea level, making the GLMz flow non-divergent. That step contained some inaccuarate statements about the coordinate transformation, while the rest of the paper contained an error on the surface dynamic boundary condition for viscous stresses. I am thankful to Mathias Delpey and Hidenori Aiki for pointing out these errors, which are corrected below.

  4. Approach to first-order exact solutions of the Ablowitz-Ladik equation.

    PubMed

    Ankiewicz, Adrian; Akhmediev, Nail; Lederer, Falk

    2011-05-01

    We derive exact solutions of the Ablowitz-Ladik (A-L) equation using a special ansatz that linearly relates the real and imaginary parts of the complex function. This ansatz allows us to derive a family of first-order solutions of the A-L equation with two independent parameters. This novel technique shows that every exact solution of the A-L equation has a direct analog among first-order solutions of the nonlinear Schrödinger equation (NLSE). © 2011 American Physical Society

  5. Equation-of-State Scaling Factors

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

    Scannapieco, Anthony J.

    2016-06-28

    Equation-of-State scaling factors are needed when using a tabular EOS in which the user de ned material isotopic fractions di er from the actual isotopic fractions used by the table. Additionally, if a material is dynamically changing its isotopic structure, then an EOS scaling will again be needed, and will vary in time and location. The procedure that allows use of a table to obtain information about a similar material with average atomic mass Ms and average atomic number Zs is described below. The procedure is exact for a fully ionized ideal gas. However, if the atomic number is replacemore » by the e ective ionization state the procedure can be applied to partially ionized material as well, which extends the applicability of the scaling approximation continuously from low to high temperatures.« less

  6. A generalized simplest equation method and its application to the Boussinesq-Burgers equation.

    PubMed

    Sudao, Bilige; Wang, Xiaomin

    2015-01-01

    In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.

  7. A Generalized Simplest Equation Method and Its Application to the Boussinesq-Burgers Equation

    PubMed Central

    Sudao, Bilige; Wang, Xiaomin

    2015-01-01

    In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method. PMID:25973605

  8. Propagation of mechanical waves through a stochastic medium with spherical symmetry

    NASA Astrophysics Data System (ADS)

    Avendaño, Carlos G.; Reyes, J. Adrián

    2018-01-01

    We theoretically analyze the propagation of outgoing mechanical waves through an infinite isotropic elastic medium possessing spherical symmetry whose Lamé coefficients and density are spatial random functions characterized by well-defined statistical parameters. We derive the differential equation that governs the average displacement for a system whose properties depend on the radial coordinate. We show that such an equation is an extended version of the well-known Bessel differential equation whose perturbative additional terms contain coefficients that depend directly on the squared noise intensities and the autocorrelation lengths in an exponential decay fashion. We numerically solve the second order differential equation for several values of noise intensities and autocorrelation lengths and compare the corresponding displacement profiles with that of the exact analytic solution for the case of absent inhomogeneities.

  9. Exact finite difference schemes for the non-linear unidirectional wave equation

    NASA Technical Reports Server (NTRS)

    Mickens, R. E.

    1985-01-01

    Attention is given to the construction of exact finite difference schemes for the nonlinear unidirectional wave equation that describes the nonlinear propagation of a wave motion in the positive x-direction. The schemes constructed for these equations are compared with those obtained by using the usual procedures of numerical analysis. It is noted that the order of the exact finite difference models is equal to the order of the differential equation.

  10. The mu-derivative and its applications to finding exact solutions of the Cahn-Hilliard, Korteveg-de Vries, and Burgers equations.

    PubMed

    Mitlin, Vlad

    2005-10-15

    A new transformation termed the mu-derivative is introduced. Applying it to the Cahn-Hilliard equation yields dynamical exact solutions. It is shown that the mu-transformed Cahn-Hilliard equation can be presented in a separable form. This transformation also yields dynamical exact solutions and separable forms for other nonlinear models such as the modified Korteveg-de Vries and the Burgers equations. The general structure of a nonlinear partial differential equation that becomes separable upon applying the mu-derivative is described.

  11. Exact solutions and low-frequency instability of the adiabatic auroral arc model

    NASA Technical Reports Server (NTRS)

    Cornwall, John M.

    1988-01-01

    The adiabatic auroral arc model couples a kinetic theory parallel current driven by mirror forces to horizontal ionospheric currents; the resulting equations are nonlinear. Some exact stationary solutions to these equations, some of them based on the Liouville equation, are developed, with both latitudinal and longitudinal spatial variations. These Liouville equation exact solutions are related to stability boundaries of low-frequency instabilities such as Kelvin-Helmholtz, as shown by a study of a simplified model.

  12. Exact solutions to the time-fractional differential equations via local fractional derivatives

    NASA Astrophysics Data System (ADS)

    Guner, Ozkan; Bekir, Ahmet

    2018-01-01

    This article utilizes the local fractional derivative and the exp-function method to construct the exact solutions of nonlinear time-fractional differential equations (FDEs). For illustrating the validity of the method, it is applied to the time-fractional Camassa-Holm equation and the time-fractional-generalized fifth-order KdV equation. Moreover, the exact solutions are obtained for the equations which are formed by different parameter values related to the time-fractional-generalized fifth-order KdV equation. This method is an reliable and efficient mathematical tool for solving FDEs and it can be applied to other non-linear FDEs.

  13. New exact solutions of the Tzitzéica-type equations in non-linear optics using the expa function method

    NASA Astrophysics Data System (ADS)

    Hosseini, K.; Ayati, Z.; Ansari, R.

    2018-04-01

    One specific class of non-linear evolution equations, known as the Tzitzéica-type equations, has received great attention from a group of researchers involved in non-linear science. In this article, new exact solutions of the Tzitzéica-type equations arising in non-linear optics, including the Tzitzéica, Dodd-Bullough-Mikhailov and Tzitzéica-Dodd-Bullough equations, are obtained using the expa function method. The integration technique actually suggests a useful and reliable method to extract new exact solutions of a wide range of non-linear evolution equations.

  14. Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

    NASA Astrophysics Data System (ADS)

    Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa

    2018-06-01

    In this work, we investigate the Lie symmetry analysis, exact solutions and conservation laws (Cls) to the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera (CDGDK) equation with Riemann-Liouville (RL) derivative. The time fractional CDGDK is reduced to nonlinear ordinary differential equation (ODE) of fractional order. New exact traveling wave solutions for the time fractional CDGDK are obtained by fractional sub-equation method. In the reduced equation, the derivative is in Erdelyi-Kober (EK) sense. Ibragimov's nonlocal conservation method is applied to construct Cls for time fractional CDGDK.

  15. On solutions of the fifth-order dispersive equations with porous medium type non-linearity

    NASA Astrophysics Data System (ADS)

    Kocak, Huseyin; Pinar, Zehra

    2018-07-01

    In this work, we focus on obtaining the exact solutions of the fifth-order semi-linear and non-linear dispersive partial differential equations, which have the second-order diffusion-like (porous-type) non-linearity. The proposed equations were not studied in the literature in the sense of the exact solutions. We reveal solutions of the proposed equations using the classical Riccati equations method. The obtained exact solutions, which can play a key role to simulate non-linear waves in the medium with dispersion and diffusion, are illustrated and discussed in details.

  16. Alternate solution to generalized Bernoulli equations via an integrating factor: an exact differential equation approach

    NASA Astrophysics Data System (ADS)

    Tisdell, C. C.

    2017-08-01

    Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem through a substitution. The purpose of this note is to present an alternative approach using 'exact methods', illustrating that a substitution and linearization of the problem is unnecessary. The ideas may be seen as forming a complimentary and arguably simpler approach to Azevedo and Valentino that have the potential to be assimilated and adapted to pedagogical needs of those learning and teaching exact differential equations in schools, colleges, universities and polytechnics. We illustrate how to apply the ideas through an analysis of the Gompertz equation, which is of interest in biomathematical models of tumour growth.

  17. New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods

    NASA Astrophysics Data System (ADS)

    S Saha, Ray

    2016-04-01

    In this paper, new exact solutions of the time fractional KdV-Khokhlov-Zabolotskaya-Kuznetsov (KdV-KZK) equation are obtained by the classical Kudryashov method and modified Kudryashov method respectively. For this purpose, the modified Riemann-Liouville derivative is used to convert the nonlinear time fractional KdV-KZK equation into the nonlinear ordinary differential equation. In the present analysis, the classical Kudryashov method and modified Kudryashov method are both used successively to compute the analytical solutions of the time fractional KdV-KZK equation. As a result, new exact solutions involving the symmetrical Fibonacci function, hyperbolic function and exponential function are obtained for the first time. The methods under consideration are reliable and efficient, and can be used as an alternative to establish new exact solutions of different types of fractional differential equations arising from mathematical physics. The obtained results are exhibited graphically in order to demonstrate the efficiencies and applicabilities of these proposed methods of solving the nonlinear time fractional KdV-KZK equation.

  18. More exact solutions of the constant astigmatism equation

    NASA Astrophysics Data System (ADS)

    Hlaváč, Adam

    2018-01-01

    By using Bäcklund transformation for the sine-Gordon equation, new periodic exact solutions of the constant astigmatism equation zyy +(1 / z) xx + 2 = 0 are generated from a seed which corresponds to Lipschitz surfaces of constant astigmatism.

  19. The Finite-Surface Method for incompressible flow: a step beyond staggered grid

    NASA Astrophysics Data System (ADS)

    Hokpunna, Arpiruk; Misaka, Takashi; Obayashi, Shigeru

    2017-11-01

    We present a newly developed higher-order finite surface method for the incompressible Navier-Stokes equations (NSE). This method defines the velocities as a surface-averaged value on the surfaces of the pressure cells. Consequently, the mass conservation on the pressure cells becomes an exact equation. The only things left to approximate is the momentum equation and the pressure at the new time step. At certain conditions, the exact mass conservation enables the explicit n-th order accurate NSE solver to be used with the pressure treatment that is two or four order less accurate without loosing the apparent convergence rate. This feature was not possible with finite volume of finite difference methods. We use Fourier analysis with a model spectrum to determine the condition and found that the range covers standard boundary layer flows. The formal convergence and the performance of the proposed scheme is compared with a sixth-order finite volume method. Finally, the accuracy and performance of the method is evaluated in turbulent channel flows. This work is partially funded by a research colloaboration from IFS, Tohoku university and ASEAN+3 funding scheme from CMUIC, Chiang Mai University.

  20. Dynamics of fictitious earth's satellites with possible past values of the ecliptic.

    NASA Astrophysics Data System (ADS)

    Callegari, N., Jr.; Yokoyama, T.; Marinho, E. P.

    The dynamics of some fictitious Earth satellites is studied considering initially, the secular perturbations due to Sun and the oblateness of the planet. With only these two disturbers, the problem is shown to be very similar to the case of fictitious satellites of Mars and Venus (already studied by one of us). Then, since the obliquities of the inner planets could have varied in a large interval, the eccentricities of the satellites were strongly disturbed, reaching some high prohibitive values. These values might have caused collision of the satellite with the planet or with some internal orbits. Moreover we can show that, provided that the semi major axis is near a critical value, a chaotic zone may appear and again the eccentricity can suffer very strong variation. However, as shown by Laskar et all (1993) in the case of Earth, large variation of the obliquity occurs only in the absence of the Moon. Although we do not know the origin and the past evolution of the M! oon, we decided to include it as a new disturber. Some information about past inclination of the Moon and the obliquity for several distances from the planet are given in Goldreich (1966), Touma and Wisdom (1993). The integration of the full equations, even for a massless satellite, takes too long computer time. In order to safe time and to get a preliminary idea, we derived the standard averaged equations for the gravitational part in the spatial case. However the present value of the Moon's mass is too much large for this first order averaged system. In spite of this, starting with some smaller masses and convenient ratio of semi major axes, it is shown that the qualitative behaviour of the averaged equations matches well with the results of the exact equation. Then it is easy to search and confirm the existence of some unstable regions where satellites placed on the equator in almost circular orbits can suffer very big increase of the eccentricity in a few hundred years. For the present value of the Moon's mass, only exact differential equations can be used and as expected, the previous region of instability not only persists but is much more pronounced for the present mass. Due to these unstable regions, satellites cannot survive in a large domain of the phase space.

  1. Some new traveling wave exact solutions of the (2+1)-dimensional Boiti-Leon-Pempinelli equations.

    PubMed

    Qi, Jian-ming; Zhang, Fu; Yuan, Wen-jun; Huang, Zi-feng

    2014-01-01

    We employ the complex method to obtain all meromorphic exact solutions of complex (2+1)-dimensional Boiti-Leon-Pempinelli equations (BLP system of equations). The idea introduced in this paper can be applied to other nonlinear evolution equations. Our results show that all rational and simply periodic traveling wave exact solutions of the equations (BLP) are solitary wave solutions, the complex method is simpler than other methods, and there exist some rational solutions ur,2 (z) and simply periodic solutions us,2-6(z) which are not only new but also not degenerated successively by the elliptic function solutions. We believe that this method should play an important role for finding exact solutions in the mathematical physics. For these new traveling wave solutions, we give some computer simulations to illustrate our main results.

  2. Alternate Solution to Generalized Bernoulli Equations via an Integrating Factor: An Exact Differential Equation Approach

    ERIC Educational Resources Information Center

    Tisdell, C. C.

    2017-01-01

    Solution methods to exact differential equations via integrating factors have a rich history dating back to Euler (1740) and the ideas enjoy applications to thermodynamics and electromagnetism. Recently, Azevedo and Valentino presented an analysis of the generalized Bernoulli equation, constructing a general solution by linearizing the problem…

  3. A procedure to construct exact solutions of nonlinear fractional differential equations.

    PubMed

    Güner, Özkan; Cevikel, Adem C

    2014-01-01

    We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

  4. Exact traveling-wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional Schrödinger equation with polynomial nonlinearity of arbitrary order.

    PubMed

    Petrović, Nikola Z; Belić, Milivoj; Zhong, Wei-Ping

    2011-02-01

    We obtain exact traveling wave and spatiotemporal soliton solutions to the generalized (3+1)-dimensional nonlinear Schrödinger equation with variable coefficients and polynomial Kerr nonlinearity of an arbitrarily high order. Exact solutions, given in terms of Jacobi elliptic functions, are presented for the special cases of cubic-quintic and septic models. We demonstrate that the widely used method for finding exact solutions in terms of Jacobi elliptic functions is not applicable to the nonlinear Schrödinger equation with saturable nonlinearity. ©2011 American Physical Society

  5. A new class of exact, nonlinear solutions to the Grad-Shafranov equation

    NASA Technical Reports Server (NTRS)

    Roumeliotis, George

    1993-01-01

    We have constructed a new class of exact, nonlinear solutions to the Grad-Shafranov equation, representing force-free magnetic fields with translational symmetry. These exact solutions are pertinent to the study of magnetic structures in the solar corona that are subjected to photospheric shearing motions.

  6. Studying relaxation phenomena via effective master equations

    NASA Astrophysics Data System (ADS)

    Chan, David; Wan, Jones T. K.; Chu, L. L.; Yu, K. W.

    2000-04-01

    The real-time dynamics of various relaxation phenomena can be conveniently formulated by a master equation with the enumeration of transition rates between given classes of conformations. To study the relaxation time towards equilibrium, it suffices to solve for the second largest eigenvalue of the resulting eigenvalue equation. Generally speaking, there is no analytic solution for the dynamic equation. Mean-field approaches generally yield misleading results while the presumably exact Monte-Carlo methods require prohibitive time steps in most real systems. In this work, we propose an exact decimation procedure for reducing the number of conformations significantly, while there is no loss of information, i.e., the reduced (or effective) equation is an exact transformed version of the original one. However, we have to pay the price: the initial Markovianity of the evolution equation is lost and the reduced equation contains memory terms in the transition rates. Since the transformed equation has significantly reduced number of degrees of freedom, the systems can readily be diagonalized by iterative means, to obtain the exact second largest eigenvalue and hence the relaxation time. The decimation method has been applied to various relaxation equations with generally desirable results. The advantages and limitations of the method will be discussed.

  7. New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod

    NASA Astrophysics Data System (ADS)

    Seadawy, Aly R.; Manafian, Jalil

    2018-03-01

    This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the longitudinal wave equation (LWE) that arises in mathematical physics with dispersion caused by the transverse Poisson's effect in a magneto-electro-elastic (MEE) circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method.

  8. Perturbed Coulomb Potentials in the Klein-Gordon Equation: Quasi-Exact Solution

    NASA Astrophysics Data System (ADS)

    Baradaran, M.; Panahi, H.

    2018-05-01

    Using the Lie algebraic approach, we present the quasi-exact solutions of the relativistic Klein-Gordon equation for perturbed Coulomb potentials namely the Cornell potential, the Kratzer potential and the Killingbeck potential. We calculate the general exact expressions for the energies, corresponding wave functions and the allowed values of the parameters of the potential within the representation space of sl(2) Lie algebra. In addition, we show that the considered equations can be transformed into the Heun's differential equations and then we reproduce the results using the associated special functions. Also, we study the special case of the Coulomb potential and show that in the non-relativistic limit, the solution of the Klein-Gordon equation converges to that of Schrödinger equation.

  9. Exact Solutions to Several Nonlinear Cases of Generalized Grad-Shafranov Equation for Ideal Magnetohydrodynamic Flows in Axisymmetric Domain

    NASA Astrophysics Data System (ADS)

    Adem, Abdullahi Rashid; Moawad, Salah M.

    2018-05-01

    In this paper, the steady-state equations of ideal magnetohydrodynamic incompressible flows in axisymmetric domains are investigated. These flows are governed by a second-order elliptic partial differential equation as a type of generalized Grad-Shafranov equation. The problem of finding exact equilibria to the full governing equations in the presence of incompressible mass flows is considered. Two different types of constraints on position variables are presented to construct exact solution classes for several nonlinear cases of the governing equations. Some of the obtained results are checked for their applications to magnetic confinement plasma. Besides, they cover many previous configurations and include new considerations about the nonlinearity of magnetic flux stream variables.

  10. Discreteness-induced concentration inversion in mesoscopic chemical systems.

    PubMed

    Ramaswamy, Rajesh; González-Segredo, Nélido; Sbalzarini, Ivo F; Grima, Ramon

    2012-04-10

    Molecular discreteness is apparent in small-volume chemical systems, such as biological cells, leading to stochastic kinetics. Here we present a theoretical framework to understand the effects of discreteness on the steady state of a monostable chemical reaction network. We consider independent realizations of the same chemical system in compartments of different volumes. Rate equations ignore molecular discreteness and predict the same average steady-state concentrations in all compartments. However, our theory predicts that the average steady state of the system varies with volume: if a species is more abundant than another for large volumes, then the reverse occurs for volumes below a critical value, leading to a concentration inversion effect. The addition of extrinsic noise increases the size of the critical volume. We theoretically predict the critical volumes and verify, by exact stochastic simulations, that rate equations are qualitatively incorrect in sub-critical volumes.

  11. Exact solution of the hidden Markov processes.

    PubMed

    Saakian, David B

    2017-11-01

    We write a master equation for the distributions related to hidden Markov processes (HMPs) and solve it using a functional equation. Thus the solution of HMPs is mapped exactly to the solution of the functional equation. For a general case the latter can be solved only numerically. We derive an exact expression for the entropy of HMPs. Our expression for the entropy is an alternative to the ones given before by the solution of integral equations. The exact solution is possible because actually the model can be considered as a generalized random walk on a one-dimensional strip. While we give the solution for the two second-order matrices, our solution can be easily generalized for the L values of the Markov process and M values of observables: We should be able to solve a system of L functional equations in the space of dimension M-1.

  12. Exact solution of the hidden Markov processes

    NASA Astrophysics Data System (ADS)

    Saakian, David B.

    2017-11-01

    We write a master equation for the distributions related to hidden Markov processes (HMPs) and solve it using a functional equation. Thus the solution of HMPs is mapped exactly to the solution of the functional equation. For a general case the latter can be solved only numerically. We derive an exact expression for the entropy of HMPs. Our expression for the entropy is an alternative to the ones given before by the solution of integral equations. The exact solution is possible because actually the model can be considered as a generalized random walk on a one-dimensional strip. While we give the solution for the two second-order matrices, our solution can be easily generalized for the L values of the Markov process and M values of observables: We should be able to solve a system of L functional equations in the space of dimension M -1 .

  13. A Procedure to Construct Exact Solutions of Nonlinear Fractional Differential Equations

    PubMed Central

    Güner, Özkan; Cevikel, Adem C.

    2014-01-01

    We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions. PMID:24737972

  14. On exact traveling-wave solutions for local fractional Korteweg-de Vries equation.

    PubMed

    Yang, Xiao-Jun; Tenreiro Machado, J A; Baleanu, Dumitru; Cattani, Carlo

    2016-08-01

    This paper investigates the Korteweg-de Vries equation within the scope of the local fractional derivative formulation. The exact traveling wave solutions of non-differentiable type with the generalized functions defined on Cantor sets are analyzed. The results for the non-differentiable solutions when fractal dimension is 1 are also discussed. It is shown that the exact solutions for the local fractional Korteweg-de Vries equation characterize the fractal wave on shallow water surfaces.

  15. Applications of exact traveling wave solutions of Modified Liouville and the Symmetric Regularized Long Wave equations via two new techniques

    NASA Astrophysics Data System (ADS)

    Lu, Dianchen; Seadawy, Aly R.; Ali, Asghar

    2018-06-01

    In this current work, we employ novel methods to find the exact travelling wave solutions of Modified Liouville equation and the Symmetric Regularized Long Wave equation, which are called extended simple equation and exp(-Ψ(ξ))-expansion methods. By assigning the different values to the parameters, different types of the solitary wave solutions are derived from the exact traveling wave solutions, which shows the efficiency and precision of our methods. Some solutions have been represented by graphical. The obtained results have several applications in physical science.

  16. Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers

    NASA Astrophysics Data System (ADS)

    Javeed, Shumaila; Saif, Summaya; Waheed, Asif; Baleanu, Dumitru

    2018-06-01

    The new exact solutions of nonlinear fractional partial differential equations (FPDEs) are established by adopting first integral method (FIM). The Riemann-Liouville (R-L) derivative and the local conformable derivative definitions are used to deal with the fractional order derivatives. The proposed method is applied to get exact solutions for space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation and coupled time-fractional Boussinesq-Burgers equation. The suggested technique is easily applicable and effectual which can be implemented successfully to obtain the solutions for different types of nonlinear FPDEs.

  17. Derivation of exact master equation with stochastic description: dissipative harmonic oscillator.

    PubMed

    Li, Haifeng; Shao, Jiushu; Wang, Shikuan

    2011-11-01

    A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation, and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipation characterized by the spectral density function. For a dissipative harmonic oscillator it is observed that the correlation between the stochastic field due to the bath and the system can be decoupled, and the master equation naturally results. Such an equation possesses the Lindblad form in which time-dependent coefficients are determined by a set of integral equations. It is proved that the obtained master equation is equivalent to the well-known Hu-Paz-Zhang equation based on the path-integral technique. The procedure is also used to obtain the master equation of a dissipative harmonic oscillator in time-dependent fields.

  18. Exact Solutions of Atmospheric (2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

    NASA Astrophysics Data System (ADS)

    Liu, Ping; Wang, Ya-Xiong; Ren, Bo; Li, Jin-Hua

    2016-12-01

    Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space. Supported by the National Natural Science Foundation of China under Grant Nos. 11305031 and 11305106, and Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No. Yq2013205

  19. Soliton and periodic solutions for time-dependent coefficient non-linear equation

    NASA Astrophysics Data System (ADS)

    Guner, Ozkan

    2016-01-01

    In this article, we establish exact solutions for the generalized (3+1)-dimensional variable coefficient Kadomtsev-Petviashvili (GVCKP) equation. Using solitary wave ansatz in terms of ? functions and the modified sine-cosine method, we find exact analytical bright soliton solutions and exact periodic solutions for the considered model. The physical parameters in the soliton solutions are obtained as function of the dependent model coefficients. The effectiveness and reliability of the method are shown by its application to the GVCKP equation.

  20. Studying the validity of relativistic hydrodynamics with a new exact solution of the Boltzmann equation

    NASA Astrophysics Data System (ADS)

    Denicol, Gabriel; Heinz, Ulrich; Martinez, Mauricio; Noronha, Jorge; Strickland, Michael

    2014-12-01

    We present an exact solution to the Boltzmann equation which describes a system undergoing boost-invariant longitudinal and azimuthally symmetric radial expansion for arbitrary shear viscosity to entropy density ratio. This new solution is constructed by considering the conformal map between Minkowski space and the direct product of three-dimensional de Sitter space with a line. The resulting solution respects S O (3 )q⊗S O (1 ,1 )⊗Z2 symmetry. We compare the exact kinetic solution with exact solutions of the corresponding macroscopic equations that were obtained from the kinetic theory in ideal and second-order viscous hydrodynamic approximations. The macroscopic solutions are obtained in de Sitter space and are subject to the same symmetries used to obtain the exact kinetic solution.

  1. Maximum likelihood estimation for periodic autoregressive moving average models

    USGS Publications Warehouse

    Vecchia, A.V.

    1985-01-01

    A useful class of models for seasonal time series that cannot be filtered or standardized to achieve second-order stationarity is that of periodic autoregressive moving average (PARMA) models, which are extensions of ARMA models that allow periodic (seasonal) parameters. An approximation to the exact likelihood for Gaussian PARMA processes is developed, and a straightforward algorithm for its maximization is presented. The algorithm is tested on several periodic ARMA(1, 1) models through simulation studies and is compared to moment estimation via the seasonal Yule-Walker equations. Applicability of the technique is demonstrated through an analysis of a seasonal stream-flow series from the Rio Caroni River in Venezuela.

  2. Modulated amplitude waves in collisionally inhomogeneous Bose Einstein condensates

    NASA Astrophysics Data System (ADS)

    Porter, Mason A.; Kevrekidis, P. G.; Malomed, Boris A.; Frantzeskakis, D. J.

    2007-05-01

    We investigate the dynamics of an effectively one-dimensional Bose-Einstein condensate (BEC) with scattering length a subjected to a spatially periodic modulation, a=a(x)=a(x+L). This “collisionally inhomogeneous” BEC is described by a Gross-Pitaevskii (GP) equation whose nonlinearity coefficient is a periodic function of x. We transform this equation into a GP equation with a constant coefficient and an additional effective potential and study a class of extended wave solutions of the transformed equation. For weak underlying inhomogeneity, the effective potential takes a form resembling a superlattice, and the amplitude dynamics of the solutions of the constant-coefficient GP equation obey a nonlinear generalization of the Ince equation. In the small-amplitude limit, we use averaging to construct analytical solutions for modulated amplitude waves (MAWs), whose stability we subsequently examine using both numerical simulations of the original GP equation and fixed-point computations with the MAWs as numerically exact solutions. We show that “on-site” solutions, whose maxima correspond to maxima of a(x), are more robust and likely to be observed than their “off-site” counterparts.

  3. Traveling wave and exact solutions for the perturbed nonlinear Schrödinger equation with Kerr law nonlinearity

    NASA Astrophysics Data System (ADS)

    Akram, Ghazala; Mahak, Nadia

    2018-06-01

    The nonlinear Schrödinger equation (NLSE) with the aid of three order dispersion terms is investigated to find the exact solutions via the extended (G'/G2)-expansion method and the first integral method. Many exact traveling wave solutions, such as trigonometric, hyperbolic, rational, soliton and complex function solutions, are characterized with some free parameters of the problem studied. It is corroborated that the proposed techniques are manageable, straightforward and powerful tools to find the exact solutions of nonlinear partial differential equations (PDEs). Some figures are plotted to describe the propagation of traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and rational functions.

  4. Modeling the heliolatitudinal gradient of the solar wind parameters with exact MHD solutions

    NASA Technical Reports Server (NTRS)

    Lima, J. J. G.; Tsinganos, K.

    1995-01-01

    The heliolatitudinal dependence of observations of the solar wind macroscopic quantities such as the averaged proton speed, density and the mass and momentum flux are modeled. The published observations covering the last two and a half solar cycles, are obtained either via the technique of interplanetary scintillations for the last 2 solar cycles (1970-1990), or, from the plasma experiment aboard the ULYSSES spacecraft for the recent period 1990-1994. Exact, two dimensional solutions of the full set of the steady MHD equations are used which are obtained through a nonlinear separation of the variables in the MHD equations. The three parameters emerging from the solutions are fixed from these observations, as well as from observations of the solar rotation. It is found that near solar maximum the solar wind speed is uniformly low, around the 400 km/s over a wide range of latitudes. On the other hand, during solar minimum and the declining phase of the solar activity cycle, there is a strong heliolatitudinal gradient in proton speed between 400-800 from equator to pole. This modeling also agrees with previous findings that the gradient in wind speed with the latitude is offset by a gradient in density such that the mass and momentum flux vary relatively little.

  5. New exact periodic solitary-wave solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in multi-temperature electron plasmas

    NASA Astrophysics Data System (ADS)

    Liu, Jian-Guo; Tian, Yu; Zeng, Zhi-Fang

    2017-10-01

    In this paper, we aim to introduce a new form of the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation for the long waves of small amplitude with slow dependence on the transverse coordinate. By using the Hirota's bilinear form and the extended homoclinic test approach, new exact periodic solitary-wave solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation are presented. Moreover, the properties and characteristics for these new exact periodic solitary-wave solutions are discussed with some figures.

  6. Kadomtsev−Petviashvili equation for a flow of highly nonisothermal collisionless plasma

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

    Movsesyants, Yu. B., E-mail: yumovsesyants@gmail.com; Rukhadze, A. A., E-mail: rukh@fpl.gpi.ru; Tyuryukanov, P. M.

    2016-01-15

    It is shown that the equations of two-fluid electrodynamics for a cold ions flow and Boltzmann electrons in the vicinity of the ion-sound point can be reduced to the Kadomtsev−Petviashvili equation. Examples of two-dimensional equilibria with pole singularities obtained by exactly solving the equations are presented. An exact self-similar solution describing a two-dimensional transonic flow and having no pole singularities is found.

  7. Kadomtsev-Petviashvili equation for a flow of highly nonisothermal collisionless plasma

    NASA Astrophysics Data System (ADS)

    Movsesyants, Yu. B.; Rukhadze, A. A.; Tyuryukanov, P. M.

    2016-01-01

    It is shown that the equations of two-fluid electrodynamics for a cold ions flow and Boltzmann electrons in the vicinity of the ion-sound point can be reduced to the Kadomtsev-Petviashvili equation. Examples of two-dimensional equilibria with pole singularities obtained by exactly solving the equations are presented. An exact self-similar solution describing a two-dimensional transonic flow and having no pole singularities is found.

  8. An improved two-dimensional depth-integrated flow equation for rough-walled fractures

    NASA Astrophysics Data System (ADS)

    Mallikamas, Wasin; Rajaram, Harihar

    2010-08-01

    We present the development of an improved 2-D flow equation for rough-walled fractures. Our improved equation accounts for the influence of midsurface tortuosity and the fact that the aperture normal to the midsurface is in general smaller than the vertical aperture. It thus improves upon the well-known Reynolds equation that is widely used for modeling flow in fractures. Unlike the Reynolds equation, our approach begins from the lubrication approximation applied in an inclined local coordinate system tangential to the fracture midsurface. The local flow equation thus obtained is rigorously transformed to an arbitrary global Cartesian coordinate system, invoking the concepts of covariant and contravariant transformations for vectors defined on surfaces. Unlike previously proposed improvements to the Reynolds equation, our improved flow equation accounts for tortuosity both along and perpendicular to a flow path. Our approach also leads to a well-defined anisotropic local transmissivity tensor relating the representations of the flux and head gradient vectors in a global Cartesian coordinate system. We show that the principal components of the transmissivity tensor and the orientation of its principal axes depend on the directional local midsurface slopes. In rough-walled fractures, the orientations of the principal axes of the local transmissivity tensor will vary from point to point. The local transmissivity tensor also incorporates the influence of the local normal aperture, which is uniquely defined at each point in the fracture. Our improved flow equation is a rigorous statement of mass conservation in any global Cartesian coordinate system. We present three examples of simple geometries to compare our flow equation to analytical solutions obtained using the exact Stokes equations: an inclined parallel plate, and circumferential and axial flows in an incomplete annulus. The effective transmissivities predicted by our flow equation agree very well with values obtained using the exact Stokes equations in all these cases. We discuss potential limitations of our depth-integrated equation, which include the neglect of convergence/divergence and the inaccuracies implicit in any depth-averaging process near sharp corners where the wall and midsurface curvatures are large.

  9. On the time-splitting scheme used in the Princeton Ocean Model

    NASA Astrophysics Data System (ADS)

    Kamenkovich, V. M.; Nechaev, D. A.

    2009-05-01

    The analysis of the time-splitting procedure implemented in the Princeton Ocean Model (POM) is presented. The time-splitting procedure uses different time steps to describe the evolution of interacting fast and slow propagating modes. In the general case the exact separation of the fast and slow modes is not possible. The main idea of the analyzed procedure is to split the system of primitive equations into two systems of equations for interacting external and internal modes. By definition, the internal mode varies slowly and the crux of the problem is to determine the proper filter, which excludes the fast component of the external mode variables in the relevant equations. The objective of this paper is to examine properties of the POM time-splitting procedure applied to equations governing the simplest linear non-rotating two-layer model of constant depth. The simplicity of the model makes it possible to study these properties analytically. First, the time-split system of differential equations is examined for two types of the determination of the slow component based on an asymptotic approach or time-averaging. Second, the differential-difference scheme is developed and some criteria of its stability are discussed for centered, forward, or backward time-averaging of the external mode variables. Finally, the stability of the POM time-splitting schemes with centered and forward time-averaging is analyzed. The effect of the Asselin filter on solutions of the considered schemes is studied. It is assumed that questions arising in the analysis of the simplest model are inherent in the general model as well.

  10. On the exact solutions of high order wave equations of KdV type (I)

    NASA Astrophysics Data System (ADS)

    Bulut, Hasan; Pandir, Yusuf; Baskonus, Haci Mehmet

    2014-12-01

    In this paper, by means of a proper transformation and symbolic computation, we study high order wave equations of KdV type (I). We obtained classification of exact solutions that contain soliton, rational, trigonometric and elliptic function solutions by using the extended trial equation method. As a result, the motivation of this paper is to utilize the extended trial equation method to explore new solutions of high order wave equation of KdV type (I). This method is confirmed by applying it to this kind of selected nonlinear equations.

  11. Some Exact Solutions of a Nonintegrable Toda-type Equation

    NASA Astrophysics Data System (ADS)

    Kim, Chanju

    2018-05-01

    We study a Toda-type equation with two scalar fields which is not integrable and construct two families of exact solutions which are expressed in terms of rational functions. The equation appears in U(1) Chern-Simons theories coupled to two nonrelativistic matter fields with opposite charges. One family of solutions is a trivial embedding of Liouville-type solutions. The other family is obtained by transforming the equation into the Taubes vortex equation on the hyperbolic space. Though the Taubes equation is not integrable, a trivial vacuum solution provides nontrivial solutions to the original Toda-type equation.

  12. Algebraic Construction of Exact Difference Equations from Symmetry of Equations

    NASA Astrophysics Data System (ADS)

    Itoh, Toshiaki

    2009-09-01

    Difference equations or exact numerical integrations, which have general solutions, are treated algebraically. Eliminating the symmetries of the equation, we can construct difference equations (DCE) or numerical integrations equivalent to some ODEs or PDEs that means both have the same solution functions. When arbitrary functions are given, whether we can construct numerical integrations that have solution functions equal to given function or not are treated in this work. Nowadays, Lie's symmetries solver for ODE and PDE has been implemented in many symbolic software. Using this solver we can construct algebraic DCEs or numerical integrations which are correspond to some ODEs or PDEs. In this work, we treated exact correspondence between ODE or PDE and DCE or numerical integration with Gröbner base and Janet base from the view of Lie's symmetries.

  13. Budget of Turbulent Kinetic Energy in a Shock Wave Boundary-Layer Interaction

    NASA Technical Reports Server (NTRS)

    Vyas, Manan; Waindim, Mbu; Gaitonde, Datta

    2016-01-01

    Implicit large-eddy simulation (ILES) of a shock wave boundary-layer interaction (SBLI) was performed. Quantities present in the exact equation of the turbulent kinetic energy (TKE) transport were accumulated. These quantities will be used to calculate the components of TKE-like production, dissipation, transport, and dilatation. Correlations of these terms will be presented to study the growth and interaction between various terms. A comparison with its RANS (Reynolds-Averaged Navier-Stokes) counterpart will also be presented.

  14. Explicit and exact nontraveling wave solutions of the (3+1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation

    NASA Astrophysics Data System (ADS)

    Yuan, Na

    2018-04-01

    With the aid of the symbolic computation, we present an improved ( G ‧ / G ) -expansion method, which can be applied to seek more types of exact solutions for certain nonlinear evolution equations. In illustration, we choose the (3 + 1)-dimensional potential Yu-Toda-Sasa-Fukuyama equation to demonstrate the validity and advantages of the method. As a result, abundant explicit and exact nontraveling wave solutions are obtained including two solitary waves solutions, nontraveling wave solutions and dromion soliton solutions. Some particular localized excitations and the interactions between two solitary waves are researched. The method can be also applied to other nonlinear partial differential equations.

  15. Conformally symmetric traversable wormholes

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

    Boehmer, Christian G.; Harko, Tiberiu; Lobo, Francisco S. N.

    2007-10-15

    Exact solutions of traversable wormholes are found under the assumption of spherical symmetry and the existence of a nonstatic conformal symmetry, which presents a more systematic approach in searching for exact wormhole solutions. In this work, a wide variety of solutions are deduced by considering choices for the form function, a specific linear equation of state relating the energy density and the pressure anisotropy, and various phantom wormhole geometries are explored. A large class of solutions impose that the spatial distribution of the exotic matter is restricted to the throat neighborhood, with a cutoff of the stress-energy tensor at amore » finite junction interface, although asymptotically flat exact solutions are also found. Using the 'volume integral quantifier', it is found that the conformally symmetric phantom wormhole geometries may, in principle, be constructed by infinitesimally small amounts of averaged null energy condition violating matter. Considering the tidal acceleration traversability conditions for the phantom wormhole geometry, specific wormhole dimensions and the traversal velocity are also deduced.« less

  16. Exact master equation and non-Markovian decoherence dynamics of Majorana zero modes under gate-induced charge fluctuations

    NASA Astrophysics Data System (ADS)

    Lai, Hon-Lam; Yang, Pei-Yun; Huang, Yu-Wei; Zhang, Wei-Min

    2018-02-01

    In this paper, we use the exact master equation approach to investigate the decoherence dynamics of Majorana zero modes in the Kitaev model, a 1D p -wave spinless topological superconducting chain (TSC) that is disturbed by gate-induced charge fluctuations. The exact master equation is derived by extending Feynman-Vernon influence functional technique to fermionic open systems involving pairing excitations. We obtain the exact master equation for the zero-energy Bogoliubov quasiparticle (bogoliubon) in the TSC, and then transfer it into the master equation for the Majorana zero modes. Within this exact master equation formalism, we can describe in detail the non-Markovian decoherence dynamics of the zero-energy bogoliubon as well as Majorana zero modes under local perturbations. We find that at zero temperature, local charge fluctuations induce level broadening to one of the Majorana zero modes but there is an isolated peak (localized bound state) located at zero energy that partially protects the Majorana zero mode from decoherence. At finite temperatures, the zero-energy localized bound state does not precisely exist, but the coherence of the Majorana zero mode can still be partially but weakly protected, due to the sharp dip of the spectral density near the zero frequency. The decoherence will be enhanced as one increases the charge fluctuations and/or the temperature of the gate.

  17. Modified method of simplest equation: Powerful tool for obtaining exact and approximate traveling-wave solutions of nonlinear PDEs

    NASA Astrophysics Data System (ADS)

    Vitanov, Nikolay K.

    2011-03-01

    We discuss the class of equations ∑i,j=0mAij(u){∂iu}/{∂ti}∂+∑k,l=0nBkl(u){∂ku}/{∂xk}∂=C(u) where Aij( u), Bkl( u) and C( u) are functions of u( x, t) as follows: (i) Aij, Bkl and C are polynomials of u; or (ii) Aij, Bkl and C can be reduced to polynomials of u by means of Taylor series for small values of u. For these two cases the above-mentioned class of equations consists of nonlinear PDEs with polynomial nonlinearities. We show that the modified method of simplest equation is powerful tool for obtaining exact traveling-wave solution of this class of equations. The balance equations for the sub-class of traveling-wave solutions of the investigated class of equations are obtained. We illustrate the method by obtaining exact traveling-wave solutions (i) of the Swift-Hohenberg equation and (ii) of the generalized Rayleigh equation for the cases when the extended tanh-equation or the equations of Bernoulli and Riccati are used as simplest equations.

  18. Classes of exact Einstein Maxwell solutions

    NASA Astrophysics Data System (ADS)

    Komathiraj, K.; Maharaj, S. D.

    2007-12-01

    We find new classes of exact solutions to the Einstein Maxwell system of equations for a charged sphere with a particular choice of the electric field intensity and one of the gravitational potentials. The condition of pressure isotropy is reduced to a linear, second order differential equation which can be solved in general. Consequently we can find exact solutions to the Einstein Maxwell field equations corresponding to a static spherically symmetric gravitational potential in terms of hypergeometric functions. It is possible to find exact solutions which can be written explicitly in terms of elementary functions, namely polynomials and product of polynomials and algebraic functions. Uncharged solutions are regainable with our choice of electric field intensity; in particular we generate the Einstein universe for particular parameter values.

  19. Assessment of the further improved (G'/G)-expansion method and the extended tanh-method in probing exact solutions of nonlinear PDEs.

    PubMed

    Akbar, M Ali; Ali, Norhashidah Hj Mohd; Mohyud-Din, Syed Tauseef

    2013-01-01

    The (G'/G)-expansion method is one of the most direct and effective method for obtaining exact solutions of nonlinear partial differential equations (PDEs). In the present article, we construct the exact traveling wave solutions of nonlinear evolution equations in mathematical physics via the (2 + 1)-dimensional breaking soliton equation by using two methods: namely, a further improved (G'/G)-expansion method, where G(ξ) satisfies the auxiliary ordinary differential equation (ODE) [G'(ξ)](2) = p G (2)(ξ) + q G (4)(ξ) + r G (6)(ξ); p, q and r are constants and the well known extended tanh-function method. We demonstrate, nevertheless some of the exact solutions bring out by these two methods are analogous, but they are not one and the same. It is worth mentioning that the first method has not been exercised anybody previously which gives further exact solutions than the second one. PACS numbers 02.30.Jr, 05.45.Yv, 02.30.Ik.

  20. A Large Class of Exact Solutions to the One-Dimensional Schrodinger Equation

    ERIC Educational Resources Information Center

    Karaoglu, Bekir

    2007-01-01

    A remarkable property of a large class of functions is exploited to generate exact solutions to the one-dimensional Schrodinger equation. The method is simple and easy to implement. (Contains 1 table and 1 figure.)

  1. An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov-Darwin particle-in-cell algorithm

    NASA Astrophysics Data System (ADS)

    Chen, G.; Chacón, L.

    2014-10-01

    A recent proof-of-principle study proposes a nonlinear electrostatic implicit particle-in-cell (PIC) algorithm in one dimension (Chen et al., 2011). The algorithm employs a kinetically enslaved Jacobian-free Newton-Krylov (JFNK) method, and conserves energy and charge to numerical round-off. In this study, we generalize the method to electromagnetic simulations in 1D using the Darwin approximation to Maxwell's equations, which avoids radiative noise issues by ordering out the light wave. An implicit, orbit-averaged, time-space-centered finite difference scheme is employed in both the 1D Darwin field equations (in potential form) and the 1D-3V particle orbit equations to produce a discrete system that remains exactly charge- and energy-conserving. Furthermore, enabled by the implicit Darwin equations, exact conservation of the canonical momentum per particle in any ignorable direction is enforced via a suitable scattering rule for the magnetic field. We have developed a simple preconditioner that targets electrostatic waves and skin currents, and allows us to employ time steps O(√{mi /me } c /veT) larger than the explicit CFL. Several 1D numerical experiments demonstrate the accuracy, performance, and conservation properties of the algorithm. In particular, the scheme is shown to be second-order accurate, and CPU speedups of more than three orders of magnitude vs. an explicit Vlasov-Maxwell solver are demonstrated in the "cold" plasma regime (where kλD ≪ 1).

  2. Solutions of the cylindrical nonlinear Maxwell equations.

    PubMed

    Xiong, Hao; Si, Liu-Gang; Ding, Chunling; Lü, Xin-You; Yang, Xiaoxue; Wu, Ying

    2012-01-01

    Cylindrical nonlinear optics is a burgeoning research area which describes cylindrical electromagnetic wave propagation in nonlinear media. Finding new exact solutions for different types of nonlinearity and inhomogeneity to describe cylindrical electromagnetic wave propagation is of great interest and meaningful for theory and application. This paper gives exact solutions for the cylindrical nonlinear Maxwell equations and presents an interesting connection between the exact solutions for different cylindrical nonlinear Maxwell equations. We also provide some examples and discussion to show the application of the results we obtained. Our results provide the basis for solving complex systems of nonlinearity and inhomogeneity with simple systems.

  3. Density-Dependent Conformable Space-time Fractional Diffusion-Reaction Equation and Its Exact Solutions

    NASA Astrophysics Data System (ADS)

    Hosseini, Kamyar; Mayeli, Peyman; Bekir, Ahmet; Guner, Ozkan

    2018-01-01

    In this article, a special type of fractional differential equations (FDEs) named the density-dependent conformable fractional diffusion-reaction (DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the \\exp (-φ (\\varepsilon )) -expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.

  4. Exact solutions for STO and (3+1)-dimensional KdV-ZK equations using (G‧/G2) -expansion method

    NASA Astrophysics Data System (ADS)

    Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Ullah, Rahmat; Ahmed, Naveed; Khan, Umar

    This article deals with finding some exact solutions of nonlinear fractional differential equations (NLFDEs) by applying a relatively new method known as (G‧/G2) -expansion method. Solutions of space-time fractional Sharma-Tasso-Olever (STO) equation of fractional order and (3+1)-dimensional KdV-Zakharov Kuznetsov (KdV-ZK) equation of fractional order are reckoned to demonstrate the validity of this method. The fractional derivative version of modified Riemann-Liouville, linked with Fractional complex transform is employed to transform fractional differential equations into the corresponding ordinary differential equations.

  5. Analytical approach for the fractional differential equations by using the extended tanh method

    NASA Astrophysics Data System (ADS)

    Pandir, Yusuf; Yildirim, Ayse

    2018-07-01

    In this study, we consider analytical solutions of space-time fractional derivative foam drainage equation, the nonlinear Korteweg-de Vries equation with time and space-fractional derivatives and time-fractional reaction-diffusion equation by using the extended tanh method. The fractional derivatives are defined in the modified Riemann-Liouville context. As a result, various exact analytical solutions consisting of trigonometric function solutions, kink-shaped soliton solutions and new exact solitary wave solutions are obtained.

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

    Dubrovsky, V. G.; Topovsky, A. V.

    New exact solutions, nonstationary and stationary, of Veselov-Novikov (VN) equation in the forms of simple nonlinear and linear superpositions of arbitrary number N of exact special solutions u{sup (n)}, n= 1, Horizontal-Ellipsis , N are constructed via Zakharov and Manakov {partial_derivative}-dressing method. Simple nonlinear superpositions are represented up to a constant by the sums of solutions u{sup (n)} and calculated by {partial_derivative}-dressing on nonzero energy level of the first auxiliary linear problem, i.e., 2D stationary Schroedinger equation. It is remarkable that in the zero energy limit simple nonlinear superpositions convert to linear ones in the form of the sums ofmore » special solutions u{sup (n)}. It is shown that the sums u=u{sup (k{sub 1})}+...+u{sup (k{sub m})}, 1 Less-Than-Or-Slanted-Equal-To k{sub 1} < k{sub 2} < Horizontal-Ellipsis < k{sub m} Less-Than-Or-Slanted-Equal-To N of arbitrary subsets of these solutions are also exact solutions of VN equation. The presented exact solutions include as superpositions of special line solitons and also superpositions of plane wave type singular periodic solutions. By construction these exact solutions represent also new exact transparent potentials of 2D stationary Schroedinger equation and can serve as model potentials for electrons in planar structures of modern electronics.« less

  7. Complex strain fields

    NASA Astrophysics Data System (ADS)

    Bradshaw, P.

    Computational techniques for accounting for extra strain rates, abnormal distributions of delta-U/delta-y, fluctuating strain rates, and the effects of body forces in modeling shear flows are discussed. Consideration is given to simple shears where the extra strain rate does not affect turbulence, thin shear layers, moderately thin shear layers, and strongly distorted flows. Attention is given to formulations based on the exact transport equations for Reynolds stress as derived from the time-averaged Navier-Stokes equations. Extra strain rates arise from curvature, lateral divergence, and bulk compression, with Coriolis forces accounting for the first, intensification of the spanwise vorticity for the second, and compression or dilation of the shear layer producing the third. The curvature forces, e.g., buoyancy and Coriolis forces, are responsible for hurricanes and tornadoes.

  8. Reduction of the equation for lower hybrid waves in a plasma to a nonlinear Schroedinger equation

    NASA Technical Reports Server (NTRS)

    Karney, C. F. F.

    1977-01-01

    Equations describing the nonlinear propagation of waves in an anisotropic plasma are rarely exactly soluble. However it is often possible to make approximations that reduce the exact equations into a simpler equation. The use of MACSYMA to make such approximations, and so reduce the equation describing lower hybrid waves into the nonlinear Schrodinger equation which is soluble by the inverse scattering method is demonstrated. MACSYMA is used at several stages in the calculation only because there is a natural division between calculations that are easiest done by hand, and those that are easiest done by machine.

  9. Traveling wavefront solutions to nonlinear reaction-diffusion-convection equations

    NASA Astrophysics Data System (ADS)

    Indekeu, Joseph O.; Smets, Ruben

    2017-08-01

    Physically motivated modified Fisher equations are studied in which nonlinear convection and nonlinear diffusion is allowed for besides the usual growth and spread of a population. It is pointed out that in a large variety of cases separable functions in the form of exponentially decaying sharp wavefronts solve the differential equation exactly provided a co-moving point source or sink is active at the wavefront. The velocity dispersion and front steepness may differ from those of some previously studied exact smooth traveling wave solutions. For an extension of the reaction-diffusion-convection equation, featuring a memory effect in the form of a maturity delay for growth and spread, also smooth exact wavefront solutions are obtained. The stability of the solutions is verified analytically and numerically.

  10. Nonexistence of exact solutions agreeing with the Gaussian beam on the beam axis or in the focal plane

    NASA Astrophysics Data System (ADS)

    Lekner, John; Andrejic, Petar

    2018-01-01

    Solutions of the Helmholtz equation which describe electromagnetic beams (and also acoustic or particle beams) are discussed. We show that an exact solution which reproduces the Gaussian beam waveform on the beam axis does not exist. This is surprising, since the Gaussian beam is a solution of the paraxial equation, and thus supposedly accurate on and near the beam axis. Likewise, a solution of the Helmholtz equation which exactly reproduces the Gaussian beam in the focal plane does not exist. We show that the last statement also holds for Bessel-Gauss beams. However, solutions of the Helmholtz equation (one of which is discussed in detail) can approximate the Gaussian waveform within the central focal region.

  11. Exact linearized Coulomb collision operator in the moment expansion

    DOE PAGES

    Ji, Jeong -Young; Held, Eric D.

    2006-10-05

    In the moment expansion, the Rosenbluth potentials, the linearized Coulomb collision operators, and the moments of the collision operators are analytically calculated for any moment. The explicit calculation of Rosenbluth potentials converts the integro-differential form of the Coulomb collision operator into a differential operator, which enables one to express the collision operator in a simple closed form for any arbitrary mass and temperature ratios. In addition, it is shown that gyrophase averaging the collision operator acting on arbitrary distribution functions is the same as the collision operator acting on the corresponding gyrophase averaged distribution functions. The moments of the collisionmore » operator are linear combinations of the fluid moments with collision coefficients parametrized by mass and temperature ratios. Furthermore, useful forms involving the small mass-ratio approximation are easily found since the collision operators and their moments are expressed in terms of the mass ratio. As an application, the general moment equations are explicitly written and the higher order heat flux equation is derived.« less

  12. AN EXACT PEAK CAPTURING AND OSCILLATION-FREE SCHEME TO SOLVE ADVECTION-DISPERSION TRANSPORT EQUATIONS

    EPA Science Inventory

    An exact peak capturing and essentially oscillation-free (EPCOF) algorithm, consisting of advection-dispersion decoupling, backward method of characteristics, forward node tracking, and adaptive local grid refinement, is developed to solve transport equations. This algorithm repr...

  13. The rotational motion of an earth orbiting gyroscope according to the Einstein theory of general relativity

    NASA Technical Reports Server (NTRS)

    Hoots, F. R.; Fitzpatrick, P. M.

    1979-01-01

    The classical Poisson equations of rotational motion are used to study the attitude motions of an earth orbiting, rapidly spinning gyroscope perturbed by the effects of general relativity (Einstein theory). The center of mass of the gyroscope is assumed to move about a rotating oblate earth in an evolving elliptic orbit which includes all first-order oblateness effects produced by the earth. A method of averaging is used to obtain a transformation of variables, for the nonresonance case, which significantly simplifies the Poisson differential equations of motion of the gyroscope. Long-term solutions are obtained by an exact analytical integration of the simplified transformed equations. These solutions may be used to predict both the orientation of the gyroscope and the motion of its rotational angular momentum vector as viewed from its center of mass. The results are valid for all eccentricities and all inclinations not near the critical inclination.

  14. A general mixture theory. I. Mixtures of spherical molecules

    NASA Astrophysics Data System (ADS)

    Hamad, Esam Z.

    1996-08-01

    We present a new general theory for obtaining mixture properties from the pure species equations of state. The theory addresses the composition and the unlike interactions dependence of mixture equation of state. The density expansion of the mixture equation gives the exact composition dependence of all virial coefficients. The theory introduces multiple-index parameters that can be calculated from binary unlike interaction parameters. In this first part of the work, details are presented for the first and second levels of approximations for spherical molecules. The second order model is simple and very accurate. It predicts the compressibility factor of additive hard spheres within simulation uncertainty (equimolar with size ratio of three). For nonadditive hard spheres, comparison with compressibility factor simulation data over a wide range of density, composition, and nonadditivity parameter, gave an average error of 2%. For mixtures of Lennard-Jones molecules, the model predictions are better than the Weeks-Chandler-Anderson perturbation theory.

  15. Average expansion rate and light propagation in a cosmological Tardis spacetime

    NASA Astrophysics Data System (ADS)

    Lavinto, Mikko; Räsänen, Syksy; Szybka, Sebastian J.

    2013-12-01

    We construct the first exact statistically homogeneous and isotropic cosmological solution in which inhomogeneity has a significant effect on the expansion rate. The universe is modelled as a Swiss Cheese, with dust FRW background and inhomogeneous holes. We show that if the holes are described by the quasispherical Szekeres solution, their average expansion rate is close to the background under certain rather general conditions. We specialise to spherically symmetric holes and violate one of these conditions. As a result, the average expansion rate at late times grows relative to the background, ie backreaction is significant. The holes fit smoothly into the background, but are larger on the inside than a corresponding background domain: we call them Tardis regions. We study light propagation, find the effective equations of state and consider the relation of the spatially averaged expansion rate to the redshift and the angular diameter distance.

  16. Elementary exact calculations of degree growth and entropy for discrete equations.

    PubMed

    Halburd, R G

    2017-05-01

    Second-order discrete equations are studied over the field of rational functions [Formula: see text], where z is a variable not appearing in the equation. The exact degree of each iterate as a function of z can be calculated easily using the standard calculations that arise in singularity confinement analysis, even when the singularities are not confined. This produces elementary yet rigorous entropy calculations.

  17. Exact periodic solutions of the sixth-order generalized Boussinesq equation

    NASA Astrophysics Data System (ADS)

    Kamenov, O. Y.

    2009-09-01

    This paper examines a class of nonlinear sixth-order generalized Boussinesq-like equations (SGBE): utt = uxx + 3(u2)xx + uxxxx + αuxxxxxx, α in R, depending on the positive parameter α. Hirota's bilinear transformation method is applied to the above class of non-integrable equations and exact periodic solutions have been obtained. The results confirmed the well-known nonlinear superposition principle.

  18. Extension of the KLI approximation toward the exact optimized effective potential.

    PubMed

    Iafrate, G J; Krieger, J B

    2013-03-07

    The integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spin-unrestricted exchange-correlation potential, Vxcσ, is obtained for any assumed orbital-dependent exchange-correlation energy functional. The method extends beyond the Krieger-Li-Iafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent first-order perturbation theory wavefunction often referred to as the "orbital shift" function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with "sum over states" numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchange-correlation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP Vxcσ, thus allowing for the reduction of the OEP equation to a self-consistent integral equation for the exact exchange-correlation potential; the exact solution to this integral equation can be determined by iteration with the natural zeroth order correction given by the KLI exchange-correlation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbital-dependent exchange-correlation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

  19. Extension of the KLI approximation toward the exact optimized effective potential

    NASA Astrophysics Data System (ADS)

    Iafrate, G. J.; Krieger, J. B.

    2013-03-01

    The integral equation for the optimized effective potential (OEP) is utilized in a compact form from which an accurate OEP solution for the spin-unrestricted exchange-correlation potential, Vxcσ, is obtained for any assumed orbital-dependent exchange-correlation energy functional. The method extends beyond the Krieger-Li-Iafrate (KLI) approximation toward the exact OEP result. The compact nature of the OEP equation arises by replacing the integrals involving the Green's function terms in the traditional OEP equation by an equivalent first-order perturbation theory wavefunction often referred to as the "orbital shift" function. Significant progress is then obtained by solving the equation for the first order perturbation theory wavefunction by use of Dalgarno functions which are determined from well known methods of partial differential equations. The use of Dalgarno functions circumvents the need to explicitly address the Green's functions and the associated problems with "sum over states" numerics; as well, the Dalgarno functions provide ease in dealing with inherent singularities arising from the origin and the zeros of the occupied orbital wavefunctions. The Dalgarno approach for finding a solution to the OEP equation is described herein, and a detailed illustrative example is presented for the special case of a spherically symmetric exchange-correlation potential. For the case of spherical symmetry, the relevant Dalgarno function is derived by direct integration of the appropriate radial equation while utilizing a user friendly method which explicitly treats the singular behavior at the origin and at the nodal singularities arising from the zeros of the occupied states. The derived Dalgarno function is shown to be an explicit integral functional of the exact OEP Vxcσ, thus allowing for the reduction of the OEP equation to a self-consistent integral equation for the exact exchange-correlation potential; the exact solution to this integral equation can be determined by iteration with the natural zeroth order correction given by the KLI exchange-correlation potential. Explicit analytic results are provided to illustrate the first order iterative correction beyond the KLI approximation. The derived correction term to the KLI potential explicitly involves spatially weighted products of occupied orbital densities in any assumed orbital-dependent exchange-correlation energy functional; as well, the correction term is obtained with no adjustable parameters. Moreover, if the equation for the exact optimized effective potential is further iterated, one can obtain the OEP as accurately as desired.

  20. Exactly Solvable Multidimensional Nonlinear Equations and Inverse Scattering,

    DTIC Science & Technology

    1986-12-01

    time dimension. Here the prototype euQation is 1 the Kadomtsev - Petviashvili (K-P) equation : .0 6u , x , x - )3,:’u ,’ which is the cop,patliil ity...AD-R193 274 EXACTLY SOLVABLE MULTIDIMENSIONAL NONLINEAR EQUATIONS L/1 AND INVERSE SCATTERING(U) CLARKSON UNIV POTSDAM MY A J MBLOUITZ DEC 86 NSOSI4...ecuations by associating thnm with appropriate compatible linear equations , -ne of which is identified as a Scattering prooD,, ne others(s) serves to

  1. Bright and singular soliton solutions of the conformable time-fractional Klein-Gordon equations with different nonlinearities

    NASA Astrophysics Data System (ADS)

    Hosseini, Kamyar; Mayeli, Peyman; Ansari, Reza

    2018-07-01

    Finding the exact solutions of nonlinear fractional differential equations has gained considerable attention, during the past two decades. In this paper, the conformable time-fractional Klein-Gordon equations with quadratic and cubic nonlinearities are studied. Several exact soliton solutions, including the bright (non-topological) and singular soliton solutions are formally extracted by making use of the ansatz method. Results demonstrate that the method can efficiently handle the time-fractional Klein-Gordon equations with different nonlinearities.

  2. Continual Lie algebras and noncommutative counterparts of exactly solvable models

    NASA Astrophysics Data System (ADS)

    Zuevsky, A.

    2004-01-01

    Noncommutative counterparts of exactly solvable models are introduced on the basis of a generalization of Saveliev-Vershik continual Lie algebras. Examples of noncommutative Liouville and sin/h-Gordon equations are given. The simplest soliton solution to the noncommutative sine-Gordon equation is found.

  3. Exact solutions for postbuckling of a graded porous beam

    NASA Astrophysics Data System (ADS)

    Ma, L. S.; Ou, Z. Y.

    2018-06-01

    An exact, closed-form solution for the postbuckling responses of graded porous beams subjected to axially loading is obtained. It was assumed that the properties of the graded porous materials vary continuously through thickness of the beams, the equations governing the axial and transverse deformations are derived based on the classical beam theory and the physical neutral surface concept. The two equations are reduced to a single nonlinear fourth-order integral-differential equation governing the transverse deformations. The nonlinear equation is directly solved without any use of approximation and a closed-form solution for postbuckled deformation is obtained as a function of the applied load. The exact solutions explicitly describe the nonlinear equilibrium paths of the buckled beam and thus are able to provide insight into deformation problems. Based on the exact solutions obtained herein, the effects of various factors such as porosity distribution pattern, porosity coefficient and boundary conditions on postbuckling behavior of graded porous beams have been investigated.

  4. Collisionless tearing instability of a bi-Maxwellian neutral sheet - An integrodifferential treatment with exact particle orbits

    NASA Technical Reports Server (NTRS)

    Burkhart, G. R.; Chen, J.

    1989-01-01

    The integrodifferential equation describing the linear tearing instability in the bi-Maxwellian neutral sheet is solved without approximating the particle orbits or the eigenfunction psi. Results of this calculation are presented. Comparison between the exact solution and the three-region approximation motivates the piecewise-straight-line approximation, a simplification that allows faster solution of the integrodifferential equation, yet retains the important features of the exact solution.

  5. Weight of fitness deviation governs strict physical chaos in replicator dynamics.

    PubMed

    Pandit, Varun; Mukhopadhyay, Archan; Chakraborty, Sagar

    2018-03-01

    Replicator equation-a paradigm equation in evolutionary game dynamics-mathematizes the frequency dependent selection of competing strategies vying to enhance their fitness (quantified by the average payoffs) with respect to the average fitnesses of the evolving population under consideration. In this paper, we deal with two discrete versions of the replicator equation employed to study evolution in a population where any two players' interaction is modelled by a two-strategy symmetric normal-form game. There are twelve distinct classes of such games, each typified by a particular ordinal relationship among the elements of the corresponding payoff matrix. Here, we find the sufficient conditions for the existence of asymptotic solutions of the replicator equations such that the solutions-fixed points, periodic orbits, and chaotic trajectories-are all strictly physical, meaning that the frequency of any strategy lies inside the closed interval zero to one at all times. Thus, we elaborate on which of the twelve types of games are capable of showing meaningful physical solutions and for which of the two types of replicator equation. Subsequently, we introduce the concept of the weight of fitness deviation that is the scaling factor in a positive affine transformation connecting two payoff matrices such that the corresponding one-shot games have exactly same Nash equilibria and evolutionary stable states. The weight also quantifies how much the excess of fitness of a strategy over the average fitness of the population affects the per capita change in the frequency of the strategy. Intriguingly, the weight's variation is capable of making the Nash equilibria and the evolutionary stable states, useless by introducing strict physical chaos in the replicator dynamics based on the normal-form game.

  6. Convergence of high order memory kernels in the Nakajima-Zwanzig generalized master equation and rate constants: Case study of the spin-boson model.

    PubMed

    Xu, Meng; Yan, Yaming; Liu, Yanying; Shi, Qiang

    2018-04-28

    The Nakajima-Zwanzig generalized master equation provides a formally exact framework to simulate quantum dynamics in condensed phases. Yet, the exact memory kernel is hard to obtain and calculations based on perturbative expansions are often employed. By using the spin-boson model as an example, we assess the convergence of high order memory kernels in the Nakajima-Zwanzig generalized master equation. The exact memory kernels are calculated by combining the hierarchical equation of motion approach and the Dyson expansion of the exact memory kernel. High order expansions of the memory kernels are obtained by extending our previous work to calculate perturbative expansions of open system quantum dynamics [M. Xu et al., J. Chem. Phys. 146, 064102 (2017)]. It is found that the high order expansions do not necessarily converge in certain parameter regimes where the exact kernel show a long memory time, especially in cases of slow bath, weak system-bath coupling, and low temperature. Effectiveness of the Padé and Landau-Zener resummation approaches is tested, and the convergence of higher order rate constants beyond Fermi's golden rule is investigated.

  7. Convergence of high order memory kernels in the Nakajima-Zwanzig generalized master equation and rate constants: Case study of the spin-boson model

    NASA Astrophysics Data System (ADS)

    Xu, Meng; Yan, Yaming; Liu, Yanying; Shi, Qiang

    2018-04-01

    The Nakajima-Zwanzig generalized master equation provides a formally exact framework to simulate quantum dynamics in condensed phases. Yet, the exact memory kernel is hard to obtain and calculations based on perturbative expansions are often employed. By using the spin-boson model as an example, we assess the convergence of high order memory kernels in the Nakajima-Zwanzig generalized master equation. The exact memory kernels are calculated by combining the hierarchical equation of motion approach and the Dyson expansion of the exact memory kernel. High order expansions of the memory kernels are obtained by extending our previous work to calculate perturbative expansions of open system quantum dynamics [M. Xu et al., J. Chem. Phys. 146, 064102 (2017)]. It is found that the high order expansions do not necessarily converge in certain parameter regimes where the exact kernel show a long memory time, especially in cases of slow bath, weak system-bath coupling, and low temperature. Effectiveness of the Padé and Landau-Zener resummation approaches is tested, and the convergence of higher order rate constants beyond Fermi's golden rule is investigated.

  8. Exact solution of the generalized Peierls equation for arbitrary n-fold screw dislocation

    NASA Astrophysics Data System (ADS)

    Wang, Shaofeng; Hu, Xiangsheng

    2018-05-01

    The exact solution of the generalized Peierls equation is presented and proved for arbitrary n-fold screw dislocation. The displacement field, stress field and the energy of the n-fold dislocation are also evaluated explicitly. It is found that the solution defined on each individual fold is given by the tail cut from the original Peierls solution. In viewpoint of energetics, a screw dislocation has a tendency to spread the distribution on all possible slip planes which are contained in the dislocation line zone. Based on the exact solution, the approximated solution of the improved Peierls equation is proposed for the modified γ-surface.

  9. A Numerical Scheme for Ordinary Differential Equations Having Time Varying and Nonlinear Coefficients Based on the State Transition Matrix

    NASA Technical Reports Server (NTRS)

    Bartels, Robert E.

    2002-01-01

    A variable order method of integrating initial value ordinary differential equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. While it is more complex than most other methods, it produces exact solutions at arbitrary time step size when the time variation of the system can be modeled exactly by a polynomial. Solutions to several nonlinear problems exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with an exact solution and with solutions obtained by established methods.

  10. Studies into the averaging problem: Macroscopic gravity and precision cosmology

    NASA Astrophysics Data System (ADS)

    Wijenayake, Tharake S.

    2016-08-01

    With the tremendous improvement in the precision of available astrophysical data in the recent past, it becomes increasingly important to examine some of the underlying assumptions behind the standard model of cosmology and take into consideration nonlinear and relativistic corrections which may affect it at percent precision level. Due to its mathematical rigor and fully covariant and exact nature, Zalaletdinov's macroscopic gravity (MG) is arguably one of the most promising frameworks to explore nonlinearities due to inhomogeneities in the real Universe. We study the application of MG to precision cosmology, focusing on developing a self-consistent cosmology model built on the averaging framework that adequately describes the large-scale Universe and can be used to study real data sets. We first implement an algorithmic procedure using computer algebra systems to explore new exact solutions to the MG field equations. After validating the process with an existing isotropic solution, we derive a new homogeneous, anisotropic and exact solution. Next, we use the simplest (and currently only) solvable homogeneous and isotropic model of MG and obtain an observable function for cosmological expansion using some reasonable assumptions on light propagation. We find that the principal modification to the angular diameter distance is through the change in the expansion history. We then linearize the MG field equations and derive a framework that contains large-scale structure, but the small scale inhomogeneities have been smoothed out and encapsulated into an additional cosmological parameter representing the averaging effect. We derive an expression for the evolution of the density contrast and peculiar velocities and integrate them to study the growth rate of large-scale structure. We find that increasing the magnitude of the averaging term leads to enhanced growth at late times. Thus, for the same matter content, the growth rate of large scale structure in the MG model is stronger than that of the standard model. Finally, we constrain the MG model using Cosmic Microwave Background temperature anisotropy data, the distance to supernovae data, the galaxy power spectrum, the weak lensing tomography shear-shear cross-correlations and the baryonic acoustic oscillations. We find that for this model the averaging density parameter is very small and does not cause any significant shift in the other cosmological parameters. However, it can lead to increased errors on some cosmological parameters such as the Hubble constant and the amplitude of the linear matter spectrum at the scale of 8h. {-1}Mpc. Further studiesare needed to explore other solutions and models of MG as well as their effects on precision cosmology.

  11. ADM For Solving Linear Second-Order Fredholm Integro-Differential Equations

    NASA Astrophysics Data System (ADS)

    Karim, Mohd F.; Mohamad, Mahathir; Saifullah Rusiman, Mohd; Che-Him, Norziha; Roslan, Rozaini; Khalid, Kamil

    2018-04-01

    In this paper, we apply Adomian Decomposition Method (ADM) as numerically analyse linear second-order Fredholm Integro-differential Equations. The approximate solutions of the problems are calculated by Maple package. Some numerical examples have been considered to illustrate the ADM for solving this equation. The results are compared with the existing exact solution. Thus, the Adomian decomposition method can be the best alternative method for solving linear second-order Fredholm Integro-Differential equation. It converges to the exact solution quickly and in the same time reduces computational work for solving the equation. The result obtained by ADM shows the ability and efficiency for solving these equations.

  12. New extended (G'/G)-expansion method to solve nonlinear evolution equation: the (3 + 1)-dimensional potential-YTSF equation.

    PubMed

    Roshid, Harun-Or-; Akbar, M Ali; Alam, Md Nur; Hoque, Md Fazlul; Rahman, Nizhum

    2014-01-01

    In this article, a new extended (G'/G) -expansion method has been proposed for constructing more general exact traveling wave solutions of nonlinear evolution equations with the aid of symbolic computation. In order to illustrate the validity and effectiveness of the method, we pick the (3 + 1)-dimensional potential-YTSF equation. As a result, abundant new and more general exact solutions have been achieved of this equation. It has been shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in applied mathematics, engineering and mathematical physics.

  13. Modeling of multi-band drift in nanowires using a full band Monte Carlo simulation

    NASA Astrophysics Data System (ADS)

    Hathwar, Raghuraj; Saraniti, Marco; Goodnick, Stephen M.

    2016-07-01

    We report on a new numerical approach for multi-band drift within the context of full band Monte Carlo (FBMC) simulation and apply this to Si and InAs nanowires. The approach is based on the solution of the Krieger and Iafrate (KI) equations [J. B. Krieger and G. J. Iafrate, Phys. Rev. B 33, 5494 (1986)], which gives the probability of carriers undergoing interband transitions subject to an applied electric field. The KI equations are based on the solution of the time-dependent Schrödinger equation, and previous solutions of these equations have used Runge-Kutta (RK) methods to numerically solve the KI equations. This approach made the solution of the KI equations numerically expensive and was therefore only applied to a small part of the Brillouin zone (BZ). Here we discuss an alternate approach to the solution of the KI equations using the Magnus expansion (also known as "exponential perturbation theory"). This method is more accurate than the RK method as the solution lies on the exponential map and shares important qualitative properties with the exact solution such as the preservation of the unitary character of the time evolution operator. The solution of the KI equations is then incorporated through a modified FBMC free-flight drift routine and applied throughout the nanowire BZ. The importance of the multi-band drift model is then demonstrated for the case of Si and InAs nanowires by simulating a uniform field FBMC and analyzing the average carrier energies and carrier populations under high electric fields. Numerical simulations show that the average energy of the carriers under high electric field is significantly higher when multi-band drift is taken into consideration, due to the interband transitions allowing carriers to achieve higher energies.

  14. 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.

  15. A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations.

    PubMed

    Islam, Md Shafiqul; Khan, Kamruzzaman; Akbar, M Ali; Mastroberardino, Antonio

    2014-10-01

    The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin-Bona-Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.

  16. A note on improved F-expansion method combined with Riccati equation applied to nonlinear evolution equations

    PubMed Central

    Islam, Md. Shafiqul; Khan, Kamruzzaman; Akbar, M. Ali; Mastroberardino, Antonio

    2014-01-01

    The purpose of this article is to present an analytical method, namely the improved F-expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering. PMID:26064530

  17. Exact traveling wave solutions for system of nonlinear evolution equations.

    PubMed

    Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H

    2016-01-01

    In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis.

  18. An exact sum-rule for the Hubbard model: an historical/pedagogical approach

    NASA Astrophysics Data System (ADS)

    Di Matteo, S.; Claveau, Y.

    2017-07-01

    The aim of the present article is to derive an exact integral equation for the Green function of the Hubbard model through an equation-of-motion procedure, like in the original Hubbard papers. Though our exact integral equation does not allow to solve the Hubbard model, it represents a strong constraint on its approximate solutions. An analogous sum rule has been already obtained in the literature, through the use of a spectral moment technique. We think however that our equation-of-motion procedure can be more easily related to the historical procedure of the original Hubbard papers. We also discuss examples of possible applications of the sum rule and propose and analyse a solution, fulfilling it, that can be used for a pedagogical introduction to the Mott-Hubbard metal-insulator transition.

  19. Exact models for isotropic matter

    NASA Astrophysics Data System (ADS)

    Thirukkanesh, S.; Maharaj, S. D.

    2006-04-01

    We study the Einstein-Maxwell system of equations in spherically symmetric gravitational fields for static interior spacetimes. The condition for pressure isotropy is reduced to a recurrence equation with variable, rational coefficients. We demonstrate that this difference equation can be solved in general using mathematical induction. Consequently, we can find an explicit exact solution to the Einstein-Maxwell field equations. The metric functions, energy density, pressure and the electric field intensity can be found explicitly. Our result contains models found previously, including the neutron star model of Durgapal and Bannerji. By placing restrictions on parameters arising in the general series, we show that the series terminate and there exist two linearly independent solutions. Consequently, it is possible to find exact solutions in terms of elementary functions, namely polynomials and algebraic functions.

  20. Unitary-matrix models as exactly solvable string theories

    NASA Technical Reports Server (NTRS)

    Periwal, Vipul; Shevitz, Danny

    1990-01-01

    Exact differential equations are presently found for the scaling functions of models of unitary matrices which are solved in a double-scaling limit, using orthogonal polynomials on a circle. For the case of the simplest, k = 1 model, the Painleve II equation with constant 0 is obtained; possible nonperturbative phase transitions exist for these models. Equations are presented for k = 2 and 3, and discussed with a view to asymptotic behavior.

  1. Higher symmetries and exact solutions of linear and nonlinear Schr{umlt o}dinger equation

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

    Fushchych, W.I.; Nikitin, A.G.

    1997-11-01

    A new approach for the analysis of partial differential equations is developed which is characterized by a simultaneous use of higher and conditional symmetries. Higher symmetries of the Schr{umlt o}dinger equation with an arbitrary potential are investigated. Nonlinear determining equations for potentials are solved using reductions to Weierstrass, Painlev{acute e}, and Riccati forms. Algebraic properties of higher order symmetry operators are analyzed. Combinations of higher and conditional symmetries are used to generate families of exact solutions of linear and nonlinear Schr{umlt o}dinger equations. {copyright} {ital 1997 American Institute of Physics.}

  2. Generalized nonequilibrium vertex correction method in coherent medium theory for quantum transport simulation of disordered nanoelectronics

    NASA Astrophysics Data System (ADS)

    Yan, Jiawei; Ke, Youqi

    2016-07-01

    Electron transport properties of nanoelectronics can be significantly influenced by the inevitable and randomly distributed impurities/defects. For theoretical simulation of disordered nanoscale electronics, one is interested in both the configurationally averaged transport property and its statistical fluctuation that tells device-to-device variability induced by disorder. However, due to the lack of an effective method to do disorder averaging under the nonequilibrium condition, the important effects of disorders on electron transport remain largely unexplored or poorly understood. In this work, we report a general formalism of Green's function based nonequilibrium effective medium theory to calculate the disordered nanoelectronics. In this method, based on a generalized coherent potential approximation for the Keldysh nonequilibrium Green's function, we developed a generalized nonequilibrium vertex correction method to calculate the average of a two-Keldysh-Green's-function correlator. We obtain nine nonequilibrium vertex correction terms, as a complete family, to express the average of any two-Green's-function correlator and find they can be solved by a set of linear equations. As an important result, the averaged nonequilibrium density matrix, averaged current, disorder-induced current fluctuation, and averaged shot noise, which involve different two-Green's-function correlators, can all be derived and computed in an effective and unified way. To test the general applicability of this method, we applied it to compute the transmission coefficient and its fluctuation with a square-lattice tight-binding model and compared with the exact results and other previously proposed approximations. Our results show very good agreement with the exact results for a wide range of disorder concentrations and energies. In addition, to incorporate with density functional theory to realize first-principles quantum transport simulation, we have also derived a general form of conditionally averaged nonequilibrium Green's function for multicomponent disorders.

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

    Lavinto, Mikko; Räsänen, Syksy; Szybka, Sebastian J., E-mail: mikko.lavinto@helsinki.fi, E-mail: syksy.rasanen@iki.fi, E-mail: sebastian.szybka@uj.edu.pl

    We construct the first exact statistically homogeneous and isotropic cosmological solution in which inhomogeneity has a significant effect on the expansion rate. The universe is modelled as a Swiss Cheese, with dust FRW background and inhomogeneous holes. We show that if the holes are described by the quasispherical Szekeres solution, their average expansion rate is close to the background under certain rather general conditions. We specialise to spherically symmetric holes and violate one of these conditions. As a result, the average expansion rate at late times grows relative to the background, \\ie backreaction is significant. The holes fit smoothly intomore » the background, but are larger on the inside than a corresponding background domain: we call them Tardis regions. We study light propagation, find the effective equations of state and consider the relation of the spatially averaged expansion rate to the redshift and the angular diameter distance.« less

  4. Exactly solvable field theories of closed strings

    NASA Astrophysics Data System (ADS)

    Brézin, E.; Kazakov, V. A.

    1990-02-01

    Field theories of closed strings are shown to be exactly solvable for a central charge of matter fields c=1-6/m(m+1),m=1,2, 3, .... The two-point function χ(λ,N), in which λ is the cosmological constant and N-1 is the string coupling constant, obeys a scaling law χ(λ,N=N-(m+1/2)C((λc-λ)Nm/(m+1/2)) in the limit in which N-1 goes to zero and λ goes to a critical value λc we have determined the universal non-linear differential equation satisfied by the function C. From this equation it is found that a phase transition takes place for some finite value of the scaling parameter (λc-λ)Nm/(m+1/2); this transition is a ``condensation of handles'' on the world sheet, characterized by a divergence of the averaged genus of the world sheets. The cases m=2,3 are elaborated in more details, and the case m=1, which corresponds to the embedding of a bosonic string in -2 dimensions, is reduced to explicit quadratures. Permanent address: Cybernetics Council and Academy of Sciences, ul. Vavilova 40, SU-117 333 Moscow, USSR.

  5. Assessments of a Turbulence Model Based on Menter's Modification to Rotta's Two-Equation Model

    NASA Technical Reports Server (NTRS)

    Abdol-Hamid, Khaled S.

    2013-01-01

    The main objective of this paper is to construct a turbulence model with a more reliable second equation simulating length scale. In the present paper, we assess the length scale equation based on Menter s modification to Rotta s two-equation model. Rotta shows that a reliable second equation can be formed in an exact transport equation from the turbulent length scale L and kinetic energy. Rotta s equation is well suited for a term-by-term modeling and shows some interesting features compared to other approaches. The most important difference is that the formulation leads to a natural inclusion of higher order velocity derivatives into the source terms of the scale equation, which has the potential to enhance the capability of Reynolds-averaged Navier-Stokes (RANS) to simulate unsteady flows. The model is implemented in the PAB3D solver with complete formulation, usage methodology, and validation examples to demonstrate its capabilities. The detailed studies include grid convergence. Near-wall and shear flows cases are documented and compared with experimental and Large Eddy Simulation (LES) data. The results from this formulation are as good or better than the well-known SST turbulence model and much better than k-epsilon results. Overall, the study provides useful insights into the model capability in predicting attached and separated flows.

  6. Classical and quantum dynamics of a kicked relativistic particle in a box

    NASA Astrophysics Data System (ADS)

    Yusupov, J. R.; Otajanov, D. M.; Eshniyazov, V. E.; Matrasulov, D. U.

    2018-03-01

    We study classical and quantum dynamics of a kicked relativistic particle confined in a one dimensional box. It is found that in classical case for chaotic motion the average kinetic energy grows in time, while for mixed regime the growth is suppressed. However, in case of regular motion energy fluctuates around certain value. Quantum dynamics is treated by solving the time-dependent Dirac equation with delta-kicking potential, whose exact solution is obtained for single kicking period. In quantum case, depending on the values of the kicking parameters, the average kinetic energy can be quasi periodic, or fluctuating around some value. Particle transport is studied by considering spatio-temporal evolution of the Gaussian wave packet and by analyzing the trembling motion.

  7. Study of the exact analytical solution of the equation of longitudinal waves in a liquid with account of its relaxation properties

    NASA Astrophysics Data System (ADS)

    Kudinov, I. V.; Kudinov, V. A.

    2013-09-01

    A mathematical model of elastic vibrations of an incompressible liquid has been developed based on the hypothesis on the finite velocity of propagation of field potentials in this liquid. A hyperbolic equation of vibrations of such a liquid with account of its relaxation properties has been obtained. An exact analytical solution of this equation has been found and investigated in detail.

  8. Brownian motion of massive skyrmions in magnetic thin films

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

    Troncoso, Roberto E., E-mail: r.troncoso.c@gmail.com; Núñez, Álvaro S., E-mail: alnunez@dfi.uchile.cl

    2014-12-15

    We report on the thermal effects on the motion of current-driven massive magnetic skyrmions. The reduced equation for the motion of skyrmion has the form of a stochastic generalized Thiele’s equation. We propose an ansatz for the magnetization texture of a non-rigid single skyrmion that depends linearly with the velocity. By using this ansatz it is found that the skyrmion mass tensor is closely related to intrinsic skyrmion parameters, such as Gilbert damping, skyrmion-charge and dissipative force. We have found an exact expression for the average drift velocity as well as the mean-square velocity of the skyrmion. The longitudinal andmore » transverse mobility of skyrmions for small spin-velocity of electrons is also determined and found to be independent of the skyrmion mass.« less

  9. Exact solution for a non-Markovian dissipative quantum dynamics.

    PubMed

    Ferialdi, Luca; Bassi, Angelo

    2012-04-27

    We provide the exact analytic solution of the stochastic Schrödinger equation describing a harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models for infinite-dimensional systems. We compute the Green's function; in the case of a free particle and with an exponentially correlated noise, we discuss the evolution of Gaussian wave functions.

  10. Exact Cosmological Models with Yang–Mills Fields on Lyra Manifold

    NASA Astrophysics Data System (ADS)

    Shchigolev, V. K.; Bezbatko, D. N.

    2018-04-01

    The present study deals with the Friedmann-Robertson-Walker cosmological models with Yang-Mills (YM) fields in Lyra geometry. The energy-momentum tensor of the YM fields for our models is obtained with the help of an exact solution to the YM equations with minimal coupling to gravity. Two specific exact solutions of the model are obtained regarding the effective equation of state and the exponential law of expansion. The physical and geometric behavior of the model is also discussed.

  11. Exact Solutions for the Integrable Sixth-Order Drinfeld-Sokolov-Satsuma-Hirota System by the Analytical Methods.

    PubMed

    Manafian Heris, Jalil; Lakestani, Mehrdad

    2014-01-01

    We establish exact solutions including periodic wave and solitary wave solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota system. We employ this system by using a generalized (G'/G)-expansion and the generalized tanh-coth methods. These methods are developed for searching exact travelling wave solutions of nonlinear partial differential equations. It is shown that these methods, with the help of symbolic computation, provide a straightforward and powerful mathematical tool for solving nonlinear partial differential equations.

  12. Homotopy decomposition method for solving one-dimensional time-fractional diffusion equation

    NASA Astrophysics Data System (ADS)

    Abuasad, Salah; Hashim, Ishak

    2018-04-01

    In this paper, we present the homotopy decomposition method with a modified definition of beta fractional derivative for the first time to find exact solution of one-dimensional time-fractional diffusion equation. In this method, the solution takes the form of a convergent series with easily computable terms. The exact solution obtained by the proposed method is compared with the exact solution obtained by using fractional variational homotopy perturbation iteration method via a modified Riemann-Liouville derivative.

  13. Partition-free approach to open quantum systems in harmonic environments: An exact stochastic Liouville equation

    NASA Astrophysics Data System (ADS)

    McCaul, G. M. G.; Lorenz, C. D.; Kantorovich, L.

    2017-03-01

    We present a partition-free approach to the evolution of density matrices for open quantum systems coupled to a harmonic environment. The influence functional formalism combined with a two-time Hubbard-Stratonovich transformation allows us to derive a set of exact differential equations for the reduced density matrix of an open system, termed the extended stochastic Liouville-von Neumann equation. Our approach generalizes previous work based on Caldeira-Leggett models and a partitioned initial density matrix. This provides a simple, yet exact, closed-form description for the evolution of open systems from equilibriated initial conditions. The applicability of this model and the potential for numerical implementations are also discussed.

  14. Linearly exact parallel closures for slab geometry

    NASA Astrophysics Data System (ADS)

    Ji, Jeong-Young; Held, Eric D.; Jhang, Hogun

    2013-08-01

    Parallel closures are obtained by solving a linearized kinetic equation with a model collision operator using the Fourier transform method. The closures expressed in wave number space are exact for time-dependent linear problems to within the limits of the model collision operator. In the adiabatic, collisionless limit, an inverse Fourier transform is performed to obtain integral (nonlocal) parallel closures in real space; parallel heat flow and viscosity closures for density, temperature, and flow velocity equations replace Braginskii's parallel closure relations, and parallel flow velocity and heat flow closures for density and temperature equations replace Spitzer's parallel transport relations. It is verified that the closures reproduce the exact linear response function of Hammett and Perkins [Phys. Rev. Lett. 64, 3019 (1990)] for Landau damping given a temperature gradient. In contrast to their approximate closures where the vanishing viscosity coefficient numerically gives an exact response, our closures relate the heat flow and nonvanishing viscosity to temperature and flow velocity (gradients).

  15. Exact solution of a quantum forced time-dependent harmonic oscillator

    NASA Technical Reports Server (NTRS)

    Yeon, Kyu Hwang; George, Thomas F.; Um, Chung IN

    1992-01-01

    The Schrodinger equation is used to exactly evaluate the propagator, wave function, energy expectation values, uncertainty values, and coherent state for a harmonic oscillator with a time dependent frequency and an external driving time dependent force. These quantities represent the solution of the classical equation of motion for the time dependent harmonic oscillator.

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

    Diwaker, E-mail: diwakerphysics@gmail.com; Chakraborty, Aniruddha

    The Smoluchowski equation with a time-dependent sink term is solved exactly. In this method, knowing the probability distribution P(0, s) at the origin, allows deriving the probability distribution P(x, s) at all positions. Exact solutions of the Smoluchowski equation are also provided in different cases where the sink term has linear, constant, inverse, and exponential variation in time.

  17. Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents.

    PubMed

    Kundu, Anjan; Mukherjee, Abhik; Naskar, Tapan

    2014-04-08

    Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension, with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave. We propose two dimensions, exactly solvable nonlinear Schrödinger (NLS) equation derivable from the basic hydrodynamic equations and endowed with integrable structures. The proposed two-dimensional equation exhibits modulation instability and frequency correction induced by the nonlinear effect, with a directional preference, all of which can be determined through precise analytic result. The two-dimensional NLS equation allows also an exact lump soliton which can model a full-grown surface rogue wave with adjustable height and modular inclination. The lump soliton under the influence of an ocean current appears and disappears preceded by a hole state, with its dynamics controlled by the current term. These desirable properties make our exact model promising for describing ocean rogue waves.

  18. Modelling rogue waves through exact dynamical lump soliton controlled by ocean currents

    PubMed Central

    Kundu, Anjan; Mukherjee, Abhik; Naskar, Tapan

    2014-01-01

    Rogue waves are extraordinarily high and steep isolated waves, which appear suddenly in a calm sea and disappear equally fast. However, though the rogue waves are localized surface waves, their theoretical models and experimental observations are available mostly in one dimension, with the majority of them admitting only limited and fixed amplitude and modular inclination of the wave. We propose two dimensions, exactly solvable nonlinear Schrödinger (NLS) equation derivable from the basic hydrodynamic equations and endowed with integrable structures. The proposed two-dimensional equation exhibits modulation instability and frequency correction induced by the nonlinear effect, with a directional preference, all of which can be determined through precise analytic result. The two-dimensional NLS equation allows also an exact lump soliton which can model a full-grown surface rogue wave with adjustable height and modular inclination. The lump soliton under the influence of an ocean current appears and disappears preceded by a hole state, with its dynamics controlled by the current term. These desirable properties make our exact model promising for describing ocean rogue waves. PMID:24711719

  19. Growth of structure in the Szekeres class-II inhomogeneous cosmological models and the matter-dominated era

    NASA Astrophysics Data System (ADS)

    Ishak, Mustapha; Peel, Austin

    2012-04-01

    This study belongs to a series devoted to using the Szekeres inhomogeneous models in order to develop a theoretical framework where cosmological observations can be investigated with a wider range of possible interpretations. While our previous work addressed the question of cosmological distances versus redshift in these models, the current study is a start at looking into the growth rate of large-scale structure. The Szekeres models are exact solutions to Einstein’s equations that were originally derived with no symmetries. We use here a formulation of the Szekeres models that is due to Goode and Wainwright, who considered the models as exact perturbations of a Friedmann-Lemaître-Robertson-Walker (FLRW) background. Using the Raychaudhuri equation we write, for the two classes of the models, exact growth equations in terms of the under/overdensity and measurable cosmological parameters. The new equations in the overdensity split into two informative parts. The first part, while exact, is identical to the growth equation in the usual linearly perturbed FLRW models, while the second part constitutes exact nonlinear perturbations. We integrate numerically the full exact growth rate equations for the flat and curved cases. We find that for the matter-dominated cosmic era, the Szekeres growth rate is up to a factor of three to five stronger than the usual linearly perturbed FLRW cases, reflecting the effect of exact Szekeres nonlinear perturbations. We also find that the Szekeres growth rate with an Einstein-de Sitter background is stronger than that of the well-known nonlinear spherical collapse model, and the difference between the two increases with time. This highlights the distinction when we use general inhomogeneous models where shear and a tidal gravitational field are present and contribute to the gravitational clustering. Additionally, it is worth observing that the enhancement of the growth found in the Szekeres models during the matter-dominated era could suggest a substitute to the argument that dark matter is needed when using FLRW models to explain the enhanced growth and resulting large-scale structures that we observe today.

  20. Using exact solutions to develop an implicit scheme for the baroclinic primitive equations

    NASA Technical Reports Server (NTRS)

    Marchesin, D.

    1984-01-01

    The exact solutions presently obtained by means of a novel method for nonlinear initial value problems are used in the development of numerical schemes for the computer solution of these problems. The method is applied to a new, fully implicit scheme on a vertical slice of the isentropic baroclinic equations. It was not possible to find a global scale phenomenon that could be simulated by the baroclinic primitive equations on a vertical slice.

  1. Symmetry reduction and exact solutions of two higher-dimensional nonlinear evolution equations.

    PubMed

    Gu, Yongyi; Qi, Jianming

    2017-01-01

    In this paper, symmetries and symmetry reduction of two higher-dimensional nonlinear evolution equations (NLEEs) are obtained by Lie group method. These NLEEs play an important role in nonlinear sciences. We derive exact solutions to these NLEEs via the [Formula: see text]-expansion method and complex method. Five types of explicit function solutions are constructed, which are rational, exponential, trigonometric, hyperbolic and elliptic function solutions of the variables in the considered equations.

  2. Period of vibration of axially vibrating truly nonlinear rod

    NASA Astrophysics Data System (ADS)

    Cveticanin, L.

    2016-07-01

    In this paper the axial vibration of a muscle whose fibers are parallel to the direction of muscle compression is investigated. The model is a clamped-free rod with a strongly nonlinear elastic property. Axial vibration is described by a nonlinear partial differential equation. A solution of the equation is constructed for special initial conditions by using the method of separation of variables. The partial differential equation is separated into two uncoupled strongly nonlinear second order differential equations. Both equations, with displacement function and with time function are exactly determined. Exact solutions are given in the form of inverse incomplete and inverse complete Beta function. Using boundary and initial conditions, the frequency of vibration is obtained. It has to be mentioned that the determined frequency represents the exact analytic description for the axially vibrating truly nonlinear clamped-free rod. The procedure suggested in this paper is applied for calculation of the frequency of the longissimus dorsi muscle of a cow. The influence of elasticity order and elasticity coefficient on the frequency property is tested.

  3. Role of protein fluctuation correlations in electron transfer in photosynthetic complexes.

    PubMed

    Nesterov, Alexander I; Berman, Gennady P

    2015-04-01

    We consider the dependence of the electron transfer in photosynthetic complexes on correlation properties of random fluctuations of the protein environment. The electron subsystem is modeled by a finite network of connected electron (exciton) sites. The fluctuations of the protein environment are modeled by random telegraph processes, which act either collectively (correlated) or independently (uncorrelated) on the electron sites. We derived an exact closed system of first-order linear differential equations with constant coefficients, for the average density matrix elements and for their first moments. Under some conditions, we obtained analytic expressions for the electron transfer rates and found the range of parameters for their applicability by comparing with the exact numerical simulations. We also compared the correlated and uncorrelated regimes and demonstrated numerically that the uncorrelated fluctuations of the protein environment can, under some conditions, either increase or decrease the electron transfer rates.

  4. A Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin for Diffusion

    NASA Technical Reports Server (NTRS)

    Huynh, H. T.

    2009-01-01

    We introduce a new approach to high-order accuracy for the numerical solution of diffusion problems by solving the equations in differential form using a reconstruction technique. The approach has the advantages of simplicity and economy. It results in several new high-order methods including a simplified version of discontinuous Galerkin (DG). It also leads to new definitions of common value and common gradient quantities at each interface shared by the two adjacent cells. In addition, the new approach clarifies the relations among the various choices of new and existing common quantities. Fourier stability and accuracy analyses are carried out for the resulting schemes. Extensions to the case of quadrilateral meshes are obtained via tensor products. For the two-point boundary value problem (steady state), it is shown that these schemes, which include most popular DG methods, yield exact common interface quantities as well as exact cell average solutions for nearly all cases.

  5. Synchronization scenarios in the Winfree model of coupled oscillators

    NASA Astrophysics Data System (ADS)

    Gallego, Rafael; Montbrió, Ernest; Pazó, Diego

    2017-10-01

    Fifty years ago Arthur Winfree proposed a deeply influential mean-field model for the collective synchronization of large populations of phase oscillators. Here we provide a detailed analysis of the model for some special, analytically tractable cases. Adopting the thermodynamic limit, we derive an ordinary differential equation that exactly describes the temporal evolution of the macroscopic variables in the Ott-Antonsen invariant manifold. The low-dimensional model is then thoroughly investigated for a variety of pulse types and sinusoidal phase response curves (PRCs). Two structurally different synchronization scenarios are found, which are linked via the mutation of a Bogdanov-Takens point. From our results, we infer a general rule of thumb relating pulse shape and PRC offset with each scenario. Finally, we compare the exact synchronization threshold with the prediction of the averaging approximation given by the Kuramoto-Sakaguchi model. At the leading order, the discrepancy appears to behave as an odd function of the PRC offset.

  6. Control of Stochastic Master Equation Models of Genetic Regulatory Networks by Approximating Their Average Behavior

    NASA Astrophysics Data System (ADS)

    Umut Caglar, Mehmet; Pal, Ranadip

    2010-10-01

    The central dogma of molecular biology states that ``information cannot be transferred back from protein to either protein or nucleic acid.'' However, this assumption is not exactly correct in most of the cases. There are a lot of feedback loops and interactions between different levels of systems. These types of interactions are hard to analyze due to the lack of data in the cellular level and probabilistic nature of interactions. Probabilistic models like Stochastic Master Equation (SME) or deterministic models like differential equations (DE) can be used to analyze these types of interactions. SME models based on chemical master equation (CME) can provide detailed representation of genetic regulatory system, but their use is restricted by the large data requirements and computational costs of calculations. The differential equations models on the other hand, have low calculation costs and much more adequate to generate control procedures on the system; but they are not adequate to investigate the probabilistic nature of interactions. In this work the success of the mapping between SME and DE is analyzed, and the success of a control policy generated by DE model with respect to SME model is examined. Index Terms--- Stochastic Master Equation models, Differential Equation Models, Control Policy Design, Systems biology

  7. Exact results relating spin-orbit interactions in two-dimensional strongly correlated systems

    NASA Astrophysics Data System (ADS)

    Kucska, Nóra; Gulácsi, Zsolt

    2018-06-01

    A 2D square, two-bands, strongly correlated and non-integrable system is analysed exactly in the presence of many-body spin-orbit interactions via the method of Positive Semidefinite Operators. The deduced exact ground states in the high concentration limit are strongly entangled, and given by the spin-orbit coupling are ferromagnetic and present an enhanced carrier mobility, which substantially differs for different spin projections. The described state emerges in a restricted parameter space region, which however is clearly accessible experimentally. The exact solutions are provided via the solution of a matching system of equations containing 74 coupled, non-linear and complex algebraic equations. In our knowledge, other exact results for 2D interacting systems with spin-orbit interactions are not present in the literature.

  8. Heisenberg-Langevin versus quantum master equation

    NASA Astrophysics Data System (ADS)

    Boyanovsky, Daniel; Jasnow, David

    2017-12-01

    The quantum master equation is an important tool in the study of quantum open systems. It is often derived under a set of approximations, chief among them the Born (factorization) and Markov (neglect of memory effects) approximations. In this article we study the paradigmatic model of quantum Brownian motion of a harmonic oscillator coupled to a bath of oscillators with a Drude-Ohmic spectral density. We obtain analytically the exact solution of the Heisenberg-Langevin equations, with which we study correlation functions in the asymptotic stationary state. We compare the exact correlation functions to those obtained in the asymptotic long time limit with the quantum master equation in the Born approximation with and without the Markov approximation. In the latter case we implement a systematic derivative expansion that yields the exact asymptotic limit under the factorization approximation only. We find discrepancies that could be significant when the bandwidth of the bath Λ is much larger than the typical scales of the system. We study the exact interaction energy as a proxy for the correlations missed by the Born approximation and find that its dependence on Λ is similar to the discrepancy between the exact solution and that of the quantum master equation in the Born approximation. We quantify the regime of validity of the quantum master equation in the Born approximation with or without the Markov approximation in terms of the system's relaxation rate γ , its unrenormalized natural frequency Ω and Λ : γ /Ω ≪1 and also γ Λ /Ω2≪1 . The reliability of the Born approximation is discussed within the context of recent experimental settings and more general environments.

  9. The evolution equation for the flame surface density in turbulent premixed combustion

    NASA Technical Reports Server (NTRS)

    Trouve, Arnaud

    1993-01-01

    The mean reaction rate in flamelet models for turbulent premixed combustion depends on two basic quantities: a mean chemical rate, called the flamelet speed, and the flame surface density. Our previous work had been primarily focused on the problem of the structure and topology of turbulent premixed flames, and it was then determined that the flamelet speed, when space-averaged, is only weakly sensitive to the turbulent flow field. Consequently, the flame surface density is the key quantity that conveys most of the effects of the turbulence on the rate of energy release. In flamelet models, this quantity is obtained via a modeled transport equation called the Sigma-equation. Past theoretical work has produced a rigorous approach that leads to an exact but unclosed formulation for the turbulent Sigma-equation. In the exact Sigma-equation, it appears that the dynamical properties of the flame surface density are determined by a single parameter, namely the turbulent flame stretch. Unfortunately, the turbulent flame stretch as well as the flame surface density is not available from experiments, and, in the absence of experimental data, little is known on the validity of the closure assumptions used in current flamelet models. Direct Numerical Simulation (DNS) is the alternative approach to get basic information on these fundamental quantities. In the present work, three-dimensional DNS of premixed flames in isotropic turbulent flow is used to estimate the different terms appearing in the Sigma-equation. A new methodology is proposed to provide the source and sink terms for the flame surface density, resolved both temporally and spatially throughout the turbulent flame brush. Using this methodology, our objective is to extract the turbulent flame stretch from the DNS data base and then perform extensive comparisons with flamelet models. Thanks to the detailed information produced by the DNS-based analysis, it is expected that this type of comparison will not only underscore the shortcomings of current models, but also suggest ways to improve them.

  10. Solution of the exact equations for three-dimensional atmospheric entry using directly matched asymptotic expansions

    NASA Technical Reports Server (NTRS)

    Busemann, A.; Vinh, N. X.; Culp, R. D.

    1976-01-01

    The problem of determining the trajectories, partially or wholly contained in the atmosphere of a spherical, nonrotating planet, is considered. The exact equations of motion for three-dimensional, aerodynamically affected flight are derived. Modified Chapman variables are introduced and the equations are transformed into a set suitable for analytic integration using asymptotic expansions. The trajectory is solved in two regions: the outer region, where the force may be considered a gravitational field with aerodynamic perturbations, and the inner region, where the force is predominantly aerodynamic, with gravity as a perturbation. The two solutions are matched directly. A composite solution, valid everywhere, is constructed by additive composition. This approach of directly matched asymptotic expansions applied to the exact equations of motion couched in terms of modified Chapman variables yields an analytical solution which should prove to be a powerful tool for aerodynamic orbit calculations.

  11. 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

  12. Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations.

    PubMed

    Islam, S M Rayhanul; Khan, Kamruzzaman; Akbar, M Ali

    2015-01-01

    In this paper, we implement the exp(-Φ(ξ))-expansion method to construct the exact traveling wave solutions for nonlinear evolution equations (NLEEs). Here we consider two model equations, namely the Korteweg-de Vries (KdV) equation and the time regularized long wave (TRLW) equation. These equations play significant role in nonlinear sciences. We obtained four types of explicit function solutions, namely hyperbolic, trigonometric, exponential and rational function solutions of the variables in the considered equations. It has shown that the applied method is quite efficient and is practically well suited for the aforementioned problems and so for the other NLEEs those arise in mathematical physics and engineering fields. PACS numbers: 02.30.Jr, 02.70.Wz, 05.45.Yv, 94.05.Fq.

  13. 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.

  14. Exact solution of some linear matrix equations using algebraic methods

    NASA Technical Reports Server (NTRS)

    Djaferis, T. E.; Mitter, S. K.

    1979-01-01

    Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.

  15. Closed solutions to a differential-difference equation and an associated plate solidification problem.

    PubMed

    Layeni, Olawanle P; Akinola, Adegbola P; Johnson, Jesse V

    2016-01-01

    Two distinct and novel formalisms for deriving exact closed solutions of a class of variable-coefficient differential-difference equations arising from a plate solidification problem are introduced. Thereupon, exact closed traveling wave and similarity solutions to the plate solidification problem are obtained for some special cases of time-varying plate surface temperature.

  16. On the motion of a quantum particle in the spinning cosmic string space–time

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

    Hassanabadi, H., E-mail: h.hasanabadi@shahroodut.ac.ir; Afshardoost, A.; Zarrinkamar, S.

    2015-05-15

    We analyze the energy spectrum and the wave function of a particle subjected to magnetic field in the spinning cosmic string space–time and investigate the influence of the spinning reference frame and topological defect on the system. To do this we solve Schrödinger equation in the spinning cosmic string background. In our work, instead of using an approximation in the calculations, we use the quasi-exact ansatz approach which gives the exact solutions for some primary levels. - Highlights: • Solving the Schrödinger equation in the spinning cosmic string space time. • Proposing a quasi-exact analytical solution to the general formmore » of the corresponding equation. • Generalizing the previous works.« less

  17. Time-local equation for exact time-dependent optimized effective potential in time-dependent density functional theory

    NASA Astrophysics Data System (ADS)

    Liao, Sheng-Lun; Ho, Tak-San; Rabitz, Herschel; Chu, Shih-I.

    2017-04-01

    Solving and analyzing the exact time-dependent optimized effective potential (TDOEP) integral equation has been a longstanding challenge due to its highly nonlinear and nonlocal nature. To meet the challenge, we derive an exact time-local TDOEP equation that admits a unique real-time solution in terms of time-dependent Kohn-Sham orbitals and effective memory orbitals. For illustration, the dipole evolution dynamics of a one-dimension-model chain of hydrogen atoms is numerically evaluated and examined to demonstrate the utility of the proposed time-local formulation. Importantly, it is shown that the zero-force theorem, violated by the time-dependent Krieger-Li-Iafrate approximation, is fulfilled in the current TDOEP framework. This work was partially supported by DOE.

  18. Solving Modal Equations of Motion with Initial Conditions Using MSC/NASTRAN DMAP. Part 2; Coupled Versus Uncoupled Integration

    NASA Technical Reports Server (NTRS)

    Barnett, Alan R.; Ibrahim, Omar M.; Abdallah, Ayman A.; Sullivan, Timothy L.

    1993-01-01

    By utilizing MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) in an existing NASA Lewis Research Center coupled loads methodology, solving modal equations of motion with initial conditions is possible using either coupled (Newmark-Beta) or uncoupled (exact mode superposition) integration available within module TRD1. Both the coupled and newly developed exact mode superposition methods have been used to perform transient analyses of various space systems. However, experience has shown that in most cases, significant time savings are realized when the equations of motion are integrated using the uncoupled solver instead of the coupled solver. Through the results of a real-world engineering analysis, advantages of using the exact mode superposition methodology are illustrated.

  19. Evolution of nonlinear waves in a blood-filled artery with an aneurysm

    NASA Astrophysics Data System (ADS)

    Nikolova, E. V.; Jordanov, I. P.; Dimitrova, Z. I.; Vitanov, N. K.

    2017-10-01

    We discuss propagation of traveling waves in a blood-filled hyper-elastic artery with a local dilatation (an aneurysm). The processes in the injured artery are modeled by an equation of the motion of the arterial wall and by equations of the motion of the fluid (the blood). Taking into account the specific arterial geometry and applying the reductive perturbation method in long-wave approximation we reduce the model equations to a version of the perturbed Korteweg-de Vries kind equation with variable coefficients. Exact traveling-wave solutions of this equation are obtained by the modified method of simplest equation where the differential equation of Abel is used as a simplest equation. A particular case of the obtained exact solution is numerically simulated and discussed from the point of view of arterial disease mechanics.

  20. Exact dark soliton solutions for a family of N coupled nonlinear Schrödinger equations in optical fiber media.

    PubMed

    Nakkeeran, K

    2001-10-01

    We consider a family of N coupled nonlinear Schrödinger equations which govern the simultaneous propagation of N fields in the normal dispersion regime of an optical fiber with various important physical effects. The linear eigenvalue problem associated with the integrable form of all the equations is constructed with the help of the Ablowitz-Kaup-Newell-Segur method. Using the Hirota bilinear method, exact dark soliton solutions are explicitly derived.

  1. Exact time-dependent solutions for a self-regulating gene.

    PubMed

    Ramos, A F; Innocentini, G C P; Hornos, J E M

    2011-06-01

    The exact time-dependent solution for the stochastic equations governing the behavior of a binary self-regulating gene is presented. Using the generating function technique to rephrase the master equations in terms of partial differential equations, we show that the model is totally integrable and the analytical solutions are the celebrated confluent Heun functions. Self-regulation plays a major role in the control of gene expression, and it is remarkable that such a microscopic model is completely integrable in terms of well-known complex functions.

  2. An exact solution of the Currie-Hill equations in 1 + 1 dimensional Minkowski space

    NASA Astrophysics Data System (ADS)

    Balog, János

    2014-11-01

    We present an exact two-particle solution of the Currie-Hill equations of Predictive Relativistic Mechanics in 1 + 1 dimensional Minkowski space. The instantaneous accelerations are given in terms of elementary functions depending on the relative particle position and velocities. The general solution of the equations of motion is given and by studying the global phase space of this system it is shown that this is a subspace of the full kinematic phase space.

  3. Exact solution to the Schrödinger’s equation with pseudo-Gaussian potential

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

    Iacob, Felix, E-mail: felix@physics.uvt.ro; Lute, Marina, E-mail: marina.lute@upt.ro

    2015-12-15

    We consider the radial Schrödinger equation with the pseudo-Gaussian potential. By making an ansatz to the solution of the eigenvalue equation for the associate Hamiltonian, we arrive at the general exact eigenfunction. The values of energy levels for the bound states are calculated along with their corresponding normalized wave-functions. The case of positive energy levels, known as meta-stable states, is also discussed and the magnitude of transmission coefficient through the potential barrier is evaluated.

  4. Communication: An exact bound on the bridge function in integral equation theories.

    PubMed

    Kast, Stefan M; Tomazic, Daniel

    2012-11-07

    We show that the formal solution of the general closure relation occurring in Ornstein-Zernike-type integral equation theories in terms of the Lambert W function leads to an exact relation between the bridge function and correlation functions, most notably to an inequality that bounds possible bridge values. The analytical results are illustrated on the example of the Lennard-Jones fluid for which the exact bridge function is known from computer simulations under various conditions. The inequality has consequences for the development of bridge function models and rationalizes numerical convergence issues.

  5. Exact nonparaxial beams of the scalar Helmholtz equation.

    PubMed

    Rodríguez-Morales, Gustavo; Chávez-Cerda, Sabino

    2004-03-01

    It is shown that three-dimensional nonparaxial beams are described by the oblate spheroidal exact solutions of the Helmholtz equation. For what is believed to be the first time, their beam behavior is investigated and their corresponding parameters are defined. Using the fact that the beam width of the family of paraxial Gaussian beams is described by a hyperbola, we formally establish the connection between the physical parameters of nonparaxial spheroidal beam solutions and those of paraxial beams. These results are also helpful for investigating exact vector nonparaxial beams.

  6. Classical Control System Design: A non-Graphical Method for Finding the Exact System Parameters

    NASA Astrophysics Data System (ADS)

    Hussein, Mohammed Tawfik

    2008-06-01

    The Root Locus method of control system design was developed in the 1940's. It is a set of rules that helps in sketching the path traced by the roots of the closed loop characteristic equation of the system, as a parameter such as a controller gain, k, is varied. The procedure provides approximate sketching guidelines. Designs on control systems using the method are therefore not exact. This paper aims at a non-graphical method for finding the exact system parameters to place a pair of complex conjugate poles on a specified damping ratio line. The overall procedure is based on the exact solution of complex equations on the PC using numerical methods.

  7. Exact soliton solutions and their stability control in the nonlinear Schrödinger equation with spatiotemporally modulated nonlinearity.

    PubMed

    Tian, Qing; Wu, Lei; Zhang, Jie-Fang; Malomed, Boris A; Mihalache, D; Liu, W M

    2011-01-01

    We put forward a generic transformation which helps to find exact soliton solutions of the nonlinear Schrödinger equation with a spatiotemporal modulation of the nonlinearity and external potentials. As an example, we construct exact solitons for the defocusing nonlinearity and harmonic potential. When the soliton's eigenvalue is fixed, the number of exact solutions is determined by energy levels of the linear harmonic oscillator. In addition to the stable fundamental solitons, stable higher-order modes, describing array of dark solitons nested in a finite-width background, are constructed too. We also show how to control the instability domain of the nonstationary solitons.

  8. Computational method for exact frequency-dependent rays on the basis of the solution of the Helmholtz equation

    NASA Astrophysics Data System (ADS)

    Protasov, M.; Gadylshin, K.

    2017-07-01

    A numerical method is proposed for the calculation of exact frequency-dependent rays when the solution of the Helmholtz equation is known. The properties of frequency-dependent rays are analysed and compared with classical ray theory and with the method of finite-difference modelling for the first time. In this paper, we study the dependence of these rays on the frequency of signals and show the convergence of the exact rays to the classical rays with increasing frequency. A number of numerical experiments demonstrate the distinctive features of exact frequency-dependent rays, in particular, their ability to penetrate into shadow zones that are impenetrable for classical rays.

  9. 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

  10. A Generalized Michaelis-Menten Equation in Protein Synthesis: Effects of Mis-Charged Cognate tRNA and Mis-Reading of Codon.

    PubMed

    Dutta, Annwesha; Chowdhury, Debashish

    2017-05-01

    The sequence of amino acid monomers in the primary structure of a protein is decided by the corresponding sequence of codons (triplets of nucleic acid monomers) on the template messenger RNA (mRNA). The polymerization of a protein, by incorporation of the successive amino acid monomers, is carried out by a molecular machine called ribosome. We develop a stochastic kinetic model that captures the possibilities of mis-reading of mRNA codon and prior mis-charging of a tRNA. By a combination of analytical and numerical methods, we obtain the distribution of the times taken for incorporation of the successive amino acids in the growing protein in this mathematical model. The corresponding exact analytical expression for the average rate of elongation of a nascent protein is a 'biologically motivated' generalization of the Michaelis-Menten formula for the average rate of enzymatic reactions. This generalized Michaelis-Menten-like formula (and the exact analytical expressions for a few other quantities) that we report here display the interplay of four different branched pathways corresponding to selection of four different types of tRNA.

  11. A Bivariate Chebyshev Spectral Collocation Quasilinearization Method for Nonlinear Evolution Parabolic Equations

    PubMed Central

    Motsa, S. S.; Magagula, V. M.; Sibanda, P.

    2014-01-01

    This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature. PMID:25254252

  12. A bivariate Chebyshev spectral collocation quasilinearization method for nonlinear evolution parabolic equations.

    PubMed

    Motsa, S S; Magagula, V M; Sibanda, P

    2014-01-01

    This paper presents a new method for solving higher order nonlinear evolution partial differential equations (NPDEs). The method combines quasilinearisation, the Chebyshev spectral collocation method, and bivariate Lagrange interpolation. In this paper, we use the method to solve several nonlinear evolution equations, such as the modified KdV-Burgers equation, highly nonlinear modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation, and the Fitzhugh-Nagumo equation. The results are compared with known exact analytical solutions from literature to confirm accuracy, convergence, and effectiveness of the method. There is congruence between the numerical results and the exact solutions to a high order of accuracy. Tables were generated to present the order of accuracy of the method; convergence graphs to verify convergence of the method and error graphs are presented to show the excellent agreement between the results from this study and the known results from literature.

  13. Oblique scattering from radially inhomogeneous dielectric cylinders: An exact Volterra integral equation formulation

    NASA Astrophysics Data System (ADS)

    Tsalamengas, John L.

    2018-07-01

    We study plane-wave electromagnetic scattering by radially and strongly inhomogeneous dielectric cylinders at oblique incidence. The method of analysis relies on an exact reformulation of the underlying field equations as a first-order 4 × 4 system of differential equations and on the ability to restate the associated initial-value problem in the form of a system of coupled linear Volterra integral equations of the second kind. The integral equations so derived are discretized via a sophisticated variant of the Nyström method. The proposed method yields results accurate up to machine precision without relying on approximations. Numerical results and case studies ably demonstrate the efficiency and high accuracy of the algorithms.

  14. Colonization of a territory by a stochastic population under a strong Allee effect and a low immigration pressure

    NASA Astrophysics Data System (ADS)

    Be'er, Shay; Assaf, Michael; Meerson, Baruch

    2015-06-01

    We study the dynamics of colonization of a territory by a stochastic population at low immigration pressure. We assume a sufficiently strong Allee effect that introduces, in deterministic theory, a large critical population size for colonization. At low immigration rates, the average precolonization population size is small, thus invalidating the WKB approximation to the master equation. We circumvent this difficulty by deriving an exact zero-flux solution of the master equation and matching it with an approximate nonzero-flux solution of the pertinent Fokker-Planck equation in a small region around the critical population size. This procedure provides an accurate evaluation of the quasistationary probability distribution of population sizes in the precolonization state and of the mean time to colonization, for a wide range of immigration rates. At sufficiently high immigration rates our results agree with WKB results obtained previously. At low immigration rates the results can be very different.

  15. Colonization of a territory by a stochastic population under a strong Allee effect and a low immigration pressure.

    PubMed

    Be'er, Shay; Assaf, Michael; Meerson, Baruch

    2015-06-01

    We study the dynamics of colonization of a territory by a stochastic population at low immigration pressure. We assume a sufficiently strong Allee effect that introduces, in deterministic theory, a large critical population size for colonization. At low immigration rates, the average precolonization population size is small, thus invalidating the WKB approximation to the master equation. We circumvent this difficulty by deriving an exact zero-flux solution of the master equation and matching it with an approximate nonzero-flux solution of the pertinent Fokker-Planck equation in a small region around the critical population size. This procedure provides an accurate evaluation of the quasistationary probability distribution of population sizes in the precolonization state and of the mean time to colonization, for a wide range of immigration rates. At sufficiently high immigration rates our results agree with WKB results obtained previously. At low immigration rates the results can be very different.

  16. Weight of fitness deviation governs strict physical chaos in replicator dynamics

    NASA Astrophysics Data System (ADS)

    Pandit, Varun; Mukhopadhyay, Archan; Chakraborty, Sagar

    2018-03-01

    Replicator equation—a paradigm equation in evolutionary game dynamics—mathematizes the frequency dependent selection of competing strategies vying to enhance their fitness (quantified by the average payoffs) with respect to the average fitnesses of the evolving population under consideration. In this paper, we deal with two discrete versions of the replicator equation employed to study evolution in a population where any two players' interaction is modelled by a two-strategy symmetric normal-form game. There are twelve distinct classes of such games, each typified by a particular ordinal relationship among the elements of the corresponding payoff matrix. Here, we find the sufficient conditions for the existence of asymptotic solutions of the replicator equations such that the solutions—fixed points, periodic orbits, and chaotic trajectories—are all strictly physical, meaning that the frequency of any strategy lies inside the closed interval zero to one at all times. Thus, we elaborate on which of the twelve types of games are capable of showing meaningful physical solutions and for which of the two types of replicator equation. Subsequently, we introduce the concept of the weight of fitness deviation that is the scaling factor in a positive affine transformation connecting two payoff matrices such that the corresponding one-shot games have exactly same Nash equilibria and evolutionary stable states. The weight also quantifies how much the excess of fitness of a strategy over the average fitness of the population affects the per capita change in the frequency of the strategy. Intriguingly, the weight's variation is capable of making the Nash equilibria and the evolutionary stable states, useless by introducing strict physical chaos in the replicator dynamics based on the normal-form game.

  17. General relativity exactly described in terms of Newton's laws within curved geometries

    NASA Astrophysics Data System (ADS)

    Savickas, D.

    2014-07-01

    Many years ago Milne and McCrea showed in their well-known paper that the Hubble expansion occurring in general relativity could be exactly described by the use of Newtonian mechanics. It will be shown that a similar method can be extended to, and used within, curved geometries when Newton's second law is expressed within a four-dimensional curved spacetime. The second law will be shown to yield an equation that is exactly identical to the geodesic equation of motion of general relativity. This in itself yields no new information concerning relativity since the equation is mathematically identical to the relativistic equation. However, when the time in the second law is defined to have a constant direction as effectively occurs in Newtonian mechanics, and no longer acts as a fourth dimension as exists in relativity theory, it separates into a vector equation in a curved three-dimensional space and an additional second scalar equation that describes conservation of energy. It is shown that the curved Newtonian equations of motion define the metric coefficients which occur in the Schwarzschild solution and that they also define its equations of motion. Also, because the curved Newtonian equations developed here use masses as gravitational sources, as occurs in Newtonian mechanics, they make it possible to derive the solution for other kinds of mass distributions and are used here to find the metric equation for a thin mass-rod and the equation of motion for a mass particle orbiting it in its relativistic gravitational field.

  18. Wigner functions for fermions in strong magnetic fields

    NASA Astrophysics Data System (ADS)

    Sheng, Xin-li; Rischke, Dirk H.; Vasak, David; Wang, Qun

    2018-02-01

    We compute the covariant Wigner function for spin-(1/2) fermions in an arbitrarily strong magnetic field by exactly solving the Dirac equation at non-zero fermion-number and chiral-charge densities. The Landau energy levels as well as a set of orthonormal eigenfunctions are found as solutions of the Dirac equation. With these orthonormal eigenfunctions we construct the fermion field operators and the corresponding Wigner-function operator. The Wigner function is obtained by taking the ensemble average of the Wigner-function operator in global thermodynamical equilibrium, i.e., at constant temperature T and non-zero fermion-number and chiral-charge chemical potentials μ and μ_5, respectively. Extracting the vector and axial-vector components of the Wigner function, we reproduce the currents of the chiral magnetic and separation effect in an arbitrarily strong magnetic field.

  19. Structural and functional networks in complex systems with delay.

    PubMed

    Eguíluz, Víctor M; Pérez, Toni; Borge-Holthoefer, Javier; Arenas, Alex

    2011-05-01

    Functional networks of complex systems are obtained from the analysis of the temporal activity of their components, and are often used to infer their unknown underlying connectivity. We obtain the equations relating topology and function in a system of diffusively delay-coupled elements in complex networks. We solve exactly the resulting equations in motifs (directed structures of three nodes) and in directed networks. The mean-field solution for directed uncorrelated networks shows that the clusterization of the activity is dominated by the in-degree of the nodes, and that the locking frequency decreases with increasing average degree. We find that the exponent of a power law degree distribution of the structural topology γ is related to the exponent of the associated functional network as α=(2-γ)(-1) for γ<2. © 2011 American Physical Society

  20. Thermalization of oscillator chains with onsite anharmonicity and comparison with kinetic theory

    DOE PAGES

    Mendl, Christian B.; Lu, Jianfeng; Lukkarinen, Jani

    2016-12-02

    We perform microscopic molecular dynamics simulations of particle chains with an onsite anharmonicity to study relaxation of spatially homogeneous states to equilibrium, and directly compare the simulations with the corresponding Boltzmann-Peierls kinetic theory. The Wigner function serves as a common interface between the microscopic and kinetic level. We demonstrate quantitative agreement after an initial transient time interval. In particular, besides energy conservation, we observe the additional quasiconservation of the phonon density, defined via an ensemble average of the related microscopic field variables and exactly conserved by the kinetic equations. On superkinetic time scales, density quasiconservation is lost while energy remainsmore » conserved, and we find evidence for eventual relaxation of the density to its canonical ensemble value. Furthermore, the precise mechanism remains unknown and is not captured by the Boltzmann-Peierls equations.« less

  1. Thermalization of oscillator chains with onsite anharmonicity and comparison with kinetic theory

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

    Mendl, Christian B.; Lu, Jianfeng; Lukkarinen, Jani

    We perform microscopic molecular dynamics simulations of particle chains with an onsite anharmonicity to study relaxation of spatially homogeneous states to equilibrium, and directly compare the simulations with the corresponding Boltzmann-Peierls kinetic theory. The Wigner function serves as a common interface between the microscopic and kinetic level. We demonstrate quantitative agreement after an initial transient time interval. In particular, besides energy conservation, we observe the additional quasiconservation of the phonon density, defined via an ensemble average of the related microscopic field variables and exactly conserved by the kinetic equations. On superkinetic time scales, density quasiconservation is lost while energy remainsmore » conserved, and we find evidence for eventual relaxation of the density to its canonical ensemble value. Furthermore, the precise mechanism remains unknown and is not captured by the Boltzmann-Peierls equations.« less

  2. The stationary sine-Gordon equation on metric graphs: Exact analytical solutions for simple topologies

    NASA Astrophysics Data System (ADS)

    Sabirov, K.; Rakhmanov, S.; Matrasulov, D.; Susanto, H.

    2018-04-01

    We consider the stationary sine-Gordon equation on metric graphs with simple topologies. Exact analytical solutions are obtained for different vertex boundary conditions. It is shown that the method can be extended for tree and other simple graph topologies. Applications of the obtained results to branched planar Josephson junctions and Josephson junctions with tricrystal boundaries are discussed.

  3. 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.

  4. Exact theory of freeze-out

    NASA Astrophysics Data System (ADS)

    Cannoni, Mirco

    2015-03-01

    We show that the standard theory of thermal production and chemical decoupling of WIMPs is incomplete. The hypothesis that WIMPs are produced and decouple from a thermal bath implies that the rate equation the bath particles interacting with the WIMPs is an algebraic equation that constraints the actual WIMPs abundance to have a precise analytical form down to the temperature . The point , which coincides with the stationary point of the equation for the quantity , is where the maximum departure of the WIMPs abundance from the thermal value is reached. For each mass and total annihilation cross section , the temperature and the actual WIMPs abundance are exactly known. This value provides the true initial condition for the usual differential equation that have to be integrated in the interval . The matching of the two abundances at is continuous and differentiable. The dependence of the present relic abundance on the abundance at an intermediate temperature is an exact result. The exact theory suggests a new analytical approximation that furnishes the relic abundance accurate at the level of 1-2 % in the case of -wave and -wave scattering cross sections. We conclude the paper studying the evolution of the WIMPs chemical potential and the entropy production using methods of non-equilibrium thermodynamics.

  5. Anisotropic nonequilibrium hydrodynamic attractor

    NASA Astrophysics Data System (ADS)

    Strickland, Michael; Noronha, Jorge; Denicol, Gabriel S.

    2018-02-01

    We determine the dynamical attractors associated with anisotropic hydrodynamics (aHydro) and the DNMR equations for a 0 +1 d conformal system using kinetic theory in the relaxation time approximation. We compare our results to the nonequilibrium attractor obtained from the exact solution of the 0 +1 d conformal Boltzmann equation, the Navier-Stokes theory, and the second-order Mueller-Israel-Stewart theory. We demonstrate that the aHydro attractor equation resums an infinite number of terms in the inverse Reynolds number. The resulting resummed aHydro attractor possesses a positive longitudinal-to-transverse pressure ratio and is virtually indistinguishable from the exact attractor. This suggests that an optimized hydrodynamic treatment of kinetic theory involves a resummation not only in gradients (Knudsen number) but also in the inverse Reynolds number. We also demonstrate that the DNMR result provides a better approximation of the exact kinetic theory attractor than the Mueller-Israel-Stewart theory. Finally, we introduce a new method for obtaining approximate aHydro equations which relies solely on an expansion in the inverse Reynolds number. We then carry this expansion out to the third order, and compare these third-order results to the exact kinetic theory solution.

  6. Preliminary Computational Fluid Dynamics (CFD) Simulation of EIIB Push Barge in Shallow Water

    NASA Astrophysics Data System (ADS)

    Beneš, Petr; Kollárik, Róbert

    2011-12-01

    This study presents preliminary CFD simulation of EIIb push barge in inland conditions using CFD software Ansys Fluent. The RANSE (Reynolds Averaged Navier-Stokes Equation) methods are used for the viscosity solution of turbulent flow around the ship hull. Different RANSE methods are used for the comparison of their results in ship resistance calculations, for selecting the appropriate and removing inappropriate methods. This study further familiarizes on the creation of geometrical model which considers exact water depth to vessel draft ratio in shallow water conditions, grid generation, setting mathematical model in Fluent and evaluation of the simulations results.

  7. Philosophies and fallacies in turbulence modeling

    NASA Astrophysics Data System (ADS)

    Spalart, Philippe R.

    2015-04-01

    We present a set of positions, likely to be controversial, on turbulence modeling for the Reynolds-Averaged Navier Stokes (RANS) equations. The paper has three themes. First is what we call the "fundamental paradox" of turbulence modeling, between the local character of the Partial Differential Equations strongly favored by CFD methods and the nonlocal physical nature of turbulence. Second, we oppose two philosophies. The "Systematic" philosophy attempts to model the exact transport equations for the Reynolds stresses or possibly higher moments term by term, gradually relegating the Closure Problem to higher moments and invoking the "Principle of Receding Influence" (although rarely formulating it). In contrast, the "Openly Empirical" philosophy produces models which satisfy strict constraints such as Galilean invariance, but lack an explicit connection with terms in the exact turbulence equations. The prime example is the eddy-viscosity assumption. Third, we explain a series of what we perceive as fallacies, many of them widely held and by senior observers, in turbulence knowledge, leading to turbulence models. We divide them into "hard" fallacies for which a short mathematical argument demonstrates that a particular statement is wrong or meaningless, and "soft" fallacies for which approximate physical arguments can be opposed, but we contend that a clear debate is overdue and wishful thinking has been involved. Some fallacies appear to be "intermediate." An example in the hard class is the supposed isotropy of the diagonal Reynolds stresses. Examples in the soft class are the need to match the decay rate of isotropic turbulence, and the value of realizability in a model. Our hope is to help the direct effort in this field away from simplistic and hopeless lines of work, and to foster debates.

  8. What is general relativity?

    NASA Astrophysics Data System (ADS)

    Coley, Alan A.; Wiltshire, David L.

    2017-05-01

    General relativity is a set of physical and geometric principles, which lead to a set of (Einstein) field equations that determine the gravitational field and to the geodesic equations that describe light propagation and the motion of particles on the background. But open questions remain, including: what is the scale on which matter and geometry are dynamically coupled in the Einstein equations? Are the field equations valid on small and large scales? What is the largest scale on which matter can be coarse grained while following a geodesic of a solution to Einstein’s equations? We address these questions. If the field equations are causal evolution equations, whose average on cosmological scales is not an exact solution of the Einstein equations, then some simplifying physical principle is required to explain the statistical homogeneity of the late epoch Universe. Such a principle may have its origin in the dynamical coupling between matter and geometry at the quantum level in the early Universe. This possibility is hinted at by diverse approaches to quantum gravity which find a dynamical reduction to two effective dimensions at high energies on one hand, and by cosmological observations which are beginning to strongly restrict the class of viable inflationary phenomenologies on the other. We suggest that the foundational principles of general relativity will play a central role in reformulating the theory of spacetime structure to meet the challenges of cosmology in the 21st century.

  9. A new exact method for line radiative transfer

    NASA Astrophysics Data System (ADS)

    Elitzur, Moshe; Asensio Ramos, Andrés

    2006-01-01

    We present a new method, the coupled escape probability (CEP), for exact calculation of line emission from multi-level systems, solving only algebraic equations for the level populations. The CEP formulation of the classical two-level problem is a set of linear equations, and we uncover an exact analytic expression for the emission from two-level optically thick sources that holds as long as they are in the `effectively thin' regime. In a comparative study of a number of standard problems, the CEP method outperformed the leading line transfer methods by substantial margins. The algebraic equations employed by our new method are already incorporated in numerous codes based on the escape probability approximation. All that is required for an exact solution with these existing codes is to augment the expression for the escape probability with simple zone-coupling terms. As an application, we find that standard escape probability calculations generally produce the correct cooling emission by the CII 158-μm line but not by the 3P lines of OI.

  10. Applications of the ETEM for obtaining optical soliton solutions for the Lakshmanan-Porsezian-Daniel model

    NASA Astrophysics Data System (ADS)

    Manafian, Jalil; Foroutan, Mohammadreza; Guzali, Aref

    2017-11-01

    This paper examines the effectiveness of an integration scheme which is called the extended trial equation method (ETEM) for solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the Lakshmanan-Porsezian-Daniel (LPD) equation with Kerr and power laws of nonlinearity which describes higher-order dispersion, full nonlinearity and spatiotemporal dispersion is considered, and as an achievement, a series of exact travelling-wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of LPD equation. The movement of obtained solutions is shown graphically, which helps to understand the physical phenomena of this optical soliton equation. Many other such types of nonlinear equations arising in basic fabric of communications network technology and nonlinear optics can also be solved by this method.

  11. Nonlinear anomalous diffusion equation and fractal dimension: exact generalized Gaussian solution.

    PubMed

    Pedron, I T; Mendes, R S; Malacarne, L C; Lenzi, E K

    2002-04-01

    In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the N-dimensional nonlinear diffusion equation partial differential rho/ partial differential t=nabla.(Knablarho(nu))-nabla.(muFrho)-alpharho, where K=Dr(-theta), nu, theta, mu, and D are real parameters, F is the external force, and alpha is a time-dependent source. This equation unifies the O'Shaughnessy-Procaccia anomalous diffusion equation on fractals (nu=1) and the spherical anomalous diffusion for porous media (theta=0). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.

  12. Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method

    NASA Astrophysics Data System (ADS)

    Rahmatullah; Ellahi, Rahmat; Mohyud-Din, Syed Tauseef; Khan, Umar

    2018-03-01

    We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses.

  13. Operator Spreading in Random Unitary Circuits

    NASA Astrophysics Data System (ADS)

    Nahum, Adam; Vijay, Sagar; Haah, Jeongwan

    2018-04-01

    Random quantum circuits yield minimally structured models for chaotic quantum dynamics, which are able to capture, for example, universal properties of entanglement growth. We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries. We study both 1 +1 D and higher dimensions and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems. In 1 +1 D , we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC and determines the butterfly speed vB. We find that in 1 +1 D , the "front" of the OTOC broadens diffusively, with a width scaling in time as t1 /2. We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front within a realization. Turning to higher dimensions, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a purely classical droplet growth problem. This implies that the width of the front of the averaged OTOC scales as t1 /3 in 2 +1 D and as t0.240 in 3 +1 D (exponents of the Kardar-Parisi-Zhang universality class). We support our analytic argument with simulations in 2 +1 D . We point out that, in two or higher spatial dimensions, the shape of the spreading operator at late times is affected by underlying lattice symmetries and, in general, is not spherical. However, when full spatial rotational symmetry is present in 2 +1 D , our mapping implies an exact asymptotic form for the OTOC, in terms of the Tracy-Widom distribution. For an alternative perspective on the OTOC in 1 +1 D , we map it to the partition function of an Ising-like statistical mechanics model. As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly. We also use this mapping to give exact results for entanglement growth in 1 +1 D circuits.

  14. Lie symmetry analysis and reduction for exact solution of (2+1)-dimensional Bogoyavlensky-Konopelchenko equation by geometric approach

    NASA Astrophysics Data System (ADS)

    Ray, S. Saha

    2018-04-01

    In this paper, the symmetry analysis and similarity reduction of the (2+1)-dimensional Bogoyavlensky-Konopelchenko (B-K) equation are investigated by means of the geometric approach of an invariance group, which is equivalent to the classical Lie symmetry method. Using the extended Harrison and Estabrook’s differential forms approach, the infinitesimal generators for (2+1)-dimensional B-K equation are obtained. Firstly, the vector field associated with the Lie group of transformation is derived. Then the symmetry reduction and the corresponding explicit exact solution of (2+1)-dimensional B-K equation is obtained.

  15. F-Expansion Method and New Exact Solutions of the Schrödinger-KdV Equation

    PubMed Central

    Filiz, Ali; Ekici, Mehmet; Sonmezoglu, Abdullah

    2014-01-01

    F-expansion method is proposed to seek exact solutions of nonlinear evolution equations. With the aid of symbolic computation, we choose the Schrödinger-KdV equation with a source to illustrate the validity and advantages of the proposed method. A number of Jacobi-elliptic function solutions are obtained including the Weierstrass-elliptic function solutions. When the modulus m of Jacobi-elliptic function approaches to 1 and 0, soliton-like solutions and trigonometric-function solutions are also obtained, respectively. The proposed method is a straightforward, short, promising, and powerful method for the nonlinear evolution equations in mathematical physics. PMID:24672327

  16. F-expansion method and new exact solutions of the Schrödinger-KdV equation.

    PubMed

    Filiz, Ali; Ekici, Mehmet; Sonmezoglu, Abdullah

    2014-01-01

    F-expansion method is proposed to seek exact solutions of nonlinear evolution equations. With the aid of symbolic computation, we choose the Schrödinger-KdV equation with a source to illustrate the validity and advantages of the proposed method. A number of Jacobi-elliptic function solutions are obtained including the Weierstrass-elliptic function solutions. When the modulus m of Jacobi-elliptic function approaches to 1 and 0, soliton-like solutions and trigonometric-function solutions are also obtained, respectively. The proposed method is a straightforward, short, promising, and powerful method for the nonlinear evolution equations in mathematical physics.

  17. Exact traveling wave solutions of the KP-BBM equation by using the new approach of generalized (G'/G)-expansion method.

    PubMed

    Alam, Md Nur; Akbar, M Ali

    2013-01-01

    The new approach of the generalized (G'/G)-expansion method is an effective and powerful mathematical tool in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in science, engineering and mathematical physics. In this article, the new approach of the generalized (G'/G)-expansion method is applied to construct traveling wave solutions of the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. The solutions are expressed in terms of the hyperbolic functions, the trigonometric functions and the rational functions. By means of this scheme, we found some new traveling wave solutions of the above mentioned equation.

  18. On the Singular Perturbations for Fractional Differential Equation

    PubMed Central

    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. PMID:24683357

  19. An investigation of the accuracy of the Merkel equation for evaporative cooling tower calculations. Waste heat management

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

    Yadigaroglu, G.; Pastor, E.J.

    1974-01-01

    The exact differential equations governing heat and mass transfer and air flow in an evaporative, natural-draft cooling tower are presented. The Merkel equation is then derived starting from this exact formulation and showing all the approximations involved. The Merkel formulation lumps the sensible and the latent heat transfer together and considers a single enthalpy-difference driving force for the total heat transfer. The effect of the approximations inherent in the Merkel equation is investigated and analyzed by a series of parametric numerical calculations of cooling tower performance under various ambient conditions and load conditions.

  20. Some Exact Results for the Schroedinger Wave Equation with a Time Dependent Potential

    NASA Technical Reports Server (NTRS)

    Campbell, Joel

    2009-01-01

    The time dependent Schroedinger equation with a time dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wave function at the origin, one may derive the wave function everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the p otential lead to conservation of the normalization of the probability density.

  1. Exact finite elements for conduction and convection

    NASA Technical Reports Server (NTRS)

    Thornton, E. A.; Dechaumphai, P.; Tamma, K. K.

    1981-01-01

    An approach for developing exact one dimensional conduction-convection finite elements is presented. Exact interpolation functions are derived based on solutions to the governing differential equations by employing a nodeless parameter. Exact interpolation functions are presented for combined heat transfer in several solids of different shapes, and for combined heat transfer in a flow passage. Numerical results demonstrate that exact one dimensional elements offer advantages over elements based on approximate interpolation functions.

  2. Exact renormalization group equation for the Lifshitz critical point

    NASA Astrophysics Data System (ADS)

    Bervillier, C.

    2004-10-01

    An exact renormalization equation (ERGE) accounting for an anisotropic scaling is derived. The critical and tricritical Lifshitz points are then studied at leading order of the derivative expansion which is shown to involve two differential equations. The resulting estimates of the Lifshitz critical exponents compare well with the O(ε) calculations. In the case of the Lifshitz tricritical point, it is shown that a marginally relevant coupling defies the perturbative approach since it actually makes the fixed point referred to in the previous perturbative calculations O(ε) finally unstable.

  3. Heun Polynomials and Exact Solutions for the Massless Dirac Particle in the C-Metric

    NASA Astrophysics Data System (ADS)

    Kar, Priyasri; Singh, Ritesh K.; Dasgupta, Ananda; Panigrahi, Prasanta K.

    2018-03-01

    The equation of motion of a massless Dirac particle in the C-metric leads to the general Heun equation (GHE) for the radial and the polar variables. The GHE, under certain parametric conditions, is cast in terms of a new set of su(1, 1) generators involving differential operators of degrees ±1/2 and 0. Additional Heun polynomials are obtained using this new algebraic structure and are used to construct some exact solutions for the radial and the polar parts of the Dirac equation.

  4. Bilinear, trilinear forms, and exact solution of certain fourth order integrable difference equations

    NASA Astrophysics Data System (ADS)

    Sahadevan, R.; Rajakumar, S.

    2008-03-01

    A systematic investigation of finding bilinear or trilinear representations of fourth order autonomous ordinary difference equation, x(n +4)=F(x(n),x(n+1),x(n+2),x(n+3)) or xn +4=F(xn,xn +1,xn +2,xn +3), is made. As an illustration, we consider fourth order symplectic integrable difference equations reported by [Capel and Sahadevan, Physica A 289, 86 (2001)] and derived their bilinear or trilinear forms. Also, it is shown that the obtained bilinear representations admit exact solution of rational form.

  5. Towards an exact factorization of the molecular wave function

    NASA Astrophysics Data System (ADS)

    Parashar, Shubham; Sajeev, Y.; Ghosh, Swapan K.

    2015-10-01

    An exact single-product factorisation of the molecular wave function for the timedependent Schrödinger equation is investigated by using an ansatz involving a phase factor. By using the Frenkel variational method, we obtain the Schrödinger equations for the electronic and nuclear wave functions. The concept of a potential energy surface (PES) is retained by introducing a modified Hamiltonian as suggested earlier by Cederbaum. The parameter ω in the phase factor is chosen such that the equations of motion retain the physically appealing Born- Oppenheimer-like form, and is therefore unique.

  6. An Exact Form of Lilley's Equation with a Velocity Quadrupole/Temperature Dipole Source Term

    NASA Technical Reports Server (NTRS)

    Goldstein, Marvin E.

    2001-01-01

    There have been several attempts to introduce approximations into the exact form of Lilley's equation in order to express the source term as the sum of a quadrupole whose strength is quadratic in the fluctuating velocities and a dipole whose strength is proportional to the temperature fluctuations. The purpose of this note is to show that it is possible to choose the dependent (i.e., the pressure) variable so that this type of result can be derived directly from the Euler equations without introducing any additional approximations.

  7. Dirac delta representation by exact parametric equations.. Application to impulsive vibration systems

    NASA Astrophysics Data System (ADS)

    Chicurel-Uziel, Enrique

    2007-08-01

    A pair of closed parametric equations are proposed to represent the Heaviside unit step function. Differentiating the step equations results in two additional parametric equations, that are also hereby proposed, to represent the Dirac delta function. These equations are expressed in algebraic terms and are handled by means of elementary algebra and elementary calculus. The proposed delta representation complies exactly with the values of the definition. It complies also with the sifting property and the requisite unit area and its Laplace transform coincides with the most general form given in the tables. Furthermore, it leads to a very simple method of solution of impulsive vibrating systems either linear or belonging to a large class of nonlinear problems. Two example solutions are presented.

  8. Exact solution of matricial Φ23 quantum field theory

    NASA Astrophysics Data System (ADS)

    Grosse, Harald; Sako, Akifumi; Wulkenhaar, Raimar

    2017-12-01

    We apply a recently developed method to exactly solve the Φ3 matrix model with covariance of a two-dimensional theory, also known as regularised Kontsevich model. Its correlation functions collectively describe graphs on a multi-punctured 2-sphere. We show how Ward-Takahashi identities and Schwinger-Dyson equations lead in a special large- N limit to integral equations that we solve exactly for all correlation functions. The solved model arises from noncommutative field theory in a special limit of strong deformation parameter. The limit defines ordinary 2D Schwinger functions which, however, do not satisfy reflection positivity.

  9. Exact solutions of the Wheeler–DeWitt equation and the Yamabe construction

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

    Ita III, Eyo Eyo, E-mail: ita@usna.edu; Soo, Chopin, E-mail: cpsoo@mail.ncku.edu.tw

    Exact solutions of the Wheeler–DeWitt equation of the full theory of four dimensional gravity of Lorentzian signature are obtained. They are characterized by Schrödinger wavefunctionals having support on 3-metrics of constant spatial scalar curvature, and thus contain two full physical field degrees of freedom in accordance with the Yamabe construction. These solutions are moreover Gaussians of minimum uncertainty and they are naturally associated with a rigged Hilbert space. In addition, in the limit the regulator is removed, exact 3-dimensional diffeomorphism and local gauge invariance of the solutions are recovered.

  10. Exact traveling soliton solutions for the generalized Benjamin-Bona-Mahony equation

    NASA Astrophysics Data System (ADS)

    Boudoue Hubert, Malwe; Kudryashov, Nikolai A.; Justin, Mibaile; Abbagari, Souleymanou; Betchewe, Gambo; Doka, Serge Y.

    2018-03-01

    In this paper, we investigate the generalized Benjamin-Bona-Mahony equation which better describes long waves with arbitrary power-law nonlinearity. As a result, we obtain exact travelling wave soliton solutions, such as anti-kink soliton solution, bright soliton solution, dark soliton solution and periodic solution. These solutions have many free parameters such that they may be used to simulate many experimental situations. The main contribution, in this work, is to not apply the computer codes for construction of exact solutions and not consider the integration constants as zero, because they give all variants for solutions.

  11. Asymptotically exact parabolic solutions of the generalized nonlinear Schrödinger equation with varying parameters

    NASA Astrophysics Data System (ADS)

    Kruglov, Vladimir I.; Harvey, John D.

    2006-12-01

    We present exact asymptotic similariton solutions of the generalized nonlinear Schrödinger equation (NLSE) with gain or loss terms for a normal-dispersion fiber amplifier with dispersion, nonlinearity, and gain profiles that depend on the propagation distance. Our treatment is based on the mapping of the NLSE with varying parameters to the NLSE with constant dispersion and nonlinearity coefficients and an arbitrary varying gain function. We formulate an effective procedure that leads directly, under appropriate conditions, to a wide range of exact asymptotic similariton solutions of NLSE demonstrating self-similar propagating regimes with linear chirp.

  12. On the Model-Based Bootstrap with Missing Data: Obtaining a "P"-Value for a Test of Exact Fit

    ERIC Educational Resources Information Center

    Savalei, Victoria; Yuan, Ke-Hai

    2009-01-01

    Evaluating the fit of a structural equation model via bootstrap requires a transformation of the data so that the null hypothesis holds exactly in the sample. For complete data, such a transformation was proposed by Beran and Srivastava (1985) for general covariance structure models and applied to structural equation modeling by Bollen and Stine…

  13. Steady-state solution of the semi-empirical diffusion equation for area sources. [air pollution studies

    NASA Technical Reports Server (NTRS)

    Lebedeff, S. A.; Hameed, S.

    1975-01-01

    The problem investigated can be solved exactly in a simple manner if the equations are written in terms of a similarity variable. The exact solution is used to explore two questions of interest in the modelling of urban air pollution, taking into account the distribution of surface concentration downwind of an area source and the distribution of concentration with height.

  14. Exact BPS domain walls at finite gauge coupling

    NASA Astrophysics Data System (ADS)

    Blaschke, Filip

    2017-01-01

    Bogomol'nyi-Prasad-Sommerfield solitons in models with spontaneously broken gauge symmetry have been intensively studied at the infinite gauge coupling limit, where the governing equation-the so-called master equation-is exactly solvable. Except for a handful of special solutions, the standing impression is that analytic results at finite coupling are generally unavailable. The aim of this paper is to demonstrate, using domain walls in Abelian-Higgs models as the simplest example, that exact solitons at finite gauge coupling can be readily obtained if the number of Higgs fields (NF ) is large enough. In particular, we present a family of exact solutions, describing N domain walls at arbitrary positions in models with at least NF≥2 N +1 . We have also found that adding together any pair of solutions can produce a new exact solution if the combined tension is below a certain limit.

  15. Dark energy fingerprints in the nonminimal Wu-Yang wormhole structure

    NASA Astrophysics Data System (ADS)

    Balakin, Alexander B.; Zayats, Alexei E.

    2014-08-01

    We discuss new exact solutions to nonminimally extended Einstein-Yang-Mills equations describing spherically symmetric static wormholes supported by the gauge field of the Wu-Yang type in a dark energy environment. We focus on the analysis of three types of exact solutions to the gravitational field equations. Solutions of the first type relate to the model, in which the dark energy is anisotropic; i.e., the radial and tangential pressures do not coincide. Solutions of the second type correspond to the isotropic pressure tensor; in particular, we discuss the exact solution, for which the dark energy is characterized by the equation of state for a string gas. Solutions of the third type describe the dark energy model with constant pressure and energy density. For the solutions of the third type, we consider in detail the problem of horizons and find constraints for the parameters of nonminimal coupling and for the constitutive parameters of the dark energy equation of state, which guarantee that the nonminimal wormholes are traversable.

  16. Treatment of pairing correlations based on the equations of motion for zero-coupled pair operators

    NASA Astrophysics Data System (ADS)

    Andreozzi, F.; Covello, A.; Gargano, A.; Ye, Liu Jian; Porrino, A.

    1985-07-01

    The pairing problem is treated by means of the equations of motion for zero-coupled pair operators. Exact equations for the seniority-v states of N particles are derived. These equations can be solved by a step-by-step procedure which consists of progressively adding pairs of particles to a core. The theory can be applied at several levels of approximation depending on the number of core states which are taken into account. Some numerical applications to the treatment of v=0, v=1, and v=2 states in the Ni isotopes are performed. The accuracy of various approximations is tested by comparison with exact results. For the seniority-one and seniority-two problems it turns out that the results obtained from the first-order theory are very accurate, while those of higher order calculations are practically exact. Concerning the seniority-zero problem, a fifth-order calculation reproduces quite well the three lowest states.

  17. Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory.

    PubMed

    Krüger, Matthias; Solon, Alexandre; Démery, Vincent; Rohwer, Christian M; Dean, David S

    2018-02-28

    Starting from the stochastic equation for the density operator, we formulate the exact (instantaneous) stress tensor for interacting Brownian particles and show that its average value agrees with expressions derived previously. We analyze the relation between the stress tensor and forces due to external potentials and observe that, out of equilibrium, particle currents give rise to extra forces. Next, we derive the stress tensor for a Landau-Ginzburg theory in generic, non-equilibrium situations, finding an expression analogous to that of the exact microscopic stress tensor, and discuss the computation of out-of-equilibrium (classical) Casimir forces. Subsequently, we give a general form for the stress tensor which is valid for a large variety of energy functionals and which reproduces the two mentioned cases. We then use these relations to study the spatio-temporal correlations of the stress tensor in a Brownian fluid, which we compute to leading order in the interaction potential strength. We observe that, after integration over time, the spatial correlations generally decay as power laws in space. These are expected to be of importance for driven confined systems. We also show that divergence-free parts of the stress tensor do not contribute to the Green-Kubo relation for the viscosity.

  18. Stresses in non-equilibrium fluids: Exact formulation and coarse-grained theory

    NASA Astrophysics Data System (ADS)

    Krüger, Matthias; Solon, Alexandre; Démery, Vincent; Rohwer, Christian M.; Dean, David S.

    2018-02-01

    Starting from the stochastic equation for the density operator, we formulate the exact (instantaneous) stress tensor for interacting Brownian particles and show that its average value agrees with expressions derived previously. We analyze the relation between the stress tensor and forces due to external potentials and observe that, out of equilibrium, particle currents give rise to extra forces. Next, we derive the stress tensor for a Landau-Ginzburg theory in generic, non-equilibrium situations, finding an expression analogous to that of the exact microscopic stress tensor, and discuss the computation of out-of-equilibrium (classical) Casimir forces. Subsequently, we give a general form for the stress tensor which is valid for a large variety of energy functionals and which reproduces the two mentioned cases. We then use these relations to study the spatio-temporal correlations of the stress tensor in a Brownian fluid, which we compute to leading order in the interaction potential strength. We observe that, after integration over time, the spatial correlations generally decay as power laws in space. These are expected to be of importance for driven confined systems. We also show that divergence-free parts of the stress tensor do not contribute to the Green-Kubo relation for the viscosity.

  19. The modified alternative (G'/G)-expansion method to nonlinear evolution equation: application to the (1+1)-dimensional Drinfel'd-Sokolov-Wilson equation.

    PubMed

    Akbar, M Ali; Mohd Ali, Norhashidah Hj; Mohyud-Din, Syed Tauseef

    2013-01-01

    Over the years, (G'/G)-expansion method is employed to generate traveling wave solutions to various wave equations in mathematical physics. In the present paper, the alternative (G'/G)-expansion method has been further modified by introducing the generalized Riccati equation to construct new exact solutions. In order to illustrate the novelty and advantages of this approach, the (1+1)-dimensional Drinfel'd-Sokolov-Wilson (DSW) equation is considered and abundant new exact traveling wave solutions are obtained in a uniform way. These solutions may be imperative and significant for the explanation of some practical physical phenomena. It is shown that the modified alternative (G'/G)-expansion method an efficient and advance mathematical tool for solving nonlinear partial differential equations in mathematical physics.

  20. Constitutive Modelling of Resins in the Stiffness Domain

    NASA Astrophysics Data System (ADS)

    Klasztorny, M.

    2004-09-01

    An analytic method for inverting the constitutive compliance equations of viscoelasticity for resins is developed. These equations describe the HWKK/H rheological model, which makes it possible to simulate, with a good accuracy, short-, medium- and long-term viscoelastic processes in epoxy and polyester resins. These processes are of first-rank reversible isothermal type. The time histories of deviatoric stresses are simulated with three independent strain history functions of fractional and normal exponential types. The stiffness equations are described by two elastic and six viscoelastic constants having a clear physic meaning (three long-term relaxation coefficients and three relaxation times). The time histories of axiatoric stresses are simulated as perfectly elastic. The inversion method utilizes approximate constitutive stiffness equations of viscoelasticity for the HWKK/H model. The constitutive compliance equations for the model are a basis for determining the exact complex shear stiffness, whereas the approximate constitutive stiffness equations are used for determining the approximate complex shear stiffness. The viscoelastic constants in the stiffness domain are derived by equating the exact and approximate complex shear stiffnesses. The viscoelastic constants are obtained for Epidian 53 epoxy and Polimal 109 polyester resins. The accuracy of the approximate constitutive stiffness equations are assessed by comparing the approximate and exact complex shear stiffnesses. The constitutive stiffness equations for the HWKK/H model are presented in uncoupled (shear/bulk) and coupled forms. Formulae for converting the constants of shear viscoelasticity into the constants of coupled viscoelasticity are given as well.

  1. Transient nucleation induction time from the birth-death equations

    NASA Technical Reports Server (NTRS)

    Shneidman, Vitaly A.; Weinberg, Michael C.

    1992-01-01

    For the set of finite-difference equations of Becker-Doering an exact formula for the induction time, which is expressed in terms of rapidly convergent sums, is presented. The form of the result is particularly amenable for analytical study, and the latter is carried out to obtain approximations of the exact expression in a rigorous manner and to assess its sensitivity to the choice of the nucleation model. The induction time is found to be governed by two main nucleation parameters, the normalized barrier height, and the number of molecules in the critical cluster. The ratio of these two parameters provides an assessment of the importance of discreteness effects. The exact expression is studied in both the continuous and the asymptotic limits. The accuracy of the Zeldovich equation, which is produced in the continuous limit, is discussed for several nucleation models.

  2. Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potential and nonlinearities.

    PubMed

    Yan, Zhenya; Konotop, V V

    2009-09-01

    It is shown that using the similarity transformations, a set of three-dimensional p-q nonlinear Schrödinger (NLS) equations with inhomogeneous coefficients can be reduced to one-dimensional stationary NLS equation with constant or varying coefficients, thus allowing for obtaining exact localized and periodic wave solutions. In the suggested reduction the original coordinates in the (1+3) space are mapped into a set of one-parametric coordinate surfaces, whose parameter plays the role of the coordinate of the one-dimensional equation. We describe the algorithm of finding solutions and concentrate on power (linear and nonlinear) potentials presenting a number of case examples. Generalizations of the method are also discussed.

  3. A class of traveling wave solutions for space-time fractional biological population model in mathematical physics

    NASA Astrophysics Data System (ADS)

    Akram, Ghazala; Batool, Fiza

    2017-10-01

    The (G'/G)-expansion method is utilized for a reliable treatment of space-time fractional biological population model. The method has been applied in the sense of the Jumarie's modified Riemann-Liouville derivative. Three classes of exact traveling wave solutions, hyperbolic, trigonometric and rational solutions of the associated equation are characterized with some free parameters. A generalized fractional complex transform is applied to convert the fractional equations to ordinary differential equations which subsequently resulted in number of exact solutions. It should be mentioned that the (G'/G)-expansion method is very effective and convenient for solving nonlinear partial differential equations of fractional order whose balancing number is a negative integer.

  4. Exact solitary wave solution for higher order nonlinear Schrodinger equation using He's variational iteration method

    NASA Astrophysics Data System (ADS)

    Rani, Monika; Bhatti, Harbax S.; Singh, Vikramjeet

    2017-11-01

    In optical communication, the behavior of the ultrashort pulses of optical solitons can be described through nonlinear Schrodinger equation. This partial differential equation is widely used to contemplate a number of physically important phenomena, including optical shock waves, laser and plasma physics, quantum mechanics, elastic media, etc. The exact analytical solution of (1+n)-dimensional higher order nonlinear Schrodinger equation by He's variational iteration method has been presented. Our proposed solutions are very helpful in studying the solitary wave phenomena and ensure rapid convergent series and avoid round off errors. Different examples with graphical representations have been given to justify the capability of the method.

  5. An exact solution of the van der Waals interaction between two ground-state hydrogen atoms

    NASA Astrophysics Data System (ADS)

    Koga, Toshikatsu; Matsumoto, Shinya

    1985-06-01

    A momentum space treatment shows that perturbation equations for the H(1s)-H(1s) van der Waals interaction can be exactly solved in their Schrödinger forms without invoking any variational methods. Using the Fock transformation, which projects the momentum vector of an electron from the three-dimensional hyperplane onto the four-dimensional hypersphere, we solve the third order integral-type perturbation equation with respect to the reciprocal of the internuclear distance R. An exact third order wave function is found as a linear combination of infinite number of four-dimensional spherical harmonics. The result allows us to evaluate the exact dispersion energy E6R-6, which is completely determined by the first three coefficients of the above linear combination.

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

    He, Yang; Xiao, Jianyuan; Zhang, Ruili

    Hamiltonian time integrators for the Vlasov-Maxwell equations are developed by a Hamiltonian splitting technique. The Hamiltonian functional is split into five parts, which produces five exactly solvable subsystems. Each subsystem is a Hamiltonian system equipped with the Morrison-Marsden-Weinstein Poisson bracket. Compositions of the exact solutions provide Poisson structure preserving/Hamiltonian methods of arbitrary high order for the Vlasov-Maxwell equations. They are then accurate and conservative over a long time because of the Poisson-preserving nature.

  7. Perturbational blowup solutions to the compressible Euler equations with damping.

    PubMed

    Cheung, Ka Luen

    2016-01-01

    The N-dimensional isentropic compressible Euler system with a damping term is one of the most fundamental equations in fluid dynamics. Since it does not have a general solution in a closed form for arbitrary well-posed initial value problems. Constructing exact solutions to the system is a useful way to obtain important information on the properties of its solutions. In this article, we construct two families of exact solutions for the one-dimensional isentropic compressible Euler equations with damping by the perturbational method. The two families of exact solutions found include the cases [Formula: see text] and [Formula: see text], where [Formula: see text] is the adiabatic constant. With analysis of the key ordinary differential equation, we show that the classes of solutions include both blowup type and global existence type when the parameters are suitably chosen. Moreover, in the blowup cases, we show that the singularities are of essential type in the sense that they cannot be smoothed by redefining values at the odd points. The two families of exact solutions obtained in this paper can be useful to study of related numerical methods and algorithms such as the finite difference method, the finite element method and the finite volume method that are applied by scientists to simulate the fluids for applications.

  8. Laplace-Beltrami operator and exact solutions for branes

    NASA Astrophysics Data System (ADS)

    Zheltukhin, A. A.

    2013-02-01

    Proposed is a new approach to finding exact solutions of nonlinear p-brane equations in D-dimensional Minkowski space based on the use of various initial value constraints. It is shown that the constraints Δx→=0 and Δx→=-Λ(t,σr)x→ give two sets of exact solutions.

  9. Wave-current interactions in three dimensions: why 3D radiation stresses are not practical

    NASA Astrophysics Data System (ADS)

    Ardhuin, Fabrice

    2017-04-01

    The coupling of ocean circulation and wave models is based on a wave-averaged mass and momentum conservation equations. Whereas several equivalent equations for the evolution of the current momentum have been proposed, implemented, and used, the possibility to formulate practical equations for the total momentum, which is the sum of the current and wave momenta, has been obscured by a series of publications. In a recent update on previous derivations, Mellor (J. Phys. Oceanogr. 2015) proposed a new set of wave-forced total momentum equations. Here we show that this derivation misses a term that integrates to zero over the vertical. This is because he went from his depth-integrated eq. (28) to the 3D equation (30) by simply removing the integral, but any extra zero-integrating term can be added. Corrected for this omission, the equations of motion are equivalent to the earlier equations by Mellor (2003) which are correct when expressed in terms of wave-induced pressure, horizontal velocity and vertical displacement. Namely the total momentum evolution is driven by the horizontal divergence of a horizontal momentum flux, ----- --- ∂^s- Sαβ = ^uα^uβ + δαβ ∂ς (^p- g^s) (1) and the vertical divergence of a vertical flux, Sαz = (p^-g^s)∂^s/∂xα, (2) where p is the wave-induced non-hydrostatic pressure, s is the wave-induced vertical displacement, and u^ α is the horizontal wave-induced velocity in direction α. So far, so good. Problems arise when p and s are evaluated. Indeend, Ardhuin et al. (J. Phys. Oceanogr. 2008) showed that, over a sloping bottom ∂Sαβ/∂xβ is of order of the slope, hence a consistent wave forcing requires an estimation of Sαz that must be estimated to first order in the bottom slope. For this, Airy wave theory, i.e. cosh(kz-+-kh) p ≃ ga cosh (kD ) cosψ, (3) is not enough. Ardhuin et al. (2008) has shown that using an exact solution of the Laplace equations the vertical flux can indeed be computed. The alternative of neglecting completely Sαz, as suggested by Mellor (2011) for small slopes, will always generate spurious currents because of the unbalanced forcing ∂Sαβ/∂xβ. Fortunately, there are many explicit versions of the wave-averaged equations without the wave momentum in them (Suzuki and Fox-Kemper 2016), with or without vortex force which are all consistent with the exact 3D equations of Andrews and McIntyre (1978). There is thus no need to stumble again and again on this fundamental problem of vertical momentum flux, which is a flux of wave momentum. The problem simply goes away by writing the equations for the current momentum only, without the problematic wave momentum. The current and wave momentum are coupled by forcing terms, and the wave momentum can be solved in 2D, the vertical distribution of momentum being maintained by the complex flux Sαz.

  10. Exact finite elements for conduction and convection

    NASA Technical Reports Server (NTRS)

    Thornton, E. A.; Dechaumphai, P.; Tamma, K. K.

    1981-01-01

    An appproach for developing exact one dimensional conduction-convection finite elements is presented. Exact interpolation functions are derived based on solutions to the governing differential equations by employing a nodeless parameter. Exact interpolation functions are presented for combined heat transfer in several solids of different shapes, and for combined heat transfer in a flow passage. Numerical results demonstrate that exact one dimensional elements offer advantages over elements based on approximate interpolation functions. Previously announced in STAR as N81-31507

  11. General method of solving the Schroedinger equation of atoms and molecules

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

    Nakatsuji, Hiroshi

    2005-12-15

    We propose a general method of solving the Schroedinger equation of atoms and molecules. We first construct the wave function having the exact structure, using the ICI (iterative configuration or complement interaction) method and then optimize the variables involved by the variational principle. Based on the scaled Schroedinger equation and related principles, we can avoid the singularity problem of atoms and molecules and formulate a general method of calculating the exact wave functions in an analytical expansion form. We choose initial function {psi}{sub 0} and scaling g function, and then the ICI method automatically generates the wave function that hasmore » the exact structure by using the Hamiltonian of the system. The Hamiltonian contains all the information of the system. The free ICI method provides a flexible and variationally favorable procedure of constructing the exact wave function. We explain the computational procedure of the analytical ICI method routinely performed in our laboratory. Simple examples are given using hydrogen atom for the nuclear singularity case, the Hooke's atom for the electron singularity case, and the helium atom for both cases.« less

  12. Solving modal equations of motion with initial conditions using MSC/NASTRAN DMAP. Part 1: Implementing exact mode superposition

    NASA Technical Reports Server (NTRS)

    Abdallah, Ayman A.; Barnett, Alan R.; Ibrahim, Omar M.; Manella, Richard T.

    1993-01-01

    Within the MSC/NASTRAN DMAP (Direct Matrix Abstraction Program) module TRD1, solving physical (coupled) or modal (uncoupled) transient equations of motion is performed using the Newmark-Beta or mode superposition algorithms, respectively. For equations of motion with initial conditions, only the Newmark-Beta integration routine has been available in MSC/NASTRAN solution sequences for solving physical systems and in custom DMAP sequences or alters for solving modal systems. In some cases, one difficulty with using the Newmark-Beta method is that the process of selecting suitable integration time steps for obtaining acceptable results is lengthy. In addition, when very small step sizes are required, a large amount of time can be spent integrating the equations of motion. For certain aerospace applications, a significant time savings can be realized when the equations of motion are solved using an exact integration routine instead of the Newmark-Beta numerical algorithm. In order to solve modal equations of motion with initial conditions and take advantage of efficiencies gained when using uncoupled solution algorithms (like that within TRD1), an exact mode superposition method using MSC/NASTRAN DMAP has been developed and successfully implemented as an enhancement to an existing coupled loads methodology at the NASA Lewis Research Center.

  13. Exact Riemann solutions of the Ripa model for flat and non-flat bottom topographies

    NASA Astrophysics Data System (ADS)

    Rehman, Asad; Ali, Ishtiaq; Qamar, Shamsul

    2018-03-01

    This article is concerned with the derivation of exact Riemann solutions for Ripa model considering flat and non-flat bottom topographies. The Ripa model is a system of shallow water equations accounting for horizontal temperature gradients. In the case of non-flat bottom topography, the mass, momentum and energy conservation principles are utilized to relate the left and right states across the step-type bottom topography. The resulting system of algebraic equations is solved iteratively. Different numerical case studies of physical interest are considered. The solutions obtained from developed exact Riemann solvers are compared with the approximate solutions of central upwind scheme.

  14. Simple iterative construction of the optimized effective potential for orbital functionals, including exact exchange.

    PubMed

    Kümmel, Stephan; Perdew, John P

    2003-01-31

    For exchange-correlation functionals that depend explicitly on the Kohn-Sham orbitals, the potential V(xcsigma)(r) must be obtained as the solution of the optimized effective potential (OEP) integral equation. This is very demanding and has limited the use of orbital functionals. We demonstrate that instead the OEP can be obtained iteratively by solving the partial differential equations for the orbital shifts that exactify the Krieger-Li-Iafrate approximation. Unoccupied orbitals do not need to be calculated. Accuracy and efficiency of the method are shown for atoms and clusters using the exact-exchange energy. Counterintuitive asymptotic limits of the exact OEP are presented.

  15. Time-Harmonic Gaussian Beams: Exact Solutions of the Helmhotz Equation in Free Space

    NASA Astrophysics Data System (ADS)

    Kiselev, A. P.

    2017-12-01

    An exact solution of the Helmholtz equation u xx + u yy + u zz + k 2 u = 0 is presented, which describes propagation of monochromatic waves in the free space. The solution has the form of a superposition of plane waves with a specific weight function dependent on a certain free parameter a. If ka→∞, the solution is localized in the Gaussian manner in a vicinity of a certain straight line and asymptotically coincides with the famous approximate solution known as the fundamental mode of a paraxial Gaussian beam. The asymptotics of the aforementioned exact solution does not include a backward wave.

  16. An efficient technique for higher order fractional differential equation.

    PubMed

    Ali, Ayyaz; Iqbal, Muhammad Asad; Ul-Hassan, Qazi Mahmood; Ahmad, Jamshad; Mohyud-Din, Syed Tauseef

    2016-01-01

    In this study, we establish exact solutions of fractional Kawahara equation by using the idea of [Formula: see text]-expansion method. The results of different studies show that the method is very effective and can be used as an alternative for finding exact solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The solitary wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, expedient for fractional PDEs, and could be extended to other physical problems.

  17. Exact periodic cross-kink wave solutions for the new (2+1)-dimensional KdV equation in fluid flows and plasma physics.

    PubMed

    Liu, Jian-Guo; Du, Jian-Qiang; Zeng, Zhi-Fang; Ai, Guo-Ping

    2016-10-01

    The Korteweg-de Vries (KdV)-type models have been shown to describe many important physical situations such as fluid flows, plasma physics, and solid state physics. In this paper, a new (2 + 1)-dimensional KdV equation is discussed. Based on the Hirota's bilinear form and a generalized three-wave approach, we obtain new exact solutions for the new (2 + 1)-dimensional KdV equation. With the help of symbolic computation, the properties for some new solutions are presented with some figures.

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

    Saenz, Juan A.; Chen, Qingshan; Ringler, Todd

    Recent work has shown that taking the thickness-weighted average (TWA) of the Boussinesq equations in buoyancy coordinates results in exact equations governing the prognostic residual mean flow where eddy–mean flow interactions appear in the horizontal momentum equations as the divergence of the Eliassen–Palm flux tensor (EPFT). It has been proposed that, given the mathematical tractability of the TWA equations, the physical interpretation of the EPFT, and its relation to potential vorticity fluxes, the TWA is an appropriate framework for modeling ocean circulation with parameterized eddies. The authors test the feasibility of this proposition and investigate the connections between the TWAmore » framework and the conventional framework used in models, where Eulerian mean flow prognostic variables are solved for. Using the TWA framework as a starting point, this study explores the well-known connections between vertical transfer of horizontal momentum by eddy form drag and eddy overturning by the bolus velocity, used by Greatbatch and Lamb and Gent and McWilliams to parameterize eddies. After implementing the TWA framework in an ocean general circulation model, we verify our analysis by comparing the flows in an idealized Southern Ocean configuration simulated using the TWA and conventional frameworks with the same mesoscale eddy parameterization.« less

  19. Exact solutions for kinetic models of macromolecular dynamics.

    PubMed

    Chemla, Yann R; Moffitt, Jeffrey R; Bustamante, Carlos

    2008-05-15

    Dynamic biological processes such as enzyme catalysis, molecular motor translocation, and protein and nucleic acid conformational dynamics are inherently stochastic processes. However, when such processes are studied on a nonsynchronized ensemble, the inherent fluctuations are lost, and only the average rate of the process can be measured. With the recent development of methods of single-molecule manipulation and detection, it is now possible to follow the progress of an individual molecule, measuring not just the average rate but the fluctuations in this rate as well. These fluctuations can provide a great deal of detail about the underlying kinetic cycle that governs the dynamical behavior of the system. However, extracting this information from experiments requires the ability to calculate the general properties of arbitrarily complex theoretical kinetic schemes. We present here a general technique that determines the exact analytical solution for the mean velocity and for measures of the fluctuations. We adopt a formalism based on the master equation and show how the probability density for the position of a molecular motor at a given time can be solved exactly in Fourier-Laplace space. With this analytic solution, we can then calculate the mean velocity and fluctuation-related parameters, such as the randomness parameter (a dimensionless ratio of the diffusion constant and the velocity) and the dwell time distributions, which fully characterize the fluctuations of the system, both commonly used kinetic parameters in single-molecule measurements. Furthermore, we show that this formalism allows calculation of these parameters for a much wider class of general kinetic models than demonstrated with previous methods.

  20. Localized light waves: Paraxial and exact solutions of the wave equation (a review)

    NASA Astrophysics Data System (ADS)

    Kiselev, A. P.

    2007-04-01

    Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “ X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.

  1. Accurate Exchange-Correlation Energies for the Warm Dense Electron Gas.

    PubMed

    Malone, Fionn D; Blunt, N S; Brown, Ethan W; Lee, D K K; Spencer, J S; Foulkes, W M C; Shepherd, James J

    2016-09-09

    The density matrix quantum Monte Carlo (DMQMC) method is used to sample exact-on-average N-body density matrices for uniform electron gas systems of up to 10^{124} matrix elements via a stochastic solution of the Bloch equation. The results of these calculations resolve a current debate over the accuracy of the data used to parametrize finite-temperature density functionals. Exchange-correlation energies calculated using the real-space restricted path-integral formalism and the k-space configuration path-integral formalism disagree by up to ∼10% at certain reduced temperatures T/T_{F}≤0.5 and densities r_{s}≤1. Our calculations confirm the accuracy of the configuration path-integral Monte Carlo results available at high density and bridge the gap to lower densities, providing trustworthy data in the regime typical of planetary interiors and solids subject to laser irradiation. We demonstrate that the DMQMC method can calculate free energies directly and present exact free energies for T/T_{F}≥1 and r_{s}≤2.

  2. Subharmonic response of a single-degree-of-freedom nonlinear vibro-impact system to a narrow-band random excitation.

    PubMed

    Haiwu, Rong; Wang, Xiangdong; Xu, Wei; Fang, Tong

    2009-08-01

    The subharmonic response of single-degree-of-freedom nonlinear vibro-impact oscillator with a one-sided barrier to narrow-band random excitation is investigated. The narrow-band random excitation used here is a filtered Gaussian white noise. The analysis is based on a special Zhuravlev transformation, which reduces the system to one without impacts, or velocity jumps, thereby permitting the applications of asymptotic averaging over the "fast" variables. The averaged stochastic equations are solved exactly by the method of moments for the mean-square response amplitude for the case of linear system with zero offset. A perturbation-based moment closure scheme is proposed and the formula of the mean-square amplitude is obtained approximately for the case of linear system with nonzero offset. The perturbation-based moment closure scheme is used once again to obtain the algebra equation of the mean-square amplitude of the response for the case of nonlinear system. The effects of damping, detuning, nonlinear intensity, bandwidth, and magnitudes of random excitations are analyzed. The theoretical analyses are verified by numerical results. Theoretical analyses and numerical simulations show that the peak amplitudes may be strongly reduced at large detunings or large nonlinear intensity.

  3. Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics

    NASA Astrophysics Data System (ADS)

    Farrell, Patricio; Koprucki, Thomas; Fuhrmann, Jürgen

    2017-10-01

    We compare three thermodynamically consistent numerical fluxes known in the literature, appearing in a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics. Our discussion includes an extension of the Scharfetter-Gummel scheme to non-Boltzmann (e.g. Fermi-Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

  4. Knotted optical vortices in exact solutions to Maxwell's equations

    NASA Astrophysics Data System (ADS)

    de Klerk, Albertus J. J. M.; van der Veen, Roland I.; Dalhuisen, Jan Willem; Bouwmeester, Dirk

    2017-05-01

    We construct a family of exact solutions to Maxwell's equations in which the points of zero intensity form knotted lines topologically equivalent to a given but arbitrary algebraic link. These lines of zero intensity, more commonly referred to as optical vortices, and their topology are preserved as time evolves and the fields have finite energy. To derive explicit expressions for these new electromagnetic fields that satisfy the nullness property, we make use of the Bateman variables for the Hopf field as well as complex polynomials in two variables whose zero sets give rise to algebraic links. The class of algebraic links includes not only all torus knots and links thereof, but also more intricate cable knots. While the unknot has been considered before, the solutions presented here show that more general knotted structures can also arise as optical vortices in exact solutions to Maxwell's equations.

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

    Pereira, S.H.; Pinho, A.S.S.; Silva, J.M. Hoff da

    In this work the exact Friedmann-Robertson-Walker equations for an Elko spinor field coupled to gravity in an Einstein-Cartan framework are presented. The torsion functions coupling the Elko field spin-connection to gravity can be exactly solved and the FRW equations for the system assume a relatively simple form. In the limit of a slowly varying Elko spinor field there is a relevant contribution to the field equations acting exactly as a time varying cosmological model Λ( t )=Λ{sub *}+3β H {sup 2}, where Λ{sub *} and β are constants. Observational data using distance luminosity from magnitudes of supernovae constraint the parametersmore » Ω {sub m} and β, which leads to a lower limit to the Elko mass. Such model mimics, then, the effects of a dark energy fluid, here sourced by the Elko spinor field. The density perturbations in the linear regime were also studied in the pseudo-Newtonian formalism.« less

  6. Optimum three-dimensional atmospheric entry from the analytical solution of Chapman's exact equations

    NASA Technical Reports Server (NTRS)

    Busemann, A.; Vinh, N. X.; Culp, R. D.

    1974-01-01

    The general solution for the optimum three-dimensional aerodynamic control of a lifting vehicle entering a planetary atmosphere is developed. A set of dimensionless variables, modified Chapman variables, is introduced. The resulting exact equations of motion, referred to as Chapman's exact equations, have the advantage that they are completely free of the physical characteristics of the vehicle. Furthermore, a completely general lift-drag relationship is used in the derivation. The results obtained apply to any type of vehicle of arbitrary weight, dimensions and shape, having an arbitrary drag polar, and entering any planetary atmosphere. The aerodynamic controls chosen are the lift coefficient and the bank angle. General optimum control laws for these controls are developed. Several earlier particular solutions are shown to be special cases of this general result. Results are valid for both free and constrained terminal position.

  7. The exact solutions and approximate analytic solutions of the (2 + 1)-dimensional KP equation based on symmetry method.

    PubMed

    Gai, Litao; Bilige, Sudao; Jie, Yingmo

    2016-01-01

    In this paper, we successfully obtained the exact solutions and the approximate analytic solutions of the (2 + 1)-dimensional KP equation based on the Lie symmetry, the extended tanh method and the homotopy perturbation method. In first part, we obtained the symmetries of the (2 + 1)-dimensional KP equation based on the Wu-differential characteristic set algorithm and reduced it. In the second part, we constructed the abundant exact travelling wave solutions by using the extended tanh method. These solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions respectively. It should be noted that when the parameters are taken as special values, some solitary wave solutions are derived from the hyperbolic function solutions. Finally, we apply the homotopy perturbation method to obtain the approximate analytic solutions based on four kinds of initial conditions.

  8. Kinetic energy equations for the average-passage equation system

    NASA Technical Reports Server (NTRS)

    Johnson, Richard W.; Adamczyk, John J.

    1989-01-01

    Important kinetic energy equations derived from the average-passage equation sets are documented, with a view to their interrelationships. These kinetic equations may be used for closing the average-passage equations. The turbulent kinetic energy transport equation used is formed by subtracting the mean kinetic energy equation from the averaged total instantaneous kinetic energy equation. The aperiodic kinetic energy equation, averaged steady kinetic energy equation, averaged unsteady kinetic energy equation, and periodic kinetic energy equation, are also treated.

  9. Exact Solutions of Coupled Multispecies Linear Reaction–Diffusion Equations on a Uniformly Growing Domain

    PubMed Central

    Simpson, Matthew J.; Sharp, Jesse A.; Morrow, Liam C.; Baker, Ruth E.

    2015-01-01

    Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction–diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction–diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction–diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially–confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially–confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit. PMID:26407013

  10. Exact Solutions of Coupled Multispecies Linear Reaction-Diffusion Equations on a Uniformly Growing Domain.

    PubMed

    Simpson, Matthew J; Sharp, Jesse A; Morrow, Liam C; Baker, Ruth E

    2015-01-01

    Embryonic development involves diffusion and proliferation of cells, as well as diffusion and reaction of molecules, within growing tissues. Mathematical models of these processes often involve reaction-diffusion equations on growing domains that have been primarily studied using approximate numerical solutions. Recently, we have shown how to obtain an exact solution to a single, uncoupled, linear reaction-diffusion equation on a growing domain, 0 < x < L(t), where L(t) is the domain length. The present work is an extension of our previous study, and we illustrate how to solve a system of coupled reaction-diffusion equations on a growing domain. This system of equations can be used to study the spatial and temporal distributions of different generations of cells within a population that diffuses and proliferates within a growing tissue. The exact solution is obtained by applying an uncoupling transformation, and the uncoupled equations are solved separately before applying the inverse uncoupling transformation to give the coupled solution. We present several example calculations to illustrate different types of behaviour. The first example calculation corresponds to a situation where the initially-confined population diffuses sufficiently slowly that it is unable to reach the moving boundary at x = L(t). In contrast, the second example calculation corresponds to a situation where the initially-confined population is able to overcome the domain growth and reach the moving boundary at x = L(t). In its basic format, the uncoupling transformation at first appears to be restricted to deal only with the case where each generation of cells has a distinct proliferation rate. However, we also demonstrate how the uncoupling transformation can be used when each generation has the same proliferation rate by evaluating the exact solutions as an appropriate limit.

  11. Gravitoelectromagnetic analogy based on tidal tensors

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

    Costa, L. Filipe O.; Herdeiro, Carlos A. R.

    2008-07-15

    We propose a new approach to a physical analogy between general relativity and electromagnetism, based on tidal tensors of both theories. Using this approach we write a covariant form for the gravitational analogues of the Maxwell equations, which makes transparent both the similarities and key differences between the two interactions. The following realizations of the analogy are given. The first one matches linearized gravitational tidal tensors to exact electromagnetic tidal tensors in Minkowski spacetime. The second one matches exact magnetic gravitational tidal tensors for ultrastationary metrics to exact magnetic tidal tensors of electromagnetism in curved spaces. In the third wemore » show that our approach leads to a two-step exact derivation of Papapetrou's equation describing the force exerted on a spinning test particle. Analogous scalar invariants built from tidal tensors of both theories are also discussed.« less

  12. Traveling wave solutions of the Boussinesq equation via the new approach of generalized (G'/G)-expansion method.

    PubMed

    Alam, Md Nur; Akbar, M Ali; Roshid, Harun-Or-

    2014-01-01

    Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this work, the exact traveling wave solutions of the Boussinesq equation is studied by using the new generalized (G'/G)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G'/G)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations in mathematical physics and engineering. 05.45.Yv, 02.30.Jr, 02.30.Ik.

  13. An Exact, Compressible One-Dimensional Riemann Solver for General, Convex Equations of State

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

    Kamm, James Russell

    2015-03-05

    This note describes an algorithm with which to compute numerical solutions to the one- dimensional, Cartesian Riemann problem for compressible flow with general, convex equations of state. While high-level descriptions of this approach are to be found in the literature, this note contains most of the necessary details required to write software for this problem. This explanation corresponds to the approach used in the source code that evaluates solutions for the 1D, Cartesian Riemann problem with a JWL equation of state in the ExactPack package [16, 29]. Numerical examples are given with the proposed computational approach for a polytropic equationmore » of state and for the JWL equation of state.« less

  14. Dispersive shock waves in the Kadomtsev-Petviashvili and two dimensional Benjamin-Ono equations

    NASA Astrophysics Data System (ADS)

    Ablowitz, Mark J.; Demirci, Ali; Ma, Yi-Ping

    2016-10-01

    Dispersive shock waves (DSWs) in the Kadomtsev-Petviashvili (KP) equation and two dimensional Benjamin-Ono (2DBO) equation are considered using step like initial data along a parabolic front. Employing a parabolic similarity reduction exactly reduces the study of such DSWs in two space one time (2 + 1) dimensions to finding DSW solutions of (1 + 1) dimensional equations. With this ansatz, the KP and 2DBO equations can be exactly reduced to the cylindrical Korteweg-de Vries (cKdV) and cylindrical Benjamin-Ono (cBO) equations, respectively. Whitham modulation equations which describe DSW evolution in the cKdV and cBO equations are derived and Riemann type variables are introduced. DSWs obtained from the numerical solutions of the corresponding Whitham systems and direct numerical simulations of the cKdV and cBO equations are compared with very good agreement obtained. In turn, DSWs obtained from direct numerical simulations of the KP and 2DBO equations are compared with the cKdV and cBO equations, again with good agreement. It is concluded that the (2 + 1) DSW behavior along self similar parabolic fronts can be effectively described by the DSW solutions of the reduced (1 + 1) dimensional equations.

  15. Exact results in the large system size limit for the dynamics of the chemical master equation, a one dimensional chain of equations.

    PubMed

    Martirosyan, A; Saakian, David B

    2011-08-01

    We apply the Hamilton-Jacobi equation (HJE) formalism to solve the dynamics of the chemical master equation (CME). We found exact analytical expressions (in large system-size limit) for the probability distribution, including explicit expression for the dynamics of variance of distribution. We also give the solution for some simple cases of the model with time-dependent rates. We derived the results of the Van Kampen method from the HJE approach using a special ansatz. Using the Van Kampen method, we give a system of ordinary differential equations (ODEs) to define the variance in a two-dimensional case. We performed numerics for the CME with stationary noise. We give analytical criteria for the disappearance of bistability in the case of stationary noise in one-dimensional CMEs.

  16. Laser radiation in active amplifying media treated as a transport problem - Transfer equation derived and exactly solved

    NASA Astrophysics Data System (ADS)

    Gupta, S. R. D.; Gupta, Santanu D.

    1991-10-01

    The flow of laser radiation in a plane-parallel cylindrical slab of active amplifying medium with axial symmetry is treated as a problem in radiative transfer. The appropriate one-dimensional transfer equation describing the transfer of laser radiation has been derived by an appeal to Einstein's A, B coefficients (describing the processes of stimulated line absorption, spontaneous line emission, and stimulated line emission sustained by population inversion in the medium) and considering the 'rate equations' to completely establish the rational of the transfer equation obtained. The equation is then exactly solved and the angular distribution of the emergent laser beam intensity is obtained; its numerically computed values are given in tables and plotted in graphs showing the nature of peaks of the emerging laser beam intensity about the axis of the laser cylinder.

  17. A highly accurate finite-difference method with minimum dispersion error for solving the Helmholtz equation

    NASA Astrophysics Data System (ADS)

    Wu, Zedong; Alkhalifah, Tariq

    2018-07-01

    Numerical simulation of the acoustic wave equation in either isotropic or anisotropic media is crucial to seismic modeling, imaging and inversion. Actually, it represents the core computation cost of these highly advanced seismic processing methods. However, the conventional finite-difference method suffers from severe numerical dispersion errors and S-wave artifacts when solving the acoustic wave equation for anisotropic media. We propose a method to obtain the finite-difference coefficients by comparing its numerical dispersion with the exact form. We find the optimal finite difference coefficients that share the dispersion characteristics of the exact equation with minimal dispersion error. The method is extended to solve the acoustic wave equation in transversely isotropic (TI) media without S-wave artifacts. Numerical examples show that the method is highly accurate and efficient.

  18. Exact Descriptions of General Relativity Derived from Newtonian Mechanics within Curved Geometries

    NASA Astrophysics Data System (ADS)

    Savickas, David

    2015-04-01

    General relativity and Newtonian mechanics are shown to be exactly related when Newton's second law is written in a curved geometry by using the physical components of a vector as is defined in tensor calculus. By replacing length within the momentum's velocity by the vector metric in a curved geometry the second law can then be shown to be exactly identical to the geodesic equation of motion occurring in general relativity. When time's vector direction is constant, as similarly occurs in Newtonian mechanics, this equation can be reduced to a curved three-dimensional equation of motion that yields the the Schwarzschild equations of motion for an isolated particle. They can be used to describe gravitational behavior for any array of masses for which the Newtonian gravitational potential is known, and is shown to describe a mass particle's behavior in the gravitational field of a thin mass-rod. This use of Newton's laws allows relativistic behavior to be described in a physically comprehensible manner. D. Savickas, Int. J. Mod. Phys. D 23 1430018, (2014).

  19. Exact analytic solution of position-dependent mass Schrödinger equation

    NASA Astrophysics Data System (ADS)

    Rajbongshi, Hangshadhar

    2018-03-01

    Exact analytic solution of position-dependent mass Schrödinger equation is generated by using extended transformation, a method of mapping a known system into a new system equipped with energy eigenvalues and corresponding wave functions. First order transformation is performed on D-dimensional radial Schrödinger equation with constant mass by taking trigonometric Pöschl-Teller potential as known system. The exactly solvable potentials with position-dependent mass generated for different choices of mass functions through first order transformation are also taken as known systems in the second order transformation performed on D-dimensional radial position-dependent mass Schrödinger equation. The solutions are fitted for "Zhu and Kroemer" ordering of ambiguity. All the wave functions corresponding to nonzero energy eigenvalues are normalizable. The new findings are that the normalizability condition of the wave functions remains independent of mass functions, and some of the generated potentials show a family relationship among themselves where power law potentials also get related to non-power law potentials and vice versa through the transformation.

  20. Lie Symmetry Analysis, Conservation Laws and Exact Power Series Solutions for Time-Fractional Fordy-Gibbons Equation

    NASA Astrophysics Data System (ADS)

    Feng, Lian-Li; Tian, Shou-Fu; Wang, Xiu-Bin; Zhang, Tian-Tian

    2016-09-01

    In this paper, the time fractional Fordy-Gibbons equation is investigated with Riemann-Liouville derivative. The equation can be reduced to the Caudrey-Dodd-Gibbon equation, Savada-Kotera equation and the Kaup-Kupershmidt equation, etc. By means of the Lie group analysis method, the invariance properties and symmetry reductions of the equation are derived. Furthermore, by means of the power series theory, its exact power series solutions of the equation are also constructed. Finally, two kinds of conservation laws of the equation are well obtained with aid of the self-adjoint method. Supported by the Fundamental Research Funds for Key Discipline Construction under Grant No. XZD201602, the Fundamental Research Funds for the Central Universities under Grant Nos. 2015QNA53 and 2015XKQY14, the Fundamental Research Funds for Postdoctoral at the Key Laboratory of Gas and Fire Control for Coal Mines, the General Financial Grant from the China Postdoctoral Science Foundation under Grant No. 2015M570498, and Natural Sciences Foundation of China under Grant No. 11301527

  1. An exact solution to the relativistic equation of motion of a charged particle driven by a linearly polarized electromagnetic wave

    NASA Technical Reports Server (NTRS)

    Shebalin, John V.

    1988-01-01

    An exact analytic solution is found for a basic electromagnetic wave-charged particle interaction by solving the nonlinear equations of motion. The particle position, velocity, and corresponding time are found to be explicit functions of the total phase of the wave. Particle position and velocity are thus implicit functions of time. Applications include describing the motion of a free electron driven by an intense laser beam..

  2. Exact coherent structures in an asymptotically reduced description of parallel shear flows

    NASA Astrophysics Data System (ADS)

    Beaume, Cédric; Knobloch, Edgar; Chini, Gregory P.; Julien, Keith

    2015-02-01

    A reduced description of shear flows motivated by the Reynolds number scaling of lower-branch exact coherent states in plane Couette flow (Wang J, Gibson J and Waleffe F 2007 Phys. Rev. Lett. 98 204501) is constructed. Exact time-independent nonlinear solutions of the reduced equations corresponding to both lower and upper branch states are found for a sinusoidal, body-forced shear flow. The lower branch solution is characterized by fluctuations that vary slowly along the critical layer while the upper branch solutions display a bimodal structure and are more strongly focused on the critical layer. The reduced equations provide a rational framework for investigations of subcritical spatiotemporal patterns in parallel shear flows.

  3. A new class of exact solutions of the Klein-Gordon equation of a charged particle interacting with an electromagnetic plane wave in a medium

    NASA Astrophysics Data System (ADS)

    Varró, Sándor

    2014-01-01

    Exact solutions are presented of the Klein-Gordon equation of a charged particle moving in a transverse monochromatic plasmon wave of arbitrary high amplitude, which propagates in an underdense plasma. These solutions are expressed in terms of Ince polynomials, forming a doubly infinite set, parametrized by discrete momentum components of the charged particle’s de Broglie wave along the polarization vector and along the propagation direction of the plasmon radiation. The envelope of the exact wavefunctions describes a high-contrast periodic structure of the particle density on the plasma length scale, which may have relevance in novel particle acceleration mechanisms.

  4. Correlated electron-nuclear dynamics with conditional wave functions.

    PubMed

    Albareda, Guillermo; Appel, Heiko; Franco, Ignacio; Abedi, Ali; Rubio, Angel

    2014-08-22

    The molecular Schrödinger equation is rewritten in terms of nonunitary equations of motion for the nuclei (or electrons) that depend parametrically on the configuration of an ensemble of generally defined electronic (or nuclear) trajectories. This scheme is exact and does not rely on the tracing out of degrees of freedom. Hence, the use of trajectory-based statistical techniques can be exploited to circumvent the calculation of the computationally demanding Born-Oppenheimer potential-energy surfaces and nonadiabatic coupling elements. The concept of the potential-energy surface is restored by establishing a formal connection with the exact factorization of the full wave function. This connection is used to gain insight from a simplified form of the exact propagation scheme.

  5. Calculating Relativistic Transition Matrix Elements for Hydrogenic Atoms Using Monte Carlo Methods

    NASA Astrophysics Data System (ADS)

    Alexander, Steven; Coldwell, R. L.

    2015-03-01

    The nonrelativistic transition matrix elements for hydrogen atoms can be computed exactly and these expressions are given in a number of classic textbooks. The relativistic counterparts of these equations can also be computed exactly but these expressions have been described in only a few places in the literature. In part, this is because the relativistic equations lack the elegant simplicity of the nonrelativistic equations. In this poster I will describe how variational Monte Carlo methods can be used to calculate the energy and properties of relativistic hydrogen atoms and how the wavefunctions for these systems can be used to calculate transition matrix elements.

  6. Renormalization of the fragmentation equation: exact self-similar solutions and turbulent cascades.

    PubMed

    Saveliev, V L; Gorokhovski, M A

    2012-12-01

    Using an approach developed earlier for renormalization of the Boltzmann collision integral [Saveliev and Nanbu, Phys. Rev. E 65, 051205 (2002)], we derive an exact divergence form for the fragmentation operator. Then we reduce the fragmentation equation to the continuity equation in size space, with the flux given explicitly. This allows us to obtain self-similar solutions and to find the integral of motion for these solutions (we call it the bare flux). We show how these solutions can be applied as a description of cascade processes in three- and two-dimensional turbulence. We also suggested an empirical cascade model of impact fragmentation of brittle materials.

  7. Explicit least squares system parameter identification for exact differential input/output models

    NASA Technical Reports Server (NTRS)

    Pearson, A. E.

    1993-01-01

    The equation error for a class of systems modeled by input/output differential operator equations has the potential to be integrated exactly, given the input/output data on a finite time interval, thereby opening up the possibility of using an explicit least squares estimation technique for system parameter identification. The paper delineates the class of models for which this is possible and shows how the explicit least squares cost function can be obtained in a way that obviates dealing with unknown initial and boundary conditions. The approach is illustrated by two examples: a second order chemical kinetics model and a third order system of Lorenz equations.

  8. Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

    NASA Astrophysics Data System (ADS)

    Vitanov, Nikolay K.; Dimitrova, Zlatinka I.

    2018-03-01

    We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

  9. Nonintegrable semidiscrete Hirota equation: gauge-equivalent structures and dynamical properties.

    PubMed

    Ma, Li-Yuan; Zhu, Zuo-Nong

    2014-09-01

    In this paper, we investigate nonintegrable semidiscrete Hirota equations, including the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation. We focus on the topics on gauge-equivalent structures and dynamical behaviors for the two nonintegrable semidiscrete equations. By using the concept of the prescribed discrete curvature, we show that, under the discrete gauge transformations, the nonintegrable semidiscrete Hirota(-) equation and the nonintegrable semidiscrete Hirota(+) equation are, respectively, gauge equivalent to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We prove that the two discrete gauge transformations are reversible. We study the dynamical properties for the two nonintegrable semidiscrete Hirota equations. The exact spatial period solutions of the two nonintegrable semidiscrete Hirota equations are obtained through the constructions of period orbits of the stationary discrete Hirota equations. We discuss the topic regarding whether the spatial period property of the solution to the nonintegrable semidiscrete Hirota equation is preserved to that of the corresponding gauge-equivalent nonintegrable semidiscrete equations under the action of discrete gauge transformation. By using the gauge equivalent, we obtain the exact solutions to the nonintegrable generalized semidiscrete modified Heisenberg ferromagnet equation and the nonintegrable generalized semidiscrete Heisenberg ferromagnet equation. We also give the numerical simulations for the stationary discrete Hirota equations. We find that their dynamics are much richer than the ones of stationary discrete nonlinear Schrödinger equations.

  10. Scattering of Gaussian Beams by Disordered Particulate Media

    NASA Technical Reports Server (NTRS)

    Mishchenko, Michael I.; Dlugach, Janna M.

    2016-01-01

    A frequently observed characteristic of electromagnetic scattering by a disordered particulate medium is the absence of pronounced speckles in angular patterns of the scattered light. It is known that such diffuse speckle-free scattering patterns can be caused by averaging over randomly changing particle positions and/or over a finite spectral range. To get further insight into the possible physical causes of the absence of speckles, we use the numerically exact superposition T-matrix solver of the Maxwell equations and analyze the scattering of plane-wave and Gaussian beams by representative multi-sphere groups. We show that phase and amplitude variations across an incident Gaussian beam do not serve to extinguish the pronounced speckle pattern typical of plane-wave illumination of a fixed multi-particle group. Averaging over random particle positions and/or over a finite spectral range is still required to generate the classical diffuse speckle-free regime.

  11. On exact solutions for disturbances to the asymptotic suction boundary layer: transformation of Barnes integrals to convolution integrals

    NASA Astrophysics Data System (ADS)

    Russell, John

    2000-11-01

    A modified Orr-Sommerfeld equation that applies to the asymptotic suction boundary layer was reported by Bussmann & Münz in a wartime report dated 1942 and by Hughes & Reid in J.F.M. ( 23, 1965, p715). Fundamental systems of exact solutions of the Orr-Sommerfeld equation for this mean velocity distribution were reported by D. Grohne in an unpublished typescript dated 1950. Exact solutions of the equation of Bussmann, Münz, Hughes, & Reid were reported by P. Baldwin in Mathematika ( 17, 1970, p206). Grohne and Baldwin noticed that these exact solutions may be expressed either as Barnes integrals or as convolution integrals. In a later paper (Phil. Trans. Roy. Soc. A, 399, 1985, p321), Baldwin applied the convolution integrals in the contruction of large-Reynolds number asymptotic approximations that hold uniformly. The present talk discusses the subtleties that arise in the construction of such convolution integrals, including several not reported by Grohne or Baldwin. The aim is to recover the full set of seven solutions (one well balanced, three balanced, and three dominant-recessive) postulated by W.H. Reid in various works on the uniformly valid solutions.

  12. Lie symmetries for systems of evolution equations

    NASA Astrophysics Data System (ADS)

    Paliathanasis, Andronikos; Tsamparlis, Michael

    2018-01-01

    The Lie symmetries for a class of systems of evolution equations are studied. The evolution equations are defined in a bimetric space with two Riemannian metrics corresponding to the space of the independent and dependent variables of the differential equations. The exact relation of the Lie symmetries with the collineations of the bimetric space is determined.

  13. Exact multisoliton solutions of general nonlinear Schrödinger equation with derivative.

    PubMed

    Li, Qi; Duan, Qiu-yuan; Zhang, Jian-bing

    2014-01-01

    Multisoliton solutions are derived for a general nonlinear Schrödinger equation with derivative by using Hirota's approach. The dynamics of one-soliton solution and two-soliton interactions are also illustrated. The considered equation can reduce to nonlinear Schrödinger equation with derivative as well as the solutions.

  14. Analytical Theory of the Destruction Terms in Dissipation Rate Transport Equations

    NASA Technical Reports Server (NTRS)

    Rubinstein, Robert; Zhou, Ye

    1996-01-01

    Modeled dissipation rate transport equations are often derived by invoking various hypotheses to close correlations in the corresponding exact equations. D. C. Leslie suggested that these models might be derived instead from Kraichnan's wavenumber space integrals for inertial range transport power. This suggestion is applied to the destruction terms in the dissipation rate equations for incompressible turbulence, buoyant turbulence, rotating incompressible turbulence, and rotating buoyant turbulence. Model constants like C(epsilon 2) are expressed as integrals; convergence of these integrals implies the absence of Reynolds number dependence in the corresponding destruction term. The dependence of C(epsilon 2) on rotation rate emerges naturally; sensitization of the modeled dissipation rate equation to rotation is not required. A buoyancy related effect which is absent in the exact transport equation for temperature variance dissipation, but which sometimes improves computational predictions, also arises naturally. Both the presence of this effect and the appropriate time scale in the modeled transport equation depend on whether Bolgiano or Kolmogorov inertial range scaling applies. A simple application of these methods leads to a preliminary, dissipation rate equation for rotating buoyant turbulence.

  15. Solving Equations Today.

    ERIC Educational Resources Information Center

    Shumway, Richard J.

    1989-01-01

    Illustrated is the problem of solving equations and some different strategies students might employ when using available technology. Gives illustrations for: exact solutions, approximate solutions, and approximate solutions which are graphically generated. (RT)

  16. Determining linear vibration frequencies of a ferromagnetic shell

    NASA Astrophysics Data System (ADS)

    Bagdoev, A. G.; Vardanyan, A. V.; Vardanyan, S. V.; Kukudzhanov, V. N.

    2007-10-01

    The problems of determining the roots of dispersion equations for free bending vibrations of thin magnetoelastic plates and shells are of both theoretical and practical interest, in particular, in studying vibrations of metallic structures used in controlled thermonuclear reactors. These problems were solved on the basis of the Kirchhoff hypothesis in [1-5]. In [6], an exact spatial approach to determining the vibration frequencies of thin plates was suggested, and it was shown that it completely agrees with the solution obtained according to the Kirchhoff hypothesis. In [7-9], this exact approach was used to solve the problem on vibrations of thin magnetoelastic plates, and it was shown by cumbersome calculations that the solutions obtained according to the exact theory and the Kirchhoff hypothesis differ substantially except in a single case. In [10], the equations of the dynamic theory of elasticity in the axisymmetric problem are given. In [11], the equations for the vibration frequencies of thin ferromagnetic plates with arbitrary conductivity were obtained in the exact statement. In [12], the Kirchhoff hypothesis was used to obtain dispersion relations for a magnetoelastic thin shell. In [5, 13-16], the relations for the Maxwell tensor and the ponderomotive force for magnetics were presented. In [17], the dispersion relations for thin ferromagnetic plates in the transverse field in the spatial statement were studied analytically and numerically. In the present paper, on the basis of the exact approach, we study free bending vibrations of a thin ferromagnetic cylindrical shell. We obtain the exact dispersion equation in the form of a sixth-order determinant, which can be solved numerically in the case of a magnetoelastic thin shell. The numerical results are presented in tables and compared with the results obtained by the Kirchhoff hypothesis. We show a large number of differences in the results, even for the least frequency.

  17. Stability: Conservation laws, Painlevé analysis and exact solutions for S-KP equation in coupled dusty plasma

    NASA Astrophysics Data System (ADS)

    EL-Kalaawy, O. H.; Moawad, S. M.; Wael, Shrouk

    The propagation of nonlinear waves in unmagnetized strongly coupled dusty plasma with Boltzmann distributed electrons, iso-nonthermal distributed ions and negatively charged dust grains is considered. The basic set of fluid equations is reduced to the Schamel Kadomtsev-Petviashvili (S-KP) equation by using the reductive perturbation method. The variational principle and conservation laws of S-KP equation are obtained. It is shown that the S-KP equation is non-integrable using Painlevé analysis. A set of new exact solutions are obtained by auto-Bäcklund transformations. The stability analysis is discussed for the existence of dust acoustic solitary waves (DASWs) and it is found that the physical parameters have strong effects on the stability criterion. In additional to, the electric field and the true Mach number of this solution are investigated. Finally, we will study the physical meanings of solutions.

  18. Inclusion of exact exchange in the noniterative partial-differential-equation method of electron-molecule scattering - Application to e-N2

    NASA Technical Reports Server (NTRS)

    Weatherford, C. A.; Onda, K.; Temkin, A.

    1985-01-01

    The noniterative partial-differential-equation (PDE) approach to electron-molecule scattering of Onda and Temkin (1983) is modified to account for the effects of exchange explicitly. The exchange equation is reduced to a set of inhomogeneous equations containing no integral terms and solved noniteratively in a difference form; a method for propagating the solution to large values of r is described; the changes in the polarization potential of the original PDE method required by the inclusion of exact static exchange are indicated; and the results of computations for e-N2 scattering in the fixed-nuclei approximation are presented in tables and graphs and compared with previous calculations and experimental data. Better agreement is obtained using the modified PDE method.

  19. Rogue wave solutions for the infinite integrable nonlinear Schrödinger equation hierarchy.

    PubMed

    Ankiewicz, A; Akhmediev, N

    2017-07-01

    We present rogue wave solutions of the integrable nonlinear Schrödinger equation hierarchy with an infinite number of higher-order terms. The latter include higher-order dispersion and higher-order nonlinear terms. In particular, we derive the fundamental rogue wave solutions for all orders of the hierarchy, with exact expressions for velocities, phase, and "stretching factors" in the solutions. We also present several examples of exact solutions of second-order rogue waves, including rogue wave triplets.

  20. Inclusion of transverse shear deformation in exact buckling and vibration analysis of composite plate assemblies

    NASA Technical Reports Server (NTRS)

    Anderson, Melvin S.; Kennedy, David

    1992-01-01

    The problem considered is the development of the necessary plate stiffnesses for use in a general purpose program for buckling and vibration of composite plate assemblies. The required stiffnesses are for the assumption of sinusoidal response along the plate length with transverse shear included. The method is based on the exact solution of the plate differential equations for a composite laminate having fully populated A, B, and D matrices which leads to a differential equation of tenth order.

  1. Hydrodynamics beyond Navier-Stokes: exact solution to the lattice Boltzmann hierarchy.

    PubMed

    Ansumali, S; Karlin, I V; Arcidiacono, S; Abbas, A; Prasianakis, N I

    2007-03-23

    The exact solution to the hierarchy of nonlinear lattice Boltzmann (LB) kinetic equations in the stationary planar Couette flow is found at nonvanishing Knudsen numbers. A new method of solving LB kinetic equations which combines the method of moments with boundary conditions for populations enables us to derive closed-form solutions for all higher-order moments. A convergence of results suggests that the LB hierarchy with larger velocity sets is the novel way to approximate kinetic theory.

  2. Using trees to compute approximate solutions to ordinary differential equations exactly

    NASA Technical Reports Server (NTRS)

    Grossman, Robert

    1991-01-01

    Some recent work is reviewed which relates families of trees to symbolic algorithms for the exact computation of series which approximate solutions of ordinary differential equations. It turns out that the vector space whose basis is the set of finite, rooted trees carries a natural multiplication related to the composition of differential operators, making the space of trees an algebra. This algebraic structure can be exploited to yield a variety of algorithms for manipulating vector fields and the series and algebras they generate.

  3. Multidimensional, fully implicit, exactly conserving electromagnetic particle-in-cell simulations

    NASA Astrophysics Data System (ADS)

    Chacon, Luis

    2015-09-01

    We discuss a new, conservative, fully implicit 2D-3V particle-in-cell algorithm for non-radiative, electromagnetic kinetic plasma simulations, based on the Vlasov-Darwin model. Unlike earlier linearly implicit PIC schemes and standard explicit PIC schemes, fully implicit PIC algorithms are unconditionally stable and allow exact discrete energy and charge conservation. This has been demonstrated in 1D electrostatic and electromagnetic contexts. In this study, we build on these recent algorithms to develop an implicit, orbit-averaged, time-space-centered finite difference scheme for the Darwin field and particle orbit equations for multiple species in multiple dimensions. The Vlasov-Darwin model is very attractive for PIC simulations because it avoids radiative noise issues in non-radiative electromagnetic regimes. The algorithm conserves global energy, local charge, and particle canonical-momentum exactly, even with grid packing. The nonlinear iteration is effectively accelerated with a fluid preconditioner, which allows efficient use of large timesteps, O(√{mi/me}c/veT) larger than the explicit CFL. In this presentation, we will introduce the main algorithmic components of the approach, and demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 1D and 2D. Support from the LANL LDRD program and the DOE-SC ASCR office.

  4. Calculating pH-dependent free energy of proteins by using Monte Carlo protonation probabilities of ionizable residues.

    PubMed

    Huang, Qiang; Herrmann, Andreas

    2012-03-01

    Protein folding, stability, and function are usually influenced by pH. And free energy plays a fundamental role in analysis of such pH-dependent properties. Electrostatics-based theoretical framework using dielectric solvent continuum model and solving Poisson-Boltzmann equation numerically has been shown to be very successful in understanding the pH-dependent properties. However, in this approach the exact computation of pH-dependent free energy becomes impractical for proteins possessing more than several tens of ionizable sites (e.g. > 30), because exact evaluation of the partition function requires a summation over a vast number of possible protonation microstates. Here we present a method which computes the free energy using the average energy and the protonation probabilities of ionizable sites obtained by the well-established Monte Carlo sampling procedure. The key feature is to calculate the entropy by using the protonation probabilities. We used this method to examine a well-studied protein (lysozyme) and produced results which agree very well with the exact calculations. Applications to the optimum pH of maximal stability of proteins and protein-DNA interactions have also resulted in good agreement with experimental data. These examples recommend our method for application to the elucidation of the pH-dependent properties of proteins.

  5. Real time correlation function in a single phase space integral beyond the linearized semiclassical initial value representation.

    PubMed

    Liu, Jian; Miller, William H

    2007-06-21

    It is shown how quantum mechanical time correlation functions [defined, e.g., in Eq. (1.1)] can be expressed, without approximation, in the same form as the linearized approximation of the semiclassical initial value representation (LSC-IVR), or classical Wigner model, for the correlation function [cf. Eq. (2.1)], i.e., as a phase space average (over initial conditions for trajectories) of the Wigner functions corresponding to the two operators. The difference is that the trajectories involved in the LSC-IVR evolve classically, i.e., according to the classical equations of motion, while in the exact theory they evolve according to generalized equations of motion that are derived here. Approximations to the exact equations of motion are then introduced to achieve practical methods that are applicable to complex (i.e., large) molecular systems. Four such methods are proposed in the paper--the full Wigner dynamics (full WD) and the second order WD based on "Wigner trajectories" [H. W. Lee and M. D. Scully, J. Chem. Phys. 77, 4604 (1982)] and the full Donoso-Martens dynamics (full DMD) and the second order DMD based on "Donoso-Martens trajectories" [A. Donoso and C. C. Martens, Phys. Rev. Lett. 8722, 223202 (2001)]--all of which can be viewed as generalizations of the original LSC-IVR method. Numerical tests of the four versions of this new approach are made for two anharmonic model problems, and for each the momentum autocorrelation function (i.e., operators linear in coordinate or momentum operators) and the force autocorrelation function (nonlinear operators) have been calculated. These four new approximate treatments are indeed seen to be significant improvements to the original LSC-IVR approximation.

  6. Project SQUID: The Foundations of Nonequilibrium Statistical Mechanics. Volume 1

    DTIC Science & Technology

    1963-06-01

    equations available (Boltzmann, Landau,, Bogolubov- Balescu -Lenard) are essentially exact and cannot be improved. That isv for kinetic gases (those...effects) as well as to the newly obtained kinetic equation for plasmas (8) (Bogolubov- Balescu -Lenard equation). The hope of ob- taining correctly

  7. Trajectory And Heating Of A Hypervelocity Projectile

    NASA Technical Reports Server (NTRS)

    Tauber, Michael E.

    1992-01-01

    Technical paper presents derivation of approximate, closed-form equation for relationship between velocity of projectile and density of atmosphere. Results of calculations based on approximate equation agree well with results from numerical integrations of exact equations of motion. Comparisons of results presented in series of graphs.

  8. General Relativity Exactly Described by Use of Newton's Laws within a Curved Geometry

    NASA Astrophysics Data System (ADS)

    Savickas, David

    2014-03-01

    The connection between general relativity and Newtonian mechanics is shown to be much closer than generally recognized. When Newton's second law is written in a curved geometry by using the physical components of a vector as defined in tensor calculus, and by replacing distance within the momentum's velocity by the vector metric ds in a curved geometry, the second law can then be easily shown to be exactly identical to the geodesic equation of motion occurring in general relativity. By using a time whose vector direction is constant, as similarly occurs in Newtonian mechanics, this equation can be separated into two equations one of which is a curved three-dimensional equation of motion and the other is an equation for energy. For the gravitational field of an isolated particle, they yield the Schwarzschild equations. They can be used to describe gravitation for any array of masses for which the Newtonian gravitational potential is known, and is applied here to describe motion in the gravitational field of a thin mass-rod.

  9. Bosonized Supersymmetric Sawada-Kotera Equations: Symmetries and Exact Solutions

    NASA Astrophysics Data System (ADS)

    Liu, Ping; Zeng, Bao-Qing; Liu, Li-Ming

    2015-04-01

    The Bosonized Supersymmetric Sawada-Kotera (BSSK) system is constructed by applying bosonization method to a Supersymmetric Sawada-Kotera system in this paper. The symmetries on the BSSK equations are researched and the calculation shows that the BSSK equations are invariant under the scaling transformations, the space-time translations and Galilean boosts. The one-parameter invariant subgroups and the corresponding invariant solutions are researched for the BSSK equations. Four types of reduction equations and similarity solutions are proposed. Period Cnoidal wave solutions, dark solitary wave solutions and bright solitary wave solutions of the BSSK equations are demonstrated and some evolution curves of the exact solutions are figured out. Supported by the National Natural Science Foundation of China under Grant No. 11305031, the Natural Science Foundation of Guangdong Province under Grant No. S2013010011546, the Science and Technology Project Foundation of Zhongshan under Grant Nos. 2013A3FC0264 and 2013A3FC0334, and Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No. Yq2013205

  10. Growing surfaces with anomalous diffusion: Results for the fractal Kardar-Parisi-Zhang equation

    NASA Astrophysics Data System (ADS)

    Katzav, Eytan

    2003-09-01

    In this paper I study a model for a growing surface in the presence of anomalous diffusion, also known as the fractal Kardar-Parisi-Zhang equation (FKPZ). This equation includes a fractional Laplacian that accounts for the possibility that surface transport is caused by a hopping mechanism of a Levy flight. It is shown that for a specific choice of parameters of the FKPZ equation, the equation can be solved exactly in one dimension, so that all the critical exponents, which describe the surface that grows under FKPZ, can be derived for that case. Afterwards, the self-consistent expansion (SCE) is used to predict the critical exponents for the FKPZ model for any choice of the parameters and any spatial dimension. It is then verified that the results obtained using SCE recover the exact result in one dimension. At the end a simple picture for the behavior of the fractal KPZ equation is suggested and the upper critical dimension of this model is discussed.

  11. Exact axisymmetric solutions of the Maxwell equations in a nonlinear nondispersive medium.

    PubMed

    Petrov, E Yu; Kudrin, A V

    2010-05-14

    The features of propagation of intense waves are of great interest for theory and experiment in electrodynamics and acoustics. The behavior of nonlinear waves in a bounded volume is of special importance and, at the same time, is an extremely complicated problem. It seems almost impossible to find a rigorous solution to such a problem even for any model of nonlinearity. We obtain the first exact solution of this type. We present a new method for deriving exact solutions of the Maxwell equations in a nonlinear medium without dispersion and give examples of the obtained solutions that describe propagation of cylindrical electromagnetic waves in a nonlinear nondispersive medium and free electromagnetic oscillations in a cylindrical cavity resonator filled with such a medium.

  12. Interior radiances in optically deep absorbing media. 1: Exact solutions for one-dimensional model

    NASA Technical Reports Server (NTRS)

    Kattawar, G. W.; Plass, G. N.

    1973-01-01

    The exact solutions are obtained for a one dimensional model of a scattering and absorbing medium. The results are given for both the reflected and transmitted radiance for any arbitrary surface albedo as well as for the interior radiance. These same quantities are calculated by the matrix operator method. The relative error of the solutions is obtained by comparison with the exact solutions as well as by an error analysis of the equations. The importance of an accurate starting value for the reflection and transmission operators is shown. A fourth order Runge-Kutta method can be used to solve the differential equations satisfied by these operators in order to obtain such accurate starting values.

  13. Exact Correlation Functions in S U (2 ) N =2 Superconformal QCD

    NASA Astrophysics Data System (ADS)

    Baggio, Marco; Niarchos, Vasilis; Papadodimas, Kyriakos

    2014-12-01

    We report an exact solution of 2- and 3-point functions of chiral primary fields in S U (2 ) N =2 super-Yang-Mills theory coupled to four hypermultiplets. It is shown that these correlation functions are nontrivial functions of the gauge coupling, obeying differential equations which take the form of the semi-infinite Toda chain. We solve these equations recursively in terms of the Zamolodchikov metric that can be determined exactly from supersymmetric localization on the four-sphere. Our results are verified independently in perturbation theory with a Feynman diagram computation up to 2 loops. This is a short version of a companion paper that contains detailed technical remarks, additional material, and aspects of an extension to the S U (N ) gauge group.

  14. Rotation relaxation splitting for optimizing parallel RF excitation pulses with T1 - and T2 -relaxations in MRI

    NASA Astrophysics Data System (ADS)

    Majewski, Kurt

    2018-03-01

    Exact solutions of the Bloch equations with T1 - and T2 -relaxation terms for piecewise constant magnetic fields are numerically challenging. We therefore investigate an approximation for the achieved magnetization in which rotations and relaxations are split into separate operations. We develop an estimate for its accuracy and explicit first and second order derivatives with respect to the complex excitation radio frequency voltages. In practice, the deviation between an exact solution of the Bloch equations and this rotation relaxation splitting approximation seems negligible. Its computation times are similar to exact solutions without relaxation terms. We apply the developed theory to numerically optimize radio frequency excitation waveforms with T1 - and T2 -relaxations in several examples.

  15. Rainfall-runoff response informed by exact solutions of Boussinesq equation on hillslopes

    NASA Astrophysics Data System (ADS)

    Bartlett, M. S., Jr.; Porporato, A. M.

    2017-12-01

    The Boussinesq equation offers a powerful approach forunderstanding the flow dynamics of unconfined aquifers. Though this nonlinear equation allows for concise representation of both soil and geomorphological controls on groundwater flow, it has only been solved exactly for a limited number of initial and boundary conditions. These solutions do not include source/sink terms (evapotranspiration, recharge, and seepage to bedrock) and are typically limited to horizontal aquifers. Here we present a class of exact solutions that are general to sloping aquifers and a time varying source/sink term. By incorporating the source/sink term, they may describe aquifers with both time varying recharge over seasonal or weekly time scales, as well as a loss of water from seepage to the bedrock interface, which is a common feature in hillslopes. These new solutions shed light on the hysteretic relationship between streamflow and groundwater and the behavior of the hydrograph recession curves, thus providing a robust basis for deriving a runoff curves for the partition of rainfall into infiltration and runoff.

  16. Generalized Landau Equation for a System with a Self-Consistent Mean Field - Derivation from an N-Particle Liouville Equation

    NASA Astrophysics Data System (ADS)

    Kandrup, H.

    1981-02-01

    Assume that the evolution of a system is determined by an N-particle Liouville equation. Suppose, moreover, that the particles which compose the system interact via a long range force like gravity so that the system will be spatially inhomogeneous. In this case, the mean force acting upon a test particle does not vanish, so that one wishes to isolate a self-consistent mean field and distinguish its "systematic" effects from the effects of "fluctuations." This is done here. The time-dependent projection operator formalism of Willis and Picard is used to obtain an exact equation for the time evolution of an appropriately defined one-particle probability density. If one implements the assumption that the "fluctuation" time scale is much shorter than both the relaxation and dynamical time scales, this exact equation can be approximated as a closed Markovian equation. In the limiting case of spatial homogeneity, one recovers precisely the standard Landau equation, which is customarily derived by a stochastic binary-encounter argument. This equation is contrasted with the standard heuristic equation for a mean field theory, as formulated for a Newtonian r-1 gravitational potential in stellar dynamics.

  17. A remark on fractional differential equation involving I-function

    NASA Astrophysics Data System (ADS)

    Mishra, Jyoti

    2018-02-01

    The present paper deals with the solution of the fractional differential equation using the Laplace transform operator and its corresponding properties in the fractional calculus; we derive an exact solution of a complex fractional differential equation involving a special function known as I-function. The analysis of the some fractional integral with two parameters is presented using the suggested Theorem 1. In addition, some very useful corollaries are established and their proofs presented in detail. Some obtained exact solutions are depicted to see the effect of each fractional order. Owing to the wider applicability of the I-function, we can conclude that, the obtained results in our work generalize numerous well-known results obtained by specializing the parameters.

  18. Exact differential equation for the density and ionization energy of a many-particle system

    NASA Technical Reports Server (NTRS)

    Levy, M.; Perdew, J. P.; Sahni, V.

    1984-01-01

    The present investigation is concerned with relations studied by Hohenberg and Kohn (1964) and Kohn and Sham (1965). The properties of a ground-state many-electron system are determined by the electron density. The correct differential equation for the density, as dictated by density-functional theory, is presented. It is found that the ground-state density n of a many-electron system obeys a Schroedinger-like differential equation which may be solved by standard Kohn-Sham programs. Results are connected to the traditional exact Kohn-Sham theory. It is pointed out that the results of the current investigations are readily extended to spin-density functional theory.

  19. Renormalization of the fragmentation equation: Exact self-similar solutions and turbulent cascades

    NASA Astrophysics Data System (ADS)

    Saveliev, V. L.; Gorokhovski, M. A.

    2012-12-01

    Using an approach developed earlier for renormalization of the Boltzmann collision integral [Saveliev and Nanbu, Phys. Rev. E1539-375510.1103/PhysRevE.65.051205 65, 051205 (2002)], we derive an exact divergence form for the fragmentation operator. Then we reduce the fragmentation equation to the continuity equation in size space, with the flux given explicitly. This allows us to obtain self-similar solutions and to find the integral of motion for these solutions (we call it the bare flux). We show how these solutions can be applied as a description of cascade processes in three- and two-dimensional turbulence. We also suggested an empirical cascade model of impact fragmentation of brittle materials.

  20. CTE Solvability, Exact Solutions and Nonlocal Symmetries of the Sharma-Tasso-Olver Equation

    NASA Astrophysics Data System (ADS)

    Pu, Huan; Jia, Man

    2015-12-01

    In this letter, we prove that the STO equation is CTE solvable and obtain the exact solutions of solitons fission and fusion. We also provide the nonlocal symmetries of the STO equation related to CTE. The nonlocal symmetries are localized by prolonging the related enlarged system. Supported by National Natural Science Foundation of China under Grant Nos. 11205092, 11175092 and 11435005, Ningbo Natural Science Foundation under Grant Nos. 2015A610159 and 2012A610178 and by the Opening Project of Zhejiang Provincial Top Key Discipline of Physics Sciences in Ningbo University under Grant No. xkzw11502. And the authors were sponsored by K. C. Wong Magna Fund in Ningbo University

  1. Exact solution of a ratchet with switching sawtooth potential

    NASA Astrophysics Data System (ADS)

    Saakian, David B.; Klümper, Andreas

    2018-01-01

    We consider the flashing potential ratchet model with general asymmetric potential. Using Bloch functions, we derive equations which allow for the calculation of both the ratchet's flux and higher moments of distribution for rather general potentials. We indicate how to derive the optimal transition rates for maximal velocity of the ratchet. We calculate explicitly the exact velocity of a ratchet with simple sawtooth potential from the solution of a system of 8 linear algebraic equations. Using Bloch functions, we derive the equations for the ratchet with potentials changing periodically with time. We also consider the case of the ratchet with evolution with two different potentials acting for some random periods of time.

  2. Active ideal sedimentation: exact two-dimensional steady states.

    PubMed

    Hermann, Sophie; Schmidt, Matthias

    2018-02-28

    We consider an ideal gas of active Brownian particles that undergo self-propelled motion and both translational and rotational diffusion under the influence of gravity. We solve analytically the corresponding Smoluchowski equation in two space dimensions for steady states. The resulting one-body density is given as a series, where each term is a product of an orientation-dependent Mathieu function and a height-dependent exponential. A lower hard wall is implemented as a no-flux boundary condition. Numerical evaluation of the suitably truncated analytical solution shows the formation of two different spatial regimes upon increasing Peclet number. These regimes differ in their mean particle orientation and in their variation of the orientation-averaged density with height.

  3. Axion as a Cold Dark Matter Candidate: Proof to Fully Nonlinear Order

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

    Noh, Hyerim; Hwang, Jai-chan; Park, Chan-Gyung

    2017-09-01

    We present proof of the axion as a cold dark matter (CDM) candidate to the fully nonlinear order perturbations based on Einstein’s gravity. We consider the axion as a coherently oscillating massive classical scalar field without interaction. We present the fully nonlinear and exact, except for ignoring the transverse-tracefree tensor-type perturbation, hydrodynamic equations for an axion fluid in Einstein’s gravity. We show that the axion has the characteristic pressure and anisotropic stress; the latter starts to appear from the second-order perturbation. But these terms do not directly affect the hydrodynamic equations in our axion treatment. Instead, what behaves as themore » effective pressure term in relativistic hydrodynamic equations is the perturbed lapse function and the relativistic result coincides exactly with the one known in the previous non-relativistic studies. The effective pressure term leads to a Jeans scale that is of the solar-system scale for conventional axion mass. As the fully nonlinear and relativistic hydrodynamic equations for an axion fluid coincide exactly with the ones of a zero-pressure fluid in the super-Jeans scale, we have proved the CDM nature of such an axion in that scale.« less

  4. On the failure of the quasicylindrical approximation and the connection to vortex breakdown in turbulent swirling flow

    NASA Astrophysics Data System (ADS)

    Gyllenram, W.; Nilsson, H.; Davidson, L.

    2007-04-01

    This paper analyzes the properties of viscous swirling flow in a pipe. The analysis is based on the time-averaged quasicylindrical Navier-Stokes equations and is applicable to steady, unsteady, and turbulent swirling flow. A method is developed to determine the critical level of swirl (vortex breakdown) for an arbitrary vortex. The method can also be used for an estimation of the radial velocity profile if the other components are given or measured along a single radial line. The quasicylindrical equations are rearranged to yield a single ordinary differential equation for the radial distribution of the radial velocity component. The equation is singular for certain levels of swirl. It is shown that the lowest swirl level at which the equation is singular corresponds exactly to the sufficient condition for axisymmetric vortex breakdown as derived by Wang and Rusak [J. Fluid Mech. 340, 177 (1997)] and Rusak et al. [AIAA J. 36, 1848 (1998)]. In narrow regions around the critical levels of swirl, the solution violates the quasicylindrical assumptions and the flow must undergo a drastic change of structure. The critical swirl level is determined by the sign change of the smallest eigenvalue of the discrete linear operator which relates the radial velocities to effects of viscosity and turbulence. It is shown that neither viscosity nor turbulence directly alters the critical level of swirl.

  5. A low-dispersion, exactly energy-charge-conserving semi-implicit relativistic particle-in-cell algorithm

    NASA Astrophysics Data System (ADS)

    Chen, Guangye; Luis, Chacon; Bird, Robert; Stark, David; Yin, Lin; Albright, Brian

    2017-10-01

    Leap-frog based explicit algorithms, either ``energy-conserving'' or ``momentum-conserving'', do not conserve energy discretely. Time-centered fully implicit algorithms can conserve discrete energy exactly, but introduce large dispersion errors in the light-wave modes, regardless of timestep sizes. This can lead to intolerable simulation errors where highly accurate light propagation is needed (e.g. laser-plasma interactions, LPI). In this study, we selectively combine the leap-frog and Crank-Nicolson methods to produce a low-dispersion, exactly energy-and-charge-conserving PIC algorithm. Specifically, we employ the leap-frog method for Maxwell equations, and the Crank-Nicolson method for particle equations. Such an algorithm admits exact global energy conservation, exact local charge conservation, and preserves the dispersion properties of the leap-frog method for the light wave. The algorithm has been implemented in a code named iVPIC, based on the VPIC code developed at LANL. We will present numerical results that demonstrate the properties of the scheme with sample test problems (e.g. Weibel instability run for 107 timesteps, and LPI applications.

  6. Prognostic residual mean flow in an ocean general circulation model and its relation to prognostic Eulerian mean flow

    DOE PAGES

    Saenz, Juan A.; Chen, Qingshan; Ringler, Todd

    2015-05-19

    Recent work has shown that taking the thickness-weighted average (TWA) of the Boussinesq equations in buoyancy coordinates results in exact equations governing the prognostic residual mean flow where eddy–mean flow interactions appear in the horizontal momentum equations as the divergence of the Eliassen–Palm flux tensor (EPFT). It has been proposed that, given the mathematical tractability of the TWA equations, the physical interpretation of the EPFT, and its relation to potential vorticity fluxes, the TWA is an appropriate framework for modeling ocean circulation with parameterized eddies. The authors test the feasibility of this proposition and investigate the connections between the TWAmore » framework and the conventional framework used in models, where Eulerian mean flow prognostic variables are solved for. Using the TWA framework as a starting point, this study explores the well-known connections between vertical transfer of horizontal momentum by eddy form drag and eddy overturning by the bolus velocity, used by Greatbatch and Lamb and Gent and McWilliams to parameterize eddies. After implementing the TWA framework in an ocean general circulation model, we verify our analysis by comparing the flows in an idealized Southern Ocean configuration simulated using the TWA and conventional frameworks with the same mesoscale eddy parameterization.« less

  7. Quantum transport under ac drive from the leads: A Redfield quantum master equation approach

    NASA Astrophysics Data System (ADS)

    Purkayastha, Archak; Dubi, Yonatan

    2017-08-01

    Evaluating the time-dependent dynamics of driven open quantum systems is relevant for a theoretical description of many systems, including molecular junctions, quantum dots, cavity-QED experiments, cold atoms experiments, and more. Here, we formulate a rigorous microscopic theory of an out-of-equilibrium open quantum system of noninteracting particles on a lattice weakly coupled bilinearly to multiple baths and driven by periodically varying thermodynamic parameters like temperature and chemical potential of the bath. The particles can be either bosonic or fermionic and the lattice can be of any dimension and geometry. Based on the Redfield quantum master equation under Born-Markov approximation, we derive a linear differential equation for an equal time two point correlation matrix, sometimes also called a single-particle density matrix, from which various physical observables, for example, current, can be calculated. Various interesting physical effects, such as resonance, can be directly read off from the equations. Thus, our theory is quite general and gives quite transparent and easy-to-calculate results. We validate our theory by comparing with exact numerical simulations. We apply our method to a generic open quantum system, namely, a double quantum dot coupled to leads with modulating chemical potentials. The two most important experimentally relevant insights from this are as follows: (i) Time-dependent measurements of current for symmetric oscillating voltages (with zero instantaneous voltage bias) can point to the degree of asymmetry in the system-bath coupling and (ii) under certain conditions time-dependent currents can exceed time-averaged currents by several orders of magnitude, and can therefore be detected even when the average current is below the measurement threshold.

  8. Exact Solutions for Stokes' Flow of a Non-Newtonian Nanofluid Model: A Lie Similarity Approach

    NASA Astrophysics Data System (ADS)

    Aziz, Taha; Aziz, A.; Khalique, C. M.

    2016-07-01

    The fully developed time-dependent flow of an incompressible, thermodynamically compatible non-Newtonian third-grade nanofluid is investigated. The classical Stokes model is considered in which the flow is generated due to the motion of the plate in its own plane with an impulsive velocity. The Lie symmetry approach is utilised to convert the governing nonlinear partial differential equation into different linear and nonlinear ordinary differential equations. The reduced ordinary differential equations are then solved by using the compatibility and generalised group method. Exact solutions for the model equation are deduced in the form of closed-form exponential functions which are not available in the literature before. In addition, we also derived the conservation laws associated with the governing model. Finally, the physical features of the pertinent parameters are discussed in detail through several graphs.

  9. 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.

  10. The Poisson-Boltzmann theory for the two-plates problem: some exact results.

    PubMed

    Xing, Xiang-Jun

    2011-12-01

    The general solution to the nonlinear Poisson-Boltzmann equation for two parallel charged plates, either inside a symmetric electrolyte, or inside a 2q:-q asymmetric electrolyte, is found in terms of Weierstrass elliptic functions. From this we derive some exact asymptotic results for the interaction between charged plates, as well as the exact form of the renormalized surface charge density.

  11. Application of the Parabolic Approximation to Predict Acoustical Propagation in the Ocean.

    ERIC Educational Resources Information Center

    McDaniel, Suzanne T.

    1979-01-01

    A simplified derivation of the parabolic approximation to the acoustical wave equation is presented. Exact solutions to this approximate equation are compared with solutions to the wave equation to demonstrate the applicability of this method to the study of underwater sound propagation. (Author/BB)

  12. Solving Simple Kinetics without Integrals

    ERIC Educational Resources Information Center

    de la Pen~a, Lisandro Herna´ndez

    2016-01-01

    The solution of simple kinetic equations is analyzed without referencing any topic from differential equations or integral calculus. Guided by the physical meaning of the rate equation, a systematic procedure is used to generate an approximate solution that converges uniformly to the exact solution in the case of zero, first, and second order…

  13. Susceptible-infected-susceptible epidemics on networks with general infection and cure times.

    PubMed

    Cator, E; van de Bovenkamp, R; Van Mieghem, P

    2013-06-01

    The classical, continuous-time susceptible-infected-susceptible (SIS) Markov epidemic model on an arbitrary network is extended to incorporate infection and curing or recovery times each characterized by a general distribution (rather than an exponential distribution as in Markov processes). This extension, called the generalized SIS (GSIS) model, is believed to have a much larger applicability to real-world epidemics (such as information spread in online social networks, real diseases, malware spread in computer networks, etc.) that likely do not feature exponential times. While the exact governing equations for the GSIS model are difficult to deduce due to their non-Markovian nature, accurate mean-field equations are derived that resemble our previous N-intertwined mean-field approximation (NIMFA) and so allow us to transfer the whole analytic machinery of the NIMFA to the GSIS model. In particular, we establish the criterion to compute the epidemic threshold in the GSIS model. Moreover, we show that the average number of infection attempts during a recovery time is the more natural key parameter, instead of the effective infection rate in the classical, continuous-time SIS Markov model. The relative simplicity of our mean-field results enables us to treat more general types of SIS epidemics, while offering an easier key parameter to measure the average activity of those general viral agents.

  14. Susceptible-infected-susceptible epidemics on networks with general infection and cure times

    NASA Astrophysics Data System (ADS)

    Cator, E.; van de Bovenkamp, R.; Van Mieghem, P.

    2013-06-01

    The classical, continuous-time susceptible-infected-susceptible (SIS) Markov epidemic model on an arbitrary network is extended to incorporate infection and curing or recovery times each characterized by a general distribution (rather than an exponential distribution as in Markov processes). This extension, called the generalized SIS (GSIS) model, is believed to have a much larger applicability to real-world epidemics (such as information spread in online social networks, real diseases, malware spread in computer networks, etc.) that likely do not feature exponential times. While the exact governing equations for the GSIS model are difficult to deduce due to their non-Markovian nature, accurate mean-field equations are derived that resemble our previous N-intertwined mean-field approximation (NIMFA) and so allow us to transfer the whole analytic machinery of the NIMFA to the GSIS model. In particular, we establish the criterion to compute the epidemic threshold in the GSIS model. Moreover, we show that the average number of infection attempts during a recovery time is the more natural key parameter, instead of the effective infection rate in the classical, continuous-time SIS Markov model. The relative simplicity of our mean-field results enables us to treat more general types of SIS epidemics, while offering an easier key parameter to measure the average activity of those general viral agents.

  15. Long-run growth rate in a random multiplicative model

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

    Pirjol, Dan

    2014-08-01

    We consider the long-run growth rate of the average value of a random multiplicative process x{sub i+1} = a{sub i}x{sub i} where the multipliers a{sub i}=1+ρexp(σW{sub i}₋1/2 σ²t{sub i}) have Markovian dependence given by the exponential of a standard Brownian motion W{sub i}. The average value (x{sub n}) is given by the grand partition function of a one-dimensional lattice gas with two-body linear attractive interactions placed in a uniform field. We study the Lyapunov exponent λ=lim{sub n→∞}1/n log(x{sub n}), at fixed β=1/2 σ²t{sub n}n, and show that it is given by the equation of state of the lattice gas inmore » thermodynamical equilibrium. The Lyapunov exponent has discontinuous partial derivatives along a curve in the (ρ, β) plane ending at a critical point (ρ{sub C}, β{sub C}) which is related to a phase transition in the equivalent lattice gas. Using the equivalence of the lattice gas with a bosonic system, we obtain the exact solution for the equation of state in the thermodynamical limit n → ∞.« less

  16. Maskless deposition technique for the physical vapor deposition of thin film and multilayer coatings with subnanometer precision and accuracy

    DOEpatents

    Vernon, Stephen P.; Ceglio, Natale M.

    2000-01-01

    The invention is a method for the production of axially symmetric, graded and ungraded thickness thin film and multilayer coatings that avoids the use of apertures or masks to tailor the deposition profile. A motional averaging scheme permits the deposition of uniform thickness coatings independent of the substrate radius. Coating uniformity results from an exact cancellation of substrate radius dependent terms, which occurs when the substrate moves at constant velocity. If the substrate is allowed to accelerate over the source, arbitrary coating profiles can be generated through appropriate selection and control of the substrate center of mass equation of motion. The radial symmetry of the coating profile is an artifact produced by orbiting the substrate about its center of mass; other distributions are obtained by selecting another rotation axis. Consequently there is a direct mapping between the coating thickness and substrate equation of motion which can be used to tailor the coating profile without the use of masks and apertures.

  17. Extensions of the Einstein-Schrodinger non-symmetric theory of gravity

    NASA Astrophysics Data System (ADS)

    Shifflett, James A.

    We modify the Einstein-Schrödinger theory to include a cosmological constant L z which multiplies the symmetric metric. The cosmological constant L z is assumed to be nearly cancelled by Schrödinger's cosmological constant L b which multiplies the nonsymmetric fundamental tensor, such that the total L = L z + L b matches measurement. The resulting theory becomes exactly Einstein-Maxwell theory in the limit as |L z | [arrow right] oo. For |L z | ~ 1/(Planck length) 2 the field equations match the ordinary Einstein and Maxwell equations except for extra terms which are < 10 -16 of the usual terms for worst-case field strengths and rates-of-change accessible to measurement. Additional fields can be included in the Lagrangian, and these fields may couple to the symmetric metric and the electromagnetic vector potential, just as in Einstein-Maxwell theory. The ordinary Lorentz force equation is obtained by taking the divergence of the Einstein equations when sources are included. The Einstein- Infeld-Hoffmann (EIH) equations of motion match the equations of motion for Einstein-Maxwell theory to Newtonian/Coulombian order, which proves the existence of a Lorentz force without requiring sources. An exact charged solution matches the Reissner-Nordström solution except for additional terms which are ~ 10 -66 of the usual terms for worst-case radii accessible to measurement. An exact electromagnetic plane-wave solution is identical to its counterpart in Einstein-Maxwell theory. Peri-center advance, deflection of light and time delay of light have a fractional difference of < 10 -56 compared to Einstein-Maxwell theory for worst-case parameters. When a spin-1/2 field is included in the Lagrangian, the theory gives the ordinary Dirac equation, and the charged solution results in fractional shifts of < 10 -50 in Hydrogen atom energy levels. Newman-Penrose methods are used to derive an exact solution of the connection equations, and to show that the charged solution is Petrov type- D like the Reissner-Nordström solution. The Newman-Penrose asymptotically flat [Special characters omitted.] (1/ r 2 ) expansion of the field equations is shown to match Einstein-Maxwell theory. Finally we generalize the theory to non-Abelian fields, and show that a special case of the resulting theory closely approximates Einstein-Weinberg-Salam theory.

  18. Similarity solutions of some two-space-dimensional nonlinear wave evolution equations

    NASA Technical Reports Server (NTRS)

    Redekopp, L. G.

    1980-01-01

    Similarity reductions of the two-space-dimensional versions of the Korteweg-de Vries, modified Korteweg-de Vries, Benjamin-Davis-Ono, and nonlinear Schroedinger equations are presented, and some solutions of the reduced equations are discussed. Exact dispersive solutions of the two-dimensional Korteweg-de Vries equation are obtained, and the similarity solution of this equation is shown to be reducible to the second Painleve transcendent.

  19. Some exact solutions for maximally symmetric topological defects in Anti de Sitter space

    NASA Astrophysics Data System (ADS)

    Alvarez, Orlando; Haddad, Matthew

    2018-03-01

    We obtain exact analytical solutions for a class of SO( l) Higgs field theories in a non-dynamic background n-dimensional anti de Sitter space. These finite transverse energy solutions are maximally symmetric p-dimensional topological defects where n = ( p + 1) + l. The radius of curvature of anti de Sitter space provides an extra length scale that allows us to study the equations of motion in a limit where the masses of the Higgs field and the massive vector bosons are both vanishing. We call this the double BPS limit. In anti de Sitter space, the equations of motion depend on both p and l. The exact analytical solutions are expressed in terms of standard special functions. The known exact analytical solutions are for kink-like defects ( p = 0 , 1 , 2 , . . . ; l = 1), vortex-like defects ( p = 1 , 2 , 3; l = 2), and the 't Hooft-Polyakov monopole ( p = 0; l = 3). A bonus is that the double BPS limit automatically gives a maximally symmetric classical glueball type solution. In certain cases where we did not find an analytic solution, we present numerical solutions to the equations of motion. The asymptotically exponentially increasing volume with distance of anti de Sitter space imposes different constraints than those found in the study of defects in Minkowski space.

  20. Oscillating solutions for nonlinear Helmholtz equations

    NASA Astrophysics Data System (ADS)

    Mandel, Rainer; Montefusco, Eugenio; Pellacci, Benedetta

    2017-12-01

    Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behaviour at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Klein-Gordon or Schrödinger equations with large frequencies.

  1. Adaptive Osher-type scheme for the Euler equations with highly nonlinear equations of state

    NASA Astrophysics Data System (ADS)

    Lee, Bok Jik; Toro, Eleuterio F.; Castro, Cristóbal E.; Nikiforakis, Nikolaos

    2013-08-01

    For the numerical simulation of detonation of condensed phase explosives, a complex equation of state (EOS), such as the Jones-Wilkins-Lee (JWL) EOS or the Cochran-Chan (C-C) EOS, are widely used. However, when a conservative scheme is used for solving the Euler equations with such equations of state, a spurious solution across the contact discontinuity, a well known phenomenon in multi-fluid systems, arises even for single materials. In this work, we develop a generalised Osher-type scheme in an adaptive primitive-conservative framework to overcome the aforementioned difficulties. Resulting numerical solutions are compared with the exact solutions and with the numerical solutions from the Godunov method in conjunction with the exact Riemann solver for the Euler equations with Mie-Grüneisen form of equations of state, such as the JWL and the C-C equations of state. The adaptive scheme is extended to second order and its empirical convergence rates are presented, verifying second order accuracy for smooth solutions. Through a suite of several tests problems in one and two space dimensions we illustrate the failure of conservative schemes and the capability of the methods of this paper to overcome the difficulties.

  2. Analytical study of exact solutions of the nonlinear Korteweg-de Vries equation with space-time fractional derivatives

    NASA Astrophysics Data System (ADS)

    Liu, Jiangen; Zhang, Yufeng

    2018-01-01

    This paper gives an analytical study of dynamic behavior of the exact solutions of nonlinear Korteweg-de Vries equation with space-time local fractional derivatives. By using the improved (G‧ G )-expansion method, the explicit traveling wave solutions including periodic solutions, dark soliton solutions, soliton solutions and soliton-like solutions, are obtained for the first time. They can better help us further understand the physical phenomena and provide a strong basis. Meanwhile, some solutions are presented through 3D-graphs.

  3. Inclusion of transverse shear deformation in the exact buckling and vibration analysis of composite plate assemblies

    NASA Technical Reports Server (NTRS)

    Anderson, Melvin S.; Kennedy, David

    1993-01-01

    The problem considered is the development of the necessary plate stiffnesses for use in the general purpose program VICONOPT for buckling and vibration of composite plate assemblies. The required stiffnesses include the effects of transverse shear deformation and are for sinusoidal response along the plate length as required in VICONOPT. The method is based on the exact solution of the plate differential equations for a composite laminate having fully populated A, B, and D stiffness matrices which leads to an ordinary differential equation of tenth order.

  4. Analytical solution of the optimal three dimensional reentry problem using Chapman's exact equations

    NASA Technical Reports Server (NTRS)

    Vinh, N. X.; Busemann, A.; Culp, R. D.

    1974-01-01

    This paper presents the general solution for the optimal three dimensional aerodynamic control of a lifting vehicle entering a planetary atmosphere. A set of dimensionless variables is introduced, and the resulting exact equations of motion have the distinctive advantage that they are completely free of the physical characteristics of the vehicle. Furthermore, a general lift-drag polar is used to define the aerodynamic control. Hence, the results obtained apply to any type of vehicle of arbitrary weight, dimensions and shape, having an arbitrary polar and entering any planetary atmosphere.

  5. A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov–Darwin particle-in-cell algorithm

    DOE PAGES

    Chen, G.; Chacón, L.

    2015-08-11

    For decades, the Vlasov–Darwin model has been recognized to be attractive for particle-in-cell (PIC) kinetic plasma simulations in non-radiative electromagnetic regimes, to avoid radiative noise issues and gain computational efficiency. However, the Darwin model results in an elliptic set of field equations that renders conventional explicit time integration unconditionally unstable. We explore a fully implicit PIC algorithm for the Vlasov–Darwin model in multiple dimensions, which overcomes many difficulties of traditional semi-implicit Darwin PIC algorithms. The finite-difference scheme for Darwin field equations and particle equations of motion is space–time-centered, employing particle sub-cycling and orbit-averaging. This algorithm conserves total energy, local charge,more » canonical-momentum in the ignorable direction, and preserves the Coulomb gauge exactly. An asymptotically well-posed fluid preconditioner allows efficient use of large cell sizes, which are determined by accuracy considerations, not stability, and can be orders of magnitude larger than required in a standard explicit electromagnetic PIC simulation. Finally, we demonstrate the accuracy and efficiency properties of the algorithm with various numerical experiments in 2D–3V.« less

  6. Gravitational Instantons and Minimal Surfaces

    NASA Astrophysics Data System (ADS)

    Nutku, Y.

    1996-12-01

    We show that for every minimal surface in E3 there is a gravitational instanton, an exact solution of the Einstein field equations with Euclidean signature and anti-self-dual curvature. The explicit metric establishing this correspondence is presented and a new class of exact solutions are obtained.

  7. An Exactly Solvable Model for the Spread of Disease

    ERIC Educational Resources Information Center

    Mickens, Ronald E.

    2012-01-01

    We present a new SIR epidemiological model whose exact analytical solution can be calculated. In this model, unlike previous models, the infective population becomes zero at a finite time. Remarkably, these results can be derived from only an elementary knowledge of differential equations.

  8. Exact-Output Tracking Theory for Systems with Parameter Jumps

    NASA Technical Reports Server (NTRS)

    Devasia, Santosh; Paden, Brad; Rossi, Carlo

    1996-01-01

    In this paper we consider the exact output tracking problem for systems with parameter jumps. Necessary and sufficient conditions are derived for the elimination of switching-introduced output transient. Previous works have studied this problem by developing a regulator that maintains exact tracking through parameter jumps (switches). Such techniques are, however, only applicable to minimum-phase systems. In contrast, our approach is applicable to nonminimum-phase systems and obtains bounded but possibly non-causal solutions. If the reference trajectories are generated by an exo-system, then we develop an exact-tracking controller in a feedback form. As in standard regulator theory, we obtain a linear map from the states of the exo-system to the desired system state which is defined via a matrix differential equation. The constant solution of this differential equation provides asymptotic tracking, and coincides with the feedback law used in standard regulator theory. The obtained results are applied to a simple flexible manipulator with jumps in the pay-load mass.

  9. Exact-Output Tracking Theory for Systems with Parameter Jumps

    NASA Technical Reports Server (NTRS)

    Devasia, Santosh; Paden, Brad; Rossi, Carlo

    1997-01-01

    We consider the exact output tracking problem for systems with parameter jumps. Necessary and sufficient conditions are derived for the elimination of switching-introduced output transient. Previous works have studied this problem by developing a regulator that maintains exact tracking through parameter jumps (switches). Such techniques are, however, only applicable to minimum-phase systems. In contrast, our approach is applicable to non-minimum-phase systems and it obtains bounded but possibly non-causal solutions. If the reference trajectories are generated by an exosystem, then we develop an exact-tracking controller in a feed-back form. As in standard regulator theory, we obtain a linear map from the states of the exosystem to the desired system state which is defined via a matrix differential equation. The constant solution of this differential equation provides asymptotic tracking, and coincides with the feedback law used in standard regulator theory. The obtained results are applied to a simple flexible manipulator with jumps in the pay-load mass.

  10. Towards an exact correlated orbital theory for electrons

    NASA Astrophysics Data System (ADS)

    Bartlett, Rodney J.

    2009-12-01

    The formal and computational attraction of effective one-particle theories like Hartree-Fock and density functional theory raise the question of how far such approaches can be taken to offer exact results for selected properties of electrons in atoms, molecules, and solids. Some properties can be exactly described within an effective one-particle theory, like principal ionization potentials and electron affinities. This fact can be used to develop equations for a correlated orbital theory (COT) that guarantees a correct one-particle energy spectrum. They are built upon a coupled-cluster based frequency independent self-energy operator presented here, which distinguishes the approach from Dyson theory. The COT also offers an alternative to Kohn-Sham density functional theory (DFT), whose objective is to represent the electronic density exactly as a single determinant, while paying less attention to the energy spectrum. For any estimate of two-electron terms COT offers a litmus test of its accuracy for principal Ip's and Ea's. This feature for approximating the COT equations is illustrated numerically.

  11. Exact and Approximate Statistical Inference for Nonlinear Regression and the Estimating Equation Approach.

    PubMed

    Demidenko, Eugene

    2017-09-01

    The exact density distribution of the nonlinear least squares estimator in the one-parameter regression model is derived in closed form and expressed through the cumulative distribution function of the standard normal variable. Several proposals to generalize this result are discussed. The exact density is extended to the estimating equation (EE) approach and the nonlinear regression with an arbitrary number of linear parameters and one intrinsically nonlinear parameter. For a very special nonlinear regression model, the derived density coincides with the distribution of the ratio of two normally distributed random variables previously obtained by Fieller (1932), unlike other approximations previously suggested by other authors. Approximations to the density of the EE estimators are discussed in the multivariate case. Numerical complications associated with the nonlinear least squares are illustrated, such as nonexistence and/or multiple solutions, as major factors contributing to poor density approximation. The nonlinear Markov-Gauss theorem is formulated based on the near exact EE density approximation.

  12. Front and pulse solutions for the complex Ginzburg-Landau equation with higher-order terms.

    PubMed

    Tian, Huiping; Li, Zhonghao; Tian, Jinping; Zhou, Guosheng

    2002-12-01

    We investigate one-dimensional complex Ginzburg-Landau equation with higher-order terms and discuss their influences on the multiplicity of solutions. An exact analytic front solution is presented. By stability analysis for the original partial differential equation, we derive its necessary stability condition for amplitude perturbations. This condition together with the exact front solution determine the region of parameter space where the uniformly translating front solution can exist. In addition, stable pulses, chaotic pulses, and attenuation pulses appear generally if the parameters are out of the range. Finally, applying these analysis into the optical transmission system numerically we find that the stable transmission of optical pulses can be achieved if the parameters are appropriately chosen.

  13. Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence.

    PubMed

    Sharma, Ati S; Moarref, Rashad; McKeon, Beverley J; Park, Jae Sung; Graham, Michael D; Willis, Ashley P

    2016-02-01

    We report that many exact invariant solutions of the Navier-Stokes equations for both pipe and channel flows are well represented by just a few modes of the model of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010)]. This model provides modes that act as a basis to decompose the velocity field, ordered by their amplitude of response to forcing arising from the interaction between scales. The model was originally derived from the Navier-Stokes equations to represent turbulent flows and has been used to explain coherent structure and to predict turbulent statistics. This establishes a surprising new link between the two distinct approaches to understanding turbulence.

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

    Vieira, H.S., E-mail: horacio.santana.vieira@hotmail.com; Bezerra, V.B., E-mail: valdir@fisica.ufpb.br; Muniz, C.R., E-mail: celiomuniz@yahoo.com

    This work considers the influence of the gravitational field produced by a charged and rotating black hole (Kerr–Newman spacetime) on a charged massive scalar field. We obtain exact solutions of both angular and radial parts of the Klein–Gordon equation in this spacetime, which are given in terms of the confluent Heun functions. From the radial solution, we obtain the exact wave solutions near the exterior horizon of the black hole, and discuss the Hawking radiation of charged massive scalar particles. - Highlights: • The covariant Klein–Gordon equation for a charged massive scalar field in the Kerr–Newman black hole is solved.more » • Both angular and radial parts are transformed to a Heun-type equation. • The resulting Hawking radiation spectrum of scalar particles has a thermal character.« less

  15. Low-dimensional representations of exact coherent states of the Navier-Stokes equations from the resolvent model of wall turbulence

    NASA Astrophysics Data System (ADS)

    Sharma, Ati S.; Moarref, Rashad; McKeon, Beverley J.; Park, Jae Sung; Graham, Michael D.; Willis, Ashley P.

    2016-02-01

    We report that many exact invariant solutions of the Navier-Stokes equations for both pipe and channel flows are well represented by just a few modes of the model of McKeon and Sharma [J. Fluid Mech. 658, 336 (2010), 10.1017/S002211201000176X]. This model provides modes that act as a basis to decompose the velocity field, ordered by their amplitude of response to forcing arising from the interaction between scales. The model was originally derived from the Navier-Stokes equations to represent turbulent flows and has been used to explain coherent structure and to predict turbulent statistics. This establishes a surprising new link between the two distinct approaches to understanding turbulence.

  16. 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.

  17. Asymptotically Exact Solution of the Problem of Harmonic Vibrations of an Elastic Parallelepiped

    NASA Astrophysics Data System (ADS)

    Papkov, S. O.

    2017-11-01

    An asymptotically exact solution of the classical problem of elasticity about the steadystate forced vibrations of an elastic rectangular parallelepiped is constructed. The general solution of the vibration equations is constructed in the form of double Fourier series with undetermined coefficients, and an infinite system of linear algebraic equations is obtained for determining these coefficients. An analysis of the infinite system permits determining the asymptotics of the unknowns which are used to convolve the double series in both equations of the infinite systems and the displacement and stress components. The efficiency of this approach is illustrated by numerical examples and comparison with known solutions. The spectrum of the parallelepiped symmetric vibrations is studied for various ratios of its sides.

  18. Exact Lyapunov exponent of the harmonic magnon modes of one-dimensional Heisenberg-Mattis spin glasses

    NASA Astrophysics Data System (ADS)

    Sepehrinia, Reza; Niry, M. D.; Bozorg, B.; Tabar, M. Reza Rahimi; Sahimi, Muhammad

    2008-03-01

    A mapping is developed between the linearized equation of motion for the dynamics of the transverse modes at T=0 of the Heisenberg-Mattis model of one-dimensional (1D) spin glasses and the (discretized) random wave equation. The mapping is used to derive an exact expression for the Lyapunov exponent (LE) of the magnon modes of spin glasses and to show that it follows anomalous scaling at low magnon frequencies. In addition, through numerical simulations, the differences between the LE and the density of states of the wave equation in a discrete 1D model of randomly disordered media (those with a finite correlation length) and that of continuous media (with a zero correlation length) are demonstrated and emphasized.

  19. 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

  20. Diffusion Influenced Adsorption Kinetics.

    PubMed

    Miura, Toshiaki; Seki, Kazuhiko

    2015-08-27

    When the kinetics of adsorption is influenced by the diffusive flow of solutes, the solute concentration at the surface is influenced by the surface coverage of solutes, which is given by the Langmuir-Hinshelwood adsorption equation. The diffusion equation with the boundary condition given by the Langmuir-Hinshelwood adsorption equation leads to the nonlinear integro-differential equation for the surface coverage. In this paper, we solved the nonlinear integro-differential equation using the Grünwald-Letnikov formula developed to solve fractional kinetics. Guided by the numerical results, analytical expressions for the upper and lower bounds of the exact numerical results were obtained. The upper and lower bounds were close to the exact numerical results in the diffusion- and reaction-controlled limits, respectively. We examined the validity of the two simple analytical expressions obtained in the diffusion-controlled limit. The results were generalized to include the effect of dispersive diffusion. We also investigated the effect of molecular rearrangement of anisotropic molecules on surface coverage.

  1. Analysis of the correlation dimension for inertial particles

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

    Gustavsson, Kristian; Department of Physics, Göteborg University, 41296 Gothenburg; Mehlig, Bernhard

    2015-07-15

    We obtain an implicit equation for the correlation dimension which describes clustering of inertial particles in a complex flow onto a fractal measure. Our general equation involves a propagator of a nonlinear stochastic process in which the velocity gradient of the fluid appears as additive noise. When the long-time limit of the propagator is considered our equation reduces to an existing large-deviation formalism from which it is difficult to extract concrete results. In the short-time limit, however, our equation reduces to a solvability condition on a partial differential equation. In the case where the inertial particles are much denser thanmore » the fluid, we show how this approach leads to a perturbative expansion of the correlation dimension, for which the coefficients can be obtained exactly and in principle to any order. We derive the perturbation series for the correlation dimension of inertial particles suspended in three-dimensional spatially smooth random flows with white-noise time correlations, obtaining the first 33 non-zero coefficients exactly.« less

  2. Dual exponential polynomials and linear differential equations

    NASA Astrophysics Data System (ADS)

    Wen, Zhi-Tao; Gundersen, Gary G.; Heittokangas, Janne

    2018-01-01

    We study linear differential equations with exponential polynomial coefficients, where exactly one coefficient is of order greater than all the others. The main result shows that a nontrivial exponential polynomial solution of such an equation has a certain dual relationship with the maximum order coefficient. Several examples illustrate our results and exhibit possibilities that can occur.

  3. Remarks on the Non-Linear Differential Equation the Second Derivative of Theta Plus A Sine Theta Equals 0.

    ERIC Educational Resources Information Center

    Fay, Temple H.; O'Neal, Elizabeth A.

    1985-01-01

    The authors draw together a variety of facts concerning a nonlinear differential equation and compare the exact solution with approximate solutions. Then they provide an expository introduction to the elliptic sine function suitable for presentation in undergraduate courses on differential equations. (MNS)

  4. Operator Factorization and the Solution of Second-Order Linear Ordinary Differential Equations

    ERIC Educational Resources Information Center

    Robin, W.

    2007-01-01

    The theory and application of second-order linear ordinary differential equations is reviewed from the standpoint of the operator factorization approach to the solution of ordinary differential equations (ODE). Using the operator factorization approach, the general second-order linear ODE is solved, exactly, in quadratures and the resulting…

  5. Quasi-generalized variables

    NASA Technical Reports Server (NTRS)

    Baumgarten, J.; Ostermeyer, G. P.

    1986-01-01

    The numerical solution of a system of differential and algebraic equations is difficult, due to the appearance of numerical instabilities. A method is presented here which permits numerical solutions of such a system to be obtained which satisfy the algebraic constraint equations exactly without reducing the order of the differential equations. The method is demonstrated using examples from mechanics.

  6. The Auto-Bäcklund transformations for the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

    NASA Astrophysics Data System (ADS)

    Kaplan, Melike; Akbulut, Arzu; Bekir, Ahmet

    2017-01-01

    In this work, the homogeneous balance method is used to construct Auto-Bäcklund transformation of the Boiti-Leon-Manna-Pempinelli (BLMP) equation. With the aid of the transformations founded in this paper and Maple packet programme, abundant exact and explicit solutions to the BLMP equation are constructed.

  7. A Class of Exact Solutions of the Boussinesq Equation for Horizontal and Sloping Aquifers

    NASA Astrophysics Data System (ADS)

    Bartlett, M. S.; Porporato, A.

    2018-02-01

    The nonlinear equation of Boussinesq (1877) is a foundational approach for studying groundwater flow through an unconfined aquifer, but solving the full nonlinear version of the Boussinesq equation remains a challenge. Here, we present an exact solution to the full nonlinear Boussinesq equation that not only applies to sloping aquifers but also accounts for source and sink terms such as bedrock seepage, an often significant flux in headwater catchments. This new solution captures the hysteretic relationship (a loop rating curve) between the groundwater flow rate and the water table height, which may be used to provide a more realistic representation of streamflow and groundwater dynamics in hillslopes. In addition, the solution provides an expression where the flow recession varies based on hillslope parameters such as bedrock slope, bedrock seepage, aquifer recharge, plant transpiration, and other factors that vary across landscape types.

  8. Violation of the continuity equation in the Krieger-Li-Iafrate approximation for current-density functional theory

    NASA Astrophysics Data System (ADS)

    Siegmund, Marc; Pankratov, Oleg

    2011-01-01

    We show that the exchange-correlation scalar and vector potentials obtained from the optimized effective potential (OEP) equations and from the Krieger-Li-Iafrate (KLI) approximation for the current-density functional theory (CDFT) change under a gauge transformation such that the energy functional remains invariant. This alone does not assure, however, the theory’s compliance with the continuity equation. Using the model of a quantum ring with a broken angular symmetry which is penetrated by a magnetic flux we demonstrate that the physical current density calculated with the exact-exchange CDFT in the KLI approximation violates the continuity condition. In contrast, the current found from a solution of the full OEP equations satisfies this condition. We argue that the continuity violation stems from the fact that the KLI potentials are not (in general) the exact functional derivatives of a gauge-invariant exchange-correlation functional.

  9. Laser Radiation in Active Amplifying Media Treated as a Transport Problem - Transfer Equation Derived and Exactly Solved

    NASA Astrophysics Data System (ADS)

    Das Gupta, Santanu; Das Gupta, S. R.

    1991-10-01

    The flow of laser radiation in a plane-parallel cylindrical slab of active amplifying medium with axial symmetry is treated as a problem in radiative transfer. The appropriate one-dimensional transfer equation describing the transfer of laser radiation has been derived by an appeal to Einstein'sA, B coefficients (describing the processes of stimulated line absorption, spontaneous line emission, and stimulated line emission sustained by population inversion in the medium) and considering the ‘rate equations’ to completely establish the rational of the transfer equation obtained. The equation is then exactly solved and the angular distribution of the emergent laser beam intensity is obtained; its numerically computed values are given in tables and plotted in graphs showing the nature of peaks of the emerging laser beam intensity about the axis of the laser cylinder.

  10. Khater method for nonlinear Sharma Tasso-Olever (STO) equation of fractional order

    NASA Astrophysics Data System (ADS)

    Bibi, Sadaf; Mohyud-Din, Syed Tauseef; Khan, Umar; Ahmed, Naveed

    In this work, we have implemented a direct method, known as Khater method to establish exact solutions of nonlinear partial differential equations of fractional order. Number of solutions provided by this method is greater than other traditional methods. Exact solutions of nonlinear fractional order Sharma Tasso-Olever (STO) equation are expressed in terms of kink, travelling wave, periodic and solitary wave solutions. Modified Riemann-Liouville derivative and Fractional complex transform have been used for compatibility with fractional order sense. Solutions have been graphically simulated for understanding the physical aspects and importance of the method. A comparative discussion between our established results and the results obtained by existing ones is also presented. Our results clearly reveal that the proposed method is an effective, powerful and straightforward technique to work out new solutions of various types of differential equations of non-integer order in the fields of applied sciences and engineering.

  11. Exact closed-form solution of the hyperbolic equation of string vibrations with material relaxation properties taken into account

    NASA Astrophysics Data System (ADS)

    Kudinov, I. V.; Kudinov, V. A.

    2014-09-01

    The differential equation of damped string vibrations was obtained with the finite speed of extension and strain propagation in the Hooke's law formula taken into account. In contrast to the well-known equations, the obtained equation contains the first and third time derivatives of the displacement and the mixed derivative with respect to the space and time variables. Separation of variables was used to obtain its exact closed-form solution, whose analysis showed that, for large values of the relaxation coefficient, the string return to the initial state after its escape from equilibrium is accompanied by high-frequency low-amplitude damped vibrations, which occur on the initial time interval only in the region of positive displacements. And in the limit, for some large values of the relaxation coefficient, the string return to the initial state occurs practically without any oscillatory process.

  12. Exact Solutions, Symmetry Reductions, Painlevé Test and Bäcklund Transformations of A Coupled KdV Equation

    NASA Astrophysics Data System (ADS)

    Min-Hui, XU; Man, JIA

    2017-10-01

    A coupled KdV equation is studied in this manuscript. The exact solutions, such as the periodic wave solutions and solitary wave solutions by means of the deformation and mapping approach from the solutions of the nonlinear ϕ 4 model are given. Using the symmetry theory, the Lie point symmetries and symmetry reductions of the coupled KdV equation are presented. The results show that the coupled KdV equation possesses infinitely many symmetries and may be considered as an integrable system. Also, the Painlevé test shows the coupled KdV equation possesses Painlevé property. The Bäcklund transformations of the coupled KdV equation related to Painlevé property and residual symmetry are shown. Supported by the National Natural Science Foundation of China under Grant Nos. 11675084 and 11435005, Ningbo Natural Science Foundation under Grant No. 2015A610159 and granted by the Opening Project of Zhejiang Provincial Top Key Discipline of Physics Sciences in Ningbo University under Grant No. xkzwl1502, and the authors are sponsored by K. C. Wong Magna Fund in Ningbo University

  13. Nonstationary stochastic charge fluctuations of a dust particle in plasmas.

    PubMed

    Shotorban, B

    2011-06-01

    Stochastic charge fluctuations of a dust particle that are due to discreteness of electrons and ions in plasmas can be described by a one-step process master equation [T. Matsoukas and M. Russell, J. Appl. Phys. 77, 4285 (1995)] with no exact solution. In the present work, using the system size expansion method of Van Kampen along with the linear noise approximation, a Fokker-Planck equation with an exact Gaussian solution is developed by expanding the master equation. The Gaussian solution has time-dependent mean and variance governed by two ordinary differential equations modeling the nonstationary process of dust particle charging. The model is tested via the comparison of its results to the results obtained by solving the master equation numerically. The electron and ion currents are calculated through the orbital motion limited theory. At various times of the nonstationary process of charging, the model results are in a very good agreement with the master equation results. The deviation is more significant when the standard deviation of the charge is comparable to the mean charge in magnitude.

  14. The Green's matrix and the boundary integral equations for analysis of time-harmonic dynamics of elastic helical springs.

    PubMed

    Sorokin, Sergey V

    2011-03-01

    Helical springs serve as vibration isolators in virtually any suspension system. Various exact and approximate methods may be employed to determine the eigenfrequencies of vibrations of these structural elements and their dynamic transfer functions. The method of boundary integral equations is a meaningful alternative to obtain exact solutions of problems of the time-harmonic dynamics of elastic springs in the framework of Bernoulli-Euler beam theory. In this paper, the derivations of the Green's matrix, of the Somigliana's identities, and of the boundary integral equations are presented. The vibrational power transmission in an infinitely long spring is analyzed by means of the Green's matrix. The eigenfrequencies and the dynamic transfer functions are found by solving the boundary integral equations. In the course of analysis, the essential features and advantages of the method of boundary integral equations are highlighted. The reported analytical results may be used to study the time-harmonic motion in any wave guide governed by a system of linear differential equations in a single spatial coordinate along its axis. © 2011 Acoustical Society of America

  15. A discrete model of a modified Burgers' partial differential equation

    NASA Technical Reports Server (NTRS)

    Mickens, R. E.; Shoosmith, J. N.

    1990-01-01

    A new finite-difference scheme is constructed for a modified Burger's equation. Three special cases of the equation are considered, and the 'exact' difference schemes for the space- and time-independent forms of the equation are presented, along with the diffusion-free case of Burger's equation modeled by a difference equation. The desired difference scheme is then obtained by imposing on any difference model of the initial equation the requirement that, in the appropriate limits, its difference scheme must reduce the results of the obtained equations.

  16. Zonal flows and turbulence in fluids and plasmas

    NASA Astrophysics Data System (ADS)

    Parker, Jeffrey Bok-Cheung

    In geophysical and plasma contexts, zonal flows are well known to arise out of turbulence. We elucidate the transition from statistically homogeneous turbulence without zonal flows to statistically inhomogeneous turbulence with steady zonal flows. Starting from the Hasegawa--Mima equation, we employ both the quasilinear approximation and a statistical average, which retains a great deal of the qualitative behavior of the full system. Within the resulting framework known as CE2, we extend recent understanding of the symmetry-breaking 'zonostrophic instability'. Zonostrophic instability can be understood in a very general way as the instability of some turbulent background spectrum to a zonally symmetric coherent mode. As a special case, the background spectrum can consist of only a single mode. We find that in this case the dispersion relation of zonostrophic instability from the CE2 formalism reduces exactly to that of the 4-mode truncation of generalized modulational instability. We then show that zonal flows constitute pattern formation amid a turbulent bath. Zonostrophic instability is an example of a Type I s instability of pattern-forming systems. The broken symmetry is statistical homogeneity. Near the bifurcation point, the slow dynamics of CE2 are governed by a well-known amplitude equation, the real Ginzburg-Landau equation. The important features of this amplitude equation, and therefore of the CE2 system, are multiple. First, the zonal flow wavelength is not unique. In an idealized, infinite system, there is a continuous band of zonal flow wavelengths that allow a nonlinear equilibrium. Second, of these wavelengths, only those within a smaller subband are stable. Unstable wavelengths must evolve to reach a stable wavelength; this process manifests as merging jets. These behaviors are shown numerically to hold in the CE2 system, and we calculate a stability diagram. The stability diagram is in agreement with direct numerical simulations of the quasilinear system. The use of statistically-averaged equations and the pattern formation methodology provide a path forward for further systematic investigations of zonal flows and their interactions with turbulence.

  17. On the dispersionless Kadomtsev-Petviashvili equation with arbitrary nonlinearity and dimensionality: exact solutions, longtime asymptotics of the Cauchy problem, wave breaking and shocks

    NASA Astrophysics Data System (ADS)

    Santucci, F.; Santini, P. M.

    2016-10-01

    We study the generalization of the dispersionless Kadomtsev-Petviashvili (dKP) equation in n+1 dimensions and with nonlinearity of degree m+1, a model equation describing the propagation of weakly nonlinear, quasi one-dimensional waves in the absence of dispersion and dissipation, and arising in several physical contexts, like acoustics, plasma physics, hydrodynamics and nonlinear optics. In 2 + 1 dimensions and with quadratic nonlinearity, this equation is integrable through a novel inverse scattering transform, and it has been recently shown to be a prototype model equation in the description of the two-dimensional wave breaking of localized initial data. In higher dimensions and with higher nonlinearity, the generalized dKP equations are not integrable, but their invariance under motions on the paraboloid allows one to construct in this paper a family of exact solutions describing waves constant on their paraboloidal wave front and breaking simultaneously in all points of it, developing after breaking either multivaluedness or single-valued discontinuous profiles (shocks). Then such exact solutions are used to build the longtime behavior of the solutions of the Cauchy problem, for small and localized initial data, showing that wave breaking of small initial data takes place in the longtime regime if and only if m(n-1)≤slant 2. Lastly, the analytic aspects of such wave breaking are investigated in detail in terms of the small initial data, in both cases in which the solution becomes multivalued after breaking or it develops a shock. These results, contained in the 2012 master’s thesis of one of the authors (FS) [1], generalize those obtained in [2] for the dKP equation in n+1 dimensions with quadratic nonlinearity, and are obtained following the same strategy.

  18. Exp-function method for solving fractional partial differential equations.

    PubMed

    Zheng, Bin

    2013-01-01

    We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.

  19. Generalized cable equation model for myelinated nerve fiber.

    PubMed

    Einziger, Pinchas D; Livshitz, Leonid M; Mizrahi, Joseph

    2005-10-01

    Herein, the well-known cable equation for nonmyelinated axon model is extended analytically for myelinated axon formulation. The myelinated membrane conductivity is represented via the Fourier series expansion. The classical cable equation is thereby modified into a linear second order ordinary differential equation with periodic coefficients, known as Hill's equation. The general internal source response, expressed via repeated convolutions, uniformly converges provided that the entire periodic membrane is passive. The solution can be interpreted as an extended source response in an equivalent nonmyelinated axon (i.e., the response is governed by the classical cable equation). The extended source consists of the original source and a novel activation function, replacing the periodic membrane in the myelinated axon model. Hill's equation is explicitly integrated for the specific choice of piecewise constant membrane conductivity profile, thereby resulting in an explicit closed form expression for the transmembrane potential in terms of trigonometric functions. The Floquet's modes are recognized as the nerve fiber activation modes, which are conventionally associated with the nonlinear Hodgkin-Huxley formulation. They can also be incorporated in our linear model, provided that the periodic membrane point-wise passivity constraint is properly modified. Indeed, the modified condition, enforcing the periodic membrane passivity constraint on the average conductivity only leads, for the first time, to the inclusion of the nerve fiber activation modes in our novel model. The validity of the generalized transmission-line and cable equation models for a myelinated nerve fiber, is verified herein through a rigorous Green's function formulation and numerical simulations for transmembrane potential induced in three-dimensional myelinated cylindrical cell. It is shown that the dominant pole contribution of the exact modal expansion is the transmembrane potential solution of our generalized model.

  20. Role of Turbulent Prandtl Number on Heat Flux at Hypersonic Mach Number

    NASA Technical Reports Server (NTRS)

    Xiao, X.; Edwards, J. R.; Hassan, H. A.

    2004-01-01

    Present simulation of turbulent flows involving shock wave/boundary layer interaction invariably overestimates heat flux by almost a factor of two. One possible reason for such a performance is a result of the fact that the turbulence models employed make use of Morkovin's hypothesis. This hypothesis is valid for non-hypersonic Mach numbers and moderate rates of heat transfer. At hypersonic Mach numbers, high rates of heat transfer exist in regions where shock wave/boundary layer interactions are important. As a result, one should not expect traditional turbulence models to yield accurate results. The goal of this investigation is to explore the role of a variable Prandtl number formulation in predicting heat flux in flows dominated by strong shock wave/boundary layer interactions. The intended applications involve external flows in the absence of combustion such as those encountered in supersonic inlets. This can be achieved by adding equations for the temperature variance and its dissipation rate. Such equations can be derived from the exact Navier-Stokes equations. Traditionally, modeled equations are based on the low speed energy equation where the pressure gradient term and the term responsible for energy dissipation are ignored. It is clear that such assumptions are not valid for hypersonic flows. The approach used here is based on the procedure used in deriving the k-zeta model, in which the exact equations that governed k, the variance of velocity, and zeta, the variance of vorticity, were derived and modeled. For the variable turbulent Prandtl number, the exact equations that govern the temperature variance and its dissipation rate are derived and modeled term by term. The resulting set of equations are free of damping and wall functions and are coordinate-system independent. Moreover, modeled correlations are tensorially consistent and invariant under Galilean transformation. The final set of equations will be given in the paper.

  1. Numerical solution of the exact cavity equations of motion for an unstable optical resonator.

    PubMed

    Bowers, M S; Moody, S E

    1990-09-20

    We solve numerically, we believe for the first time, the exact cavity equations of motion for a realistic unstable resonator with a simple gain saturation model. The cavity equations of motion, first formulated by Siegman ["Exact Cavity Equations for Lasers with Large Output Coupling," Appl. Phys. Lett. 36, 412-414 (1980)], and which we term the dynamic coupled modes (DCM) method of solution, solve for the full 3-D time dependent electric field inside the optical cavity by expanding the field in terms of the actual diffractive transverse eigenmodes of the bare (gain free) cavity with time varying coefficients. The spatially varying gain serves to couple the bare cavity transverse modes and to scatter power from mode to mode. We show that the DCM method numerically converges with respect to the number of eigenmodes in the basis set. The intracavity intensity in the numerical example shown reaches a steady state, and this steady state distribution is compared with that computed from the traditional Fox and Li approach using a fast Fourier transform propagation algorithm. The output wavefronts from both methods are quite similar, and the computed output powers agree to within 10%. The usefulness and advantages of using this method for predicting the output of a laser, especially pulsed lasers used for coherent detection, are discussed.

  2. Erratum: Nonlinear Dirac equation solitary waves in external fields [Phys. Rev. E 86, 046602 (2012)

    DOE PAGES

    Mertens, Franz G.; Quintero, Niurka R.; Cooper, Fred; ...

    2016-05-10

    In Sec. IV of our original paper, we assumed a particular conservation law Eq. (4.6), which was true in the absence of external potentials, to derive some particular potentials for which we obtained solutions to the nonlinear Dirac equation (NLDE). Because the conservation law of Eq. (4.6) for the component T 11 of the energy-momentum tensor is not true in the presence of these external potentials, the solutions we found do not satisfy the NLDEs in the presence of these potentials. Thus all the equations from Eq. (4.6) through Eq. (4.44) are not correct, since the exact solutions that followedmore » in that section presumed Eq. (4.6) was true. Also Eqs. (A3)–(A5) are a restatement of Eq. (4.6) and also are not correct. These latter equations are also not used in Sec. V and beyond. The rest of our original paper (starting with Sec. V) was not concerned with exact solutions, rather it was concerned with how the exact solitary-wave solutions to the NLDE in the absence of an external potential responded to being in the presence of various external potentials. This Erratum corrects this mistake.« less

  3. New conditions for obtaining the exact solutions of the general Riccati equation.

    PubMed

    Bougoffa, Lazhar

    2014-01-01

    We propose a direct method for solving the general Riccati equation y' = f(x) + g(x)y + h(x)y(2). We first reduce it into an equivalent equation, and then we formulate the relations between the coefficients functions f(x), g(x), and h(x) of the equation to obtain an equivalent separable equation from which the previous equation can be solved in closed form. Several examples are presented to demonstrate the efficiency of this method.

  4. Anomalous transport in turbulent plasmas and continuous time random walks

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

    Balescu, R.

    1995-05-01

    The possibility of a model of anomalous transport problems in a turbulent plasma by a purely stochastic process is investigated. The theory of continuous time random walks (CTRW`s) is briefly reviewed. It is shown that a particular class, called the standard long tail CTRW`s is of special interest for the description of subdiffusive transport. Its evolution is described by a non-Markovian diffusion equation that is constructed in such a way as to yield exact values for all the moments of the density profile. The concept of a CTRW model is compared to an exact solution of a simple test problem:more » transport of charged particles in a fluctuating magnetic field in the limit of infinite perpendicular correlation length. Although the well-known behavior of the mean square displacement proportional to {ital t}{sup 1/2} is easily recovered, the exact density profile cannot be modeled by a CTRW. However, the quasilinear approximation of the kinetic equation has the form of a non-Markovian diffusion equation and can thus be generated by a CTRW.« less

  5. Applications of He's semi-inverse method, ITEM and GGM to the Davey-Stewartson equation

    NASA Astrophysics Data System (ADS)

    Zinati, Reza Farshbaf; Manafian, Jalil

    2017-04-01

    We investigate the Davey-Stewartson (DS) equation. Travelling wave solutions were found. In this paper, we demonstrate the effectiveness of the analytical methods, namely, He's semi-inverse variational principle method (SIVPM), the improved tan(φ/2)-expansion method (ITEM) and generalized G'/G-expansion method (GGM) for seeking more exact solutions via the DS equation. These methods are direct, concise and simple to implement compared to other existing methods. The exact solutions containing four types solutions have been achieved. The results demonstrate that the aforementioned methods are more efficient than the Ansatz method applied by Mirzazadeh (2015). Abundant exact travelling wave solutions including solitons, kink, periodic and rational solutions have been found by the improved tan(φ/2)-expansion and generalized G'/G-expansion methods. By He's semi-inverse variational principle we have obtained dark and bright soliton wave solutions. Also, the obtained semi-inverse variational principle has profound implications in physical understandings. These solutions might play important role in engineering and physics fields. Moreover, by using Matlab, some graphical simulations were done to see the behavior of these solutions.

  6. Discrete conservation properties for shallow water flows using mixed mimetic spectral elements

    NASA Astrophysics Data System (ADS)

    Lee, D.; Palha, A.; Gerritsma, M.

    2018-03-01

    A mixed mimetic spectral element method is applied to solve the rotating shallow water equations. The mixed method uses the recently developed spectral element histopolation functions, which exactly satisfy the fundamental theorem of calculus with respect to the standard Lagrange basis functions in one dimension. These are used to construct tensor product solution spaces which satisfy the generalized Stokes theorem, as well as the annihilation of the gradient operator by the curl and the curl by the divergence. This allows for the exact conservation of first order moments (mass, vorticity), as well as higher moments (energy, potential enstrophy), subject to the truncation error of the time stepping scheme. The continuity equation is solved in the strong form, such that mass conservation holds point wise, while the momentum equation is solved in the weak form such that vorticity is globally conserved. While mass, vorticity and energy conservation hold for any quadrature rule, potential enstrophy conservation is dependent on exact spatial integration. The method possesses a weak form statement of geostrophic balance due to the compatible nature of the solution spaces and arbitrarily high order spatial error convergence.

  7. Exact solutions of the Navier-Stokes equations generalized for flow in porous media

    NASA Astrophysics Data System (ADS)

    Daly, Edoardo; Basser, Hossein; Rudman, Murray

    2018-05-01

    Flow of Newtonian fluids in porous media is often modelled using a generalized version of the full non-linear Navier-Stokes equations that include additional terms describing the resistance to flow due to the porous matrix. Because this formulation is becoming increasingly popular in numerical models, exact solutions are required as a benchmark of numerical codes. The contribution of this study is to provide a number of non-trivial exact solutions of the generalized form of the Navier-Stokes equations for parallel flow in porous media. Steady-state solutions are derived in the case of flows in a medium with constant permeability along the main direction of flow and a constant cross-stream velocity in the case of both linear and non-linear drag. Solutions are also presented for cases in which the permeability changes in the direction normal to the main flow. An unsteady solution for a flow with velocity driven by a time-periodic pressure gradient is also derived. These solutions form a basis for validating computational models across a wide range of Reynolds and Darcy numbers.

  8. 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.

  9. Exact milestoning

    PubMed Central

    Bello-Rivas, Juan M.; Elber, Ron

    2015-01-01

    A new theory and an exact computer algorithm for calculating kinetics and thermodynamic properties of a particle system are described. The algorithm avoids trapping in metastable states, which are typical challenges for Molecular Dynamics (MD) simulations on rough energy landscapes. It is based on the division of the full space into Voronoi cells. Prior knowledge or coarse sampling of space points provides the centers of the Voronoi cells. Short time trajectories are computed between the boundaries of the cells that we call milestones and are used to determine fluxes at the milestones. The flux function, an essential component of the new theory, provides a complete description of the statistical mechanics of the system at the resolution of the milestones. We illustrate the accuracy and efficiency of the exact Milestoning approach by comparing numerical results obtained on a model system using exact Milestoning with the results of long trajectories and with a solution of the corresponding Fokker-Planck equation. The theory uses an equation that resembles the approximate Milestoning method that was introduced in 2004 [A. K. Faradjian and R. Elber, J. Chem. Phys. 120(23), 10880-10889 (2004)]. However, the current formulation is exact and is still significantly more efficient than straightforward MD simulations on the system studied. PMID:25747056

  10. New exact perfect fluid solutions of Einstein's equations. II

    NASA Astrophysics Data System (ADS)

    Uggla, Claes; Rosquist, Kjell

    1990-12-01

    A family of new spatially homogeneous Bianchi type VIh perfect fluid solutions of the Einstein equations is presented. The fluid flow is orthogonal to the spatially homogeneous hypersurfaces, and the pressure is proportional to the energy density.

  11. An improved plate theory of order (1,2) for thick composite laminates

    NASA Technical Reports Server (NTRS)

    Tessler, A.

    1992-01-01

    A new (1,2)-order theory is proposed for the linear elasto-static analysis of laminated composite plates. The basic assumptions are those concerning the distribution through the laminate thickness of the displacements, transverse shear strains and the transverse normal stress, with these quantities regarded as some weighted averages of their exact elasticity theory representations. The displacement expansions are linear for the inplane components and quadratic for the transverse component, whereas the transverse shear strains and transverse normal stress are respectively quadratic and cubic through the thickness. The main distinguishing feature of the theory is that all strain and stress components are expressed in terms of the assumed displacements prior to the application of a variational principle. This is accomplished by an a priori least-square compatibility requirement for the transverse strains and by requiring exact stress boundary conditions at the top and bottom plate surfaces. Equations of equilibrium and associated Poisson boundary conditions are derived from the virtual work principle. It is shown that the theory is particularly suited for finite element discretization as it requires simple C(sup 0)- and C(sup -1)-continuous displacement interpolation fields. Analytic solutions for the problem of cylindrical bending are derived and compared with the exact elasticity solutions and those of our earlier (1,2)-order theory based on the assumed displacements and transverse strains.

  12. An Astronomical Test of CCD Photometric Precision

    NASA Technical Reports Server (NTRS)

    Koch, David; Dunham, Edward; Borucki, William; Jenkins, Jon; DeVingenzi, D. (Technical Monitor)

    1998-01-01

    This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques. we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

  13. The effect of the earth's rotation on ground water motion.

    PubMed

    Loáiciga, Hugo A

    2007-01-01

    The average pore velocity of ground water according to Darcy's law is a function of the fluid pressure gradient and the gravitational force (per unit volume of ground water) and of aquifer properties. There is also an acceleration exerted on ground water that arises from the Earth's rotation. The magnitude and direction of this rotation-induced force are determined in exact mathematical form in this article. It is calculated that the gravitational force is at least 300 times larger than the largest rotation-induced force anywhere on Earth, the latter force being maximal along the equator and approximately equal to 34 N/m(3) there. This compares with a gravitational force of approximately 10(4) N/m(3).

  14. You can run, you can hide: The epidemiology and statistical mechanics of zombies

    NASA Astrophysics Data System (ADS)

    Alemi, Alexander A.; Bierbaum, Matthew; Myers, Christopher R.; Sethna, James P.

    2015-11-01

    We use a popular fictional disease, zombies, in order to introduce techniques used in modern epidemiology modeling, and ideas and techniques used in the numerical study of critical phenomena. We consider variants of zombie models, from fully connected continuous time dynamics to a full scale exact stochastic dynamic simulation of a zombie outbreak on the continental United States. Along the way, we offer a closed form analytical expression for the fully connected differential equation, and demonstrate that the single person per site two dimensional square lattice version of zombies lies in the percolation universality class. We end with a quantitative study of the full scale US outbreak, including the average susceptibility of different geographical regions.

  15. Quantum entanglement of a harmonic oscillator with an electromagnetic field.

    PubMed

    Makarov, Dmitry N

    2018-05-29

    At present, there are many methods for obtaining quantum entanglement of particles with an electromagnetic field. Most methods have a low probability of quantum entanglement and not an exact theoretical apparatus based on an approximate solution of the Schrodinger equation. There is a need for new methods for obtaining quantum-entangled particles and mathematically accurate studies of such methods. In this paper, a quantum harmonic oscillator (for example, an electron in a magnetic field) interacting with a quantized electromagnetic field is considered. Based on the exact solution of the Schrodinger equation for this system, it is shown that for certain parameters there can be a large quantum entanglement between the electron and the electromagnetic field. Quantum entanglement is analyzed on the basis of a mathematically exact expression for the Schmidt modes and the Von Neumann entropy.

  16. Exact vibration analysis of a double-nanobeam-systems embedded in an elastic medium by a Hamiltonian-based method

    NASA Astrophysics Data System (ADS)

    Zhou, Zhenhuan; Li, Yuejie; Fan, Junhai; Rong, Dalun; Sui, Guohao; Xu, Chenghui

    2018-05-01

    A new Hamiltonian-based approach is presented for finding exact solutions for transverse vibrations of double-nanobeam-systems embedded in an elastic medium. The continuum model is established within the frameworks of the symplectic methodology and the nonlocal Euler-Bernoulli and Timoshenko beam beams. The symplectic eigenfunctions are obtained after expressing the governing equations in a Hamiltonian form. Exact frequency equations, vibration modes and displacement amplitudes are obtained by using symplectic eigenfunctions and end conditions. Comparisons with previously published work are presented to illustrate the accuracy and reliability of the proposed method. The comprehensive results for arbitrary boundary conditions could serve as benchmark results for verifying numerically obtained solutions. In addition, a study on the difference between the nonlocal beam and the nonlocal plate is also included.

  17. Early-time solution of the horizontal unconfined aquifer in the build-up phase

    NASA Astrophysics Data System (ADS)

    Gravanis, Elias; Akylas, Evangelos

    2017-04-01

    The Boussinesq equation is a dynamical equation for the free surface of saturated subsurface flows over an impervious bed. Boussinesq equation is non-linear. The non-linearity comes from the reduction of the dimensionality of the problem: The flow is assumed to be vertically homogeneous, therefore the flow rate through a cross section of the flow is proportional to the free surface height times the hydraulic gradient, which is assumed to be equal to the slope of the free surface (Dupuit approximation). In general, 'vertically' means normally on the bed; combining the Dupuit approximation with the continuity equation leads to the Boussinesq equation. There are very few transient exact solutions. Self- similar solutions have been constructed in the past by various authors. A power series type of solution was derived for a self-similar Boussinesq equation by Barenblatt in 1990. That type of solution has generated a certain amount of literature. For the unconfined flow case for zero recharge rate Boussinesq derived for the horizontal aquifer an exact solution assuming separation of variables. This is actually an exact asymptotic solution of the horizontal aquifer recession phase for late times. The kinematic wave is an interesting solution obtained by dropping the non-linear term in the Boussinesq equation. Although it is an approximate solution, and holds well only for small values of the Henderson and Wooding λ parameter (that is, for steep slopes, high conductivity or small recharge rate), it becomes less and less approximate for smaller values of the parameter, that is, it is asymptotically exact with respect to that parameter. In the present work we consider the case of the unconfined subsurface flow over horizontal bed in the build-up phase under constant recharge rate. This is a case with an infinite Henderson and Wooding parameter, that is, it is the limiting case where the non-linear term is present in the Boussinesq while the linear spatial derivative term goes away. Nonetheless, no analogue of the kinematic wave or the Boussinesq separable solution exists in this case. The late time state of the build-up phase under constant recharge rate is very simply the steady state solution. Our aim is to construct the early time asymptotic solution of this problem. The solution is expressed as a power series of a suitable similarity variable, which is constructed so that to satisfy the boundary conditions at both ends of the aquifer, that is, it is a polynomial approximation of the exact solution. The series turn out to be asymptotic and it is regularized by re-summation techniques which are used to define divergent series. The outflow rate in this regime is linear in time, and the (dimensionless) coefficient is calculated to eight significant figures. The local error of the series is quantified by its deviation from satisfying the self-similar Boussinesq equation at every point. The local error turns out to be everywhere positive, hence, so is the integrated error, which in turn quantifies the degree of convergence of the series to the exact solution.

  18. Nonlinear equation of the modes in circular slab waveguides and its application.

    PubMed

    Zhu, Jianxin; Zheng, Jia

    2013-11-20

    In this paper, circularly curved inhomogeneous waveguides are transformed into straight inhomogeneous waveguides first by a conformal mapping. Then, the differential transfer matrix method is introduced and adopted to deduce the exact dispersion relation for modes. This relation itself is complex and difficult to solve, but it can be approximated by a simpler nonlinear equation in practical applications, which is close to the exact relation and quite easy to analyze. Afterward, optimized asymptotic solutions are obtained and act as initial guesses for the following Newton's iteration. Finally, very accurate solutions are achieved in the numerical experiment.

  19. Numerical Solution of the Electron Transport Equation in the Upper Atmosphere

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

    Woods, Mark Christopher; Holmes, Mark; Sailor, William C

    A new approach for solving the electron transport equation in the upper atmosphere is derived. The problem is a very stiff boundary value problem, and to obtain an accurate numerical solution, matrix factorizations are used to decouple the fast and slow modes. A stable finite difference method is applied to each mode. This solver is applied to a simplifieed problem for which an exact solution exists using various versions of the boundary conditions that might arise in a natural auroral display. The numerical and exact solutions are found to agree with each other to at least two significant digits.

  20. The exact fundamental solution for the Benes tracking problem

    NASA Astrophysics Data System (ADS)

    Balaji, Bhashyam

    2009-05-01

    The universal continuous-discrete tracking problem requires the solution of a Fokker-Planck-Kolmogorov forward equation (FPKfe) for an arbitrary initial condition. Using results from quantum mechanics, the exact fundamental solution for the FPKfe is derived for the state model of arbitrary dimension with Benes drift that requires only the computation of elementary transcendental functions and standard linear algebra techniques- no ordinary or partial differential equations need to be solved. The measurement process may be an arbitrary, discrete-time nonlinear stochastic process, and the time step size can be arbitrary. Numerical examples are included, demonstrating its utility in practical implementation.

  1. A Time Integration Algorithm Based on the State Transition Matrix for Structures with Time Varying and Nonlinear Properties

    NASA Technical Reports Server (NTRS)

    Bartels, Robert E.

    2003-01-01

    A variable order method of integrating the structural dynamics equations that is based on the state transition matrix has been developed. The method has been evaluated for linear time variant and nonlinear systems of equations. When the time variation of the system can be modeled exactly by a polynomial it produces nearly exact solutions for a wide range of time step sizes. Solutions of a model nonlinear dynamic response exhibiting chaotic behavior have been computed. Accuracy of the method has been demonstrated by comparison with solutions obtained by established methods.

  2. Analyzing Lie symmetry and constructing conservation laws for time-fractional Benny-Lin equation

    NASA Astrophysics Data System (ADS)

    Rashidi, Saeede; Hejazi, S. Reza

    This paper investigates the invariance properties of the time fractional Benny-Lin equation with Riemann-Liouville and Caputo derivatives. This equation can be reduced to the Kawahara equation, fifth-order Kdv equation, the Kuramoto-Sivashinsky equation and Navier-Stokes equation. By using the Lie group analysis method of fractional differential equations (FDEs), we derive Lie symmetries for the Benny-Lin equation. Conservation laws for this equation are obtained with the aid of the concept of nonlinear self-adjointness and the fractional generalization of the Noether’s operators. Furthermore, by means of the invariant subspace method, exact solutions of the equation are also constructed.

  3. New envelope solitons for Gerdjikov-Ivanov model in nonlinear fiber optics

    NASA Astrophysics Data System (ADS)

    Triki, Houria; Alqahtani, Rubayyi T.; Zhou, Qin; Biswas, Anjan

    2017-11-01

    Exact soliton solutions in a class of derivative nonlinear Schrödinger equations including a pure quintic nonlinearity are investigated. By means of the coupled amplitude-phase formulation, we derive a nonlinear differential equation describing the evolution of the wave amplitude in the non-Kerr quintic media. The resulting amplitude equation is then solved to get exact analytical chirped bright, kink, antikink, and singular soliton solutions for the model. It is also shown that the nonlinear chirp associated with these solitons is crucially dependent on the wave intensity and related to self-steepening and group velocity dispersion parameters. Parametric conditions on physical parameters for the existence of chirped solitons are also presented. These localized structures exist due to a balance among quintic nonlinearity, group velocity dispersion, and self-steepening effects.

  4. Nonlinear model of a rotating hub-beams structure: Equations of motion

    NASA Astrophysics Data System (ADS)

    Warminski, Jerzy

    2018-01-01

    Dynamics of a rotating structure composed of a rigid hub and flexible beams is presented in the paper. A nonlinear model of a beam takes into account bending, extension and nonlinear curvature. The influence of geometric nonlinearity and nonconstant angular velocity on dynamics of the rotating structure is presented. The exact equations of motion and associated boundary conditions are derived on the basis of the Hamilton's principle. The simplification of the exact nonlinear mathematical model is proposed taking into account the second order approximation. The reduced partial differential equations of motion together with associated boundary conditions can be used to study natural or forced vibrations of a rotating structure considering constant or nonconstant angular speed of a rigid hub and an arbitrary number of flexible blades.

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

    Ahmed, Jawad; Shahzad, Azeem; Khan, Masood

    This article focuses on the exact solution regarding convective heat transfer of a magnetohydrodynamic (MHD) Jeffrey fluid over a stretching sheet. The effects of joule and viscous dissipation, internal heat source/sink and thermal radiation on the heat transfer characteristics are taken in account in the presence of a transverse magnetic field for two types of boundary heating process namely prescribed power law surface temperature (PST) and prescribed heat flux (PHF). Similarity transformations are used to reduce the governing non-linear momentum and thermal boundary layer equations into a set of ordinary differential equations. The exact solutions of the reduced ordinary differentialmore » equations are developed in the form of confluent hypergeometric function. The influence of the pertinent parameters on the temperature profile is examined. In addition the results for the wall temperature gradient are also discussed in detail.« less

  6. An accuracy assessment of Cartesian-mesh approaches for the Euler equations

    NASA Technical Reports Server (NTRS)

    Coirier, William J.; Powell, Kenneth G.

    1995-01-01

    A critical assessment of the accuracy of Cartesian-mesh approaches for steady, transonic solutions of the Euler equations of gas dynamics is made. An exact solution of the Euler equations (Ringleb's flow) is used not only to infer the order of the truncation error of the Cartesian-mesh approaches, but also to compare the magnitude of the discrete error directly to that obtained with a structured mesh approach. Uniformly and adaptively refined solutions using a Cartesian-mesh approach are obtained and compared to each other and to uniformly refined structured mesh results. The effect of cell merging is investigated as well as the use of two different K-exact reconstruction procedures. The solution methodology of the schemes is explained and tabulated results are presented to compare the solution accuracies.

  7. Scattering from a cylindrical reflector: modified theory of physical optics solution.

    PubMed

    Yalçin, Ugur

    2007-02-01

    The problem of scattering from a perfectly conducting cylindrical reflector is examined with the method of the modified theory of physical optics. In this technique the physical optics currents are modified by using a variable unit vector on the scatterer's surface. These current components are obtained for the reflector, which is fed by an offset electric line source. The scattering integral is expressed by using these currents and evaluated asymptotically with the stationary phase method. The results are compared numerically by using physical optics theory, geometrical optics diffraction theory, and the exact solution of the Helmholtz equation. It is found that the modified theory of physical optics scattering field equations agrees with the geometrical optics diffraction theory and the exact solution of the Helmholtz equation.

  8. Bound state solution of Dirac equation for 3D harmonics oscillator plus trigonometric scarf noncentral potential using SUSY QM approach

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

    Cari, C., E-mail: carinln@yahoo.com; Suparmi, A., E-mail: carinln@yahoo.com

    2014-09-30

    Dirac equation of 3D harmonics oscillator plus trigonometric Scarf non-central potential for spin symmetric case is solved using supersymmetric quantum mechanics approach. The Dirac equation for exact spin symmetry reduces to Schrodinger like equation. The relativistic energy and wave function for spin symmetric case are simply obtained using SUSY quantum mechanics method and idea of shape invariance.

  9. Formal expressions and corresponding expansions for the exact Kohn-Sham exchange potential

    NASA Astrophysics Data System (ADS)

    Bulat, Felipe A.; Levy, Mel

    2009-11-01

    Formal expressions and their corresponding expansions in terms of Kohn-Sham (KS) orbitals are deduced for the exchange potential vx(r) . After an alternative derivation of the basic optimized effective potential integrodifferential equations is given through a Hartree-Fock adiabatic connection perturbation theory, we present an exact infinite expansion for vx(r) that is particularly simple in structure. It contains the very same occupied-virtual quantities that appear in the well-known optimized effective potential integral equation, but in this new expression vx(r) is isolated on one side of the equation. An orbital-energy modified Slater potential is its leading term which gives encouraging numerical results. Along different lines, while the earlier Krieger-Li-Iafrate approximation truncates completely the necessary first-order perturbation orbitals, we observe that the improved localized Hartree-Fock (LHF) potential, or common energy denominator potential (CEDA), or effective local potential (ELP), incorporates the part of each first-order orbital that consists of the occupied KS orbitals. With this in mind, the exact correction to the LHF, CEDA, or ELP potential (they are all equivalent) is deduced and displayed in terms of the virtual portions of the first-order orbitals. We close by observing that the newly derived exact formal expressions and corresponding expansions apply as well for obtaining the correlation potential from an orbital-dependent correlation energy functional.

  10. Integral Equations and Scattering Solutions for a Square-Well Potential.

    ERIC Educational Resources Information Center

    Bagchi, B.; Seyler, R. G.

    1979-01-01

    Derives Green's functions and integral equations for scattering solutions subject to a variety of boundary conditions. Exact solutions are obtained for the case of a finite spherical square-well potential, and properties of these solutions are discussed. (Author/HM)

  11. Modeling tracer transport in randomly heterogeneous porous media by nonlocal moment equations: Anomalous transport

    NASA Astrophysics Data System (ADS)

    Morales-Casique, E.; Lezama-Campos, J. L.; Guadagnini, A.; Neuman, S. P.

    2013-05-01

    Modeling tracer transport in geologic porous media suffers from the corrupt characterization of the spatial distribution of hydrogeologic properties of the system and the incomplete knowledge of processes governing transport at multiple scales. Representations of transport dynamics based on a Fickian model of the kind considered in the advection-dispersion equation (ADE) fail to capture (a) the temporal variation associated with the rate of spreading of a tracer, and (b) the distribution of early and late arrival times which are often observed in field and/or laboratory scenarios and are considered as the signature of anomalous transport. Elsewhere we have presented exact stochastic moment equations to model tracer transport in randomly heterogeneous aquifers. We have also developed a closure scheme which enables one to provide numerical solutions of such moment equations at different orders of approximations. The resulting (ensemble) average and variance of concentration fields were found to display a good agreement against Monte Carlo - based simulation results for mildly heterogeneous (or well-conditioned strongly heterogeneous) media. Here we explore the ability of the moment equations approach to describe the distribution of early arrival times and late time tailing effects which can be observed in Monte-Carlo based breakthrough curves (BTCs) of the (ensemble) mean concentration. We show that BTCs of mean resident concentration calculated at a fixed space location through higher-order approximations of moment equations display long tailing features of the kind which is typically associated with anomalous transport behavior and are not represented by an ADE model with constant dispersive parameter, such as the zero-order approximation.

  12. Diffusion in random networks: Asymptotic properties, and numerical and engineering approximations

    NASA Astrophysics Data System (ADS)

    Padrino, Juan C.; Zhang, Duan Z.

    2016-11-01

    The ensemble phase averaging technique is applied to model mass transport by diffusion in random networks. The system consists of an ensemble of random networks, where each network is made of a set of pockets connected by tortuous channels. Inside a channel, we assume that fluid transport is governed by the one-dimensional diffusion equation. Mass balance leads to an integro-differential equation for the pores mass density. The so-called dual porosity model is found to be equivalent to the leading order approximation of the integration kernel when the diffusion time scale inside the channels is small compared to the macroscopic time scale. As a test problem, we consider the one-dimensional mass diffusion in a semi-infinite domain, whose solution is sought numerically. Because of the required time to establish the linear concentration profile inside a channel, for early times the similarity variable is xt- 1 / 4 rather than xt- 1 / 2 as in the traditional theory. This early time sub-diffusive similarity can be explained by random walk theory through the network. In addition, by applying concepts of fractional calculus, we show that, for small time, the governing equation reduces to a fractional diffusion equation with known solution. We recast this solution in terms of special functions easier to compute. Comparison of the numerical and exact solutions shows excellent agreement.

  13. A Posteriori Error Estimation for Discontinuous Galerkin Approximations of Hyperbolic Systems

    NASA Technical Reports Server (NTRS)

    Larson, Mats G.; Barth, Timothy J.

    1999-01-01

    This article considers a posteriori error estimation of specified functionals for first-order systems of conservation laws discretized using the discontinuous Galerkin (DG) finite element method. Using duality techniques, we derive exact error representation formulas for both linear and nonlinear functionals given an associated bilinear or nonlinear variational form. Weighted residual approximations of the exact error representation formula are then proposed and numerically evaluated for Ringleb flow, an exact solution of the 2-D Euler equations.

  14. Non-linear analysis of wave progagation using transform methods and plates and shells using integral equations

    NASA Astrophysics Data System (ADS)

    Pipkins, Daniel Scott

    Two diverse topics of relevance in modern computational mechanics are treated. The first involves the modeling of linear and non-linear wave propagation in flexible, lattice structures. The technique used combines the Laplace Transform with the Finite Element Method (FEM). The procedure is to transform the governing differential equations and boundary conditions into the transform domain where the FEM formulation is carried out. For linear problems, the transformed differential equations can be solved exactly, hence the method is exact. As a result, each member of the lattice structure is modeled using only one element. In the non-linear problem, the method is no longer exact. The approximation introduced is a spatial discretization of the transformed non-linear terms. The non-linear terms are represented in the transform domain by making use of the complex convolution theorem. A weak formulation of the resulting transformed non-linear equations yields a set of element level matrix equations. The trial and test functions used in the weak formulation correspond to the exact solution of the linear part of the transformed governing differential equation. Numerical results are presented for both linear and non-linear systems. The linear systems modeled are longitudinal and torsional rods and Bernoulli-Euler and Timoshenko beams. For non-linear systems, a viscoelastic rod and Von Karman type beam are modeled. The second topic is the analysis of plates and shallow shells under-going finite deflections by the Field/Boundary Element Method. Numerical results are presented for two plate problems. The first is the bifurcation problem associated with a square plate having free boundaries which is loaded by four, self equilibrating corner forces. The results are compared to two existing numerical solutions of the problem which differ substantially.

  15. Finite Difference Schemes as Algebraic Correspondences between Layers

    NASA Astrophysics Data System (ADS)

    Malykh, Mikhail; Sevastianov, Leonid

    2018-02-01

    For some differential equations, especially for Riccati equation, new finite difference schemes are suggested. These schemes define protective correspondences between the layers. Calculation using these schemes can be extended to the area beyond movable singularities of exact solution without any error accumulation.

  16. Exact Theory of Compressible Fluid Turbulence

    NASA Astrophysics Data System (ADS)

    Drivas, Theodore; Eyink, Gregory

    2017-11-01

    We obtain exact results for compressible turbulence with any equation of state, using coarse-graining/filtering. We find two mechanisms of turbulent kinetic energy dissipation: scale-local energy cascade and ``pressure-work defect'', or pressure-work at viscous scales exceeding that in the inertial-range. Planar shocks in an ideal gas dissipate all kinetic energy by pressure-work defect, but the effect is omitted by standard LES modeling of pressure-dilatation. We also obtain a novel inverse cascade of thermodynamic entropy, injected by microscopic entropy production, cascaded upscale, and removed by large-scale cooling. This nonlinear process is missed by the Kovasznay linear mode decomposition, treating entropy as a passive scalar. For small Mach number we recover the incompressible ``negentropy cascade'' predicted by Obukhov. We derive exact Kolmogorov 4/5th-type laws for energy and entropy cascades, constraining scaling exponents of velocity, density, and internal energy to sub-Kolmogorov values. Although precise exponents and detailed physics are Mach-dependent, our exact results hold at all Mach numbers. Flow realizations at infinite Reynolds are ``dissipative weak solutions'' of compressible Euler equations, similarly as Onsager proposed for incompressible turbulence.

  17. The method of Ritz applied to the equation of Hamilton. [for pendulum systems

    NASA Technical Reports Server (NTRS)

    Bailey, C. D.

    1976-01-01

    Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

  18. Symmetry analysis of the high-order equations for the description of the Fermi - Pasta - Ulam problem

    NASA Astrophysics Data System (ADS)

    Kudryashov, N. A.; Volkov, A. K.

    2017-01-01

    Recently some new nonlinear equations for the description of the Fermi - Pasta - Ulam problem have been derived. The main aim of this work is to use the symmetry test to investigate these equations. We consider equations for the description of the α and α + β Fermi - Pasta - Ulam model. We find the infinitesimal operators and Lie groups, admitted by the equations. Using the groups we find the self-similar variables as well as the reductions to the ordinary differential equations. Some exact solutions are also constructed.

  19. Reorientational versus Kerr dark and gray solitary waves using modulation theory.

    PubMed

    Assanto, Gaetano; Marchant, T R; Minzoni, Antonmaria A; Smyth, Noel F

    2011-12-01

    We develop a modulation theory model based on a Lagrangian formulation to investigate the evolution of dark and gray optical spatial solitary waves for both the defocusing nonlinear Schrödinger (NLS) equation and the nematicon equations describing nonlinear beams, nematicons, in self-defocusing nematic liquid crystals. Since it has an exact soliton solution, the defocusing NLS equation is used as a test bed for the modulation theory applied to the nematicon equations, which have no exact solitary wave solution. We find that the evolution of dark and gray NLS solitons, as well as nematicons, is entirely driven by the emission of diffractive radiation, in contrast to the evolution of bright NLS solitons and bright nematicons. Moreover, the steady nematicon profile is nonmonotonic due to the long-range nonlocality associated with the perturbation of the optic axis. Excellent agreement is obtained with numerical solutions of both the defocusing NLS and nematicon equations. The comparisons for the nematicon solutions raise a number of subtle issues relating to the definition and measurement of the width of a dark or gray nematicon.

  20. On multisoliton solutions of the constant astigmatism equation

    NASA Astrophysics Data System (ADS)

    Hlaváč, Adam

    2015-09-01

    We introduce an algebraic formula producing infinitely many exact solutions of the constant astigmatism equation {z}{yy}+{(1/z)}{xx}+2=0 from a given seed. A construction of corresponding surfaces of constant astigmatism is then a matter of routine. As a special case, we consider multisoliton solutions of the constant astigmatism equation defined as counterparts of famous multisoliton solutions of the sine-Gordon equation. A few particular examples are surveyed as well.

  1. Modulation stability analysis of exact multidimensional solutions to the generalized nonlinear Schrödinger equation and the Gross-Pitaevskii equation using a variational approach.

    PubMed

    Petrović, Nikola Z; Aleksić, Najdan B; Belić, Milivoj

    2015-04-20

    We analyze the modulation stability of spatiotemporal solitary and traveling wave solutions to the multidimensional nonlinear Schrödinger equation and the Gross-Pitaevskii equation with variable coefficients that were obtained using Jacobi elliptic functions. For all the solutions we obtain either unconditional stability, or a conditional stability that can be furnished through the use of dispersion management.

  2. Stochastic differential equation model for linear growth birth and death processes with immigration and emigration

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

    Granita, E-mail: granitafc@gmail.com; Bahar, A.

    This paper discusses on linear birth and death with immigration and emigration (BIDE) process to stochastic differential equation (SDE) model. Forward Kolmogorov equation in continuous time Markov chain (CTMC) with a central-difference approximation was used to find Fokker-Planckequation corresponding to a diffusion process having the stochastic differential equation of BIDE process. The exact solution, mean and variance function of BIDE process was found.

  3. Wave Functions for Time-Dependent Dirac Equation under GUP

    NASA Astrophysics Data System (ADS)

    Zhang, Meng-Yao; Long, Chao-Yun; Long, Zheng-Wen

    2018-04-01

    In this work, the time-dependent Dirac equation is investigated under generalized uncertainty principle (GUP) framework. It is possible to construct the exact solutions of Dirac equation when the time-dependent potentials satisfied the proper conditions. In (1+1) dimensions, the analytical wave functions of the Dirac equation under GUP have been obtained for the two kinds time-dependent potentials. Supported by the National Natural Science Foundation of China under Grant No. 11565009

  4. A modification of Einstein-Schrödinger theory that contains both general relativity and electrodynamics

    NASA Astrophysics Data System (ADS)

    Shifflett, J. A.

    2008-08-01

    We modify the Einstein-Schrödinger theory to include a cosmological constant Λ z which multiplies the symmetric metric, and we show how the theory can be easily coupled to additional fields. The cosmological constant Λ z is assumed to be nearly cancelled by Schrödinger’s cosmological constant Λ b which multiplies the nonsymmetric fundamental tensor, such that the total Λ = Λ z + Λ b matches measurement. The resulting theory becomes exactly Einstein-Maxwell theory in the limit as | Λ z | → ∞. For | Λ z | ~ 1/(Planck length)2 the field equations match the ordinary Einstein and Maxwell equations except for extra terms which are < 10-16 of the usual terms for worst-case field strengths and rates-of-change accessible to measurement. Additional fields can be included in the Lagrangian, and these fields may couple to the symmetric metric and the electromagnetic vector potential, just as in Einstein-Maxwell theory. The ordinary Lorentz force equation is obtained by taking the divergence of the Einstein equations when sources are included. The Einstein-Infeld-Hoffmann (EIH) equations of motion match the equations of motion for Einstein-Maxwell theory to Newtonian/Coulombian order, which proves the existence of a Lorentz force without requiring sources. This fixes a problem of the original Einstein-Schrödinger theory, which failed to predict a Lorentz force. An exact charged solution matches the Reissner-Nordström solution except for additional terms which are ~10-66 of the usual terms for worst-case radii accessible to measurement. An exact electromagnetic plane-wave solution is identical to its counterpart in Einstein-Maxwell theory.

  5. Second-order numerical solution of time-dependent, first-order hyperbolic equations

    NASA Technical Reports Server (NTRS)

    Shah, Patricia L.; Hardin, Jay

    1995-01-01

    A finite difference scheme is developed to find an approximate solution of two similar hyperbolic equations, namely a first-order plane wave and spherical wave problem. Finite difference approximations are made for both the space and time derivatives. The result is a conditionally stable equation yielding an exact solution when the Courant number is set to one.

  6. Bukhvostov-Lipatov model and quantum-classical duality

    NASA Astrophysics Data System (ADS)

    Bazhanov, Vladimir V.; Lukyanov, Sergei L.; Runov, Boris A.

    2018-02-01

    The Bukhvostov-Lipatov model is an exactly soluble model of two interacting Dirac fermions in 1 + 1 dimensions. The model describes weakly interacting instantons and anti-instantons in the O (3) non-linear sigma model. In our previous work [arxiv:arXiv:1607.04839] we have proposed an exact formula for the vacuum energy of the Bukhvostov-Lipatov model in terms of special solutions of the classical sinh-Gordon equation, which can be viewed as an example of a remarkable duality between integrable quantum field theories and integrable classical field theories in two dimensions. Here we present a complete derivation of this duality based on the classical inverse scattering transform method, traditional Bethe ansatz techniques and analytic theory of ordinary differential equations. In particular, we show that the Bethe ansatz equations defining the vacuum state of the quantum theory also define connection coefficients of an auxiliary linear problem for the classical sinh-Gordon equation. Moreover, we also present details of the derivation of the non-linear integral equations determining the vacuum energy and other spectral characteristics of the model in the case when the vacuum state is filled by 2-string solutions of the Bethe ansatz equations.

  7. Numerical simulation of vortical ideal fluid flow through curved channel

    NASA Astrophysics Data System (ADS)

    Moshkin, N. P.; Mounnamprang, P.

    2003-04-01

    A numerical algorithm to study the boundary-value problem in which the governing equations are the steady Euler equations and the vorticity is given on the inflow parts of the domain boundary is developed. The Euler equations are implemented in terms of the stream function and vorticity. An irregular physical domain is transformed into a rectangle in the computational domain and the Euler equations are rewritten with respect to a curvilinear co-ordinate system. The convergence of the finite-difference equations to the exact solution is shown experimentally for the test problems by comparing the computational results with the exact solutions on the sequence of grids. To find the pressure from the known vorticity and stream function, the Euler equations are utilized in the Gromeka-Lamb form. The numerical algorithm is illustrated with several examples of steady flow through a two-dimensional channel with curved walls. The analysis of calculations shows strong dependence of the pressure field on the vorticity given at the inflow parts of the boundary. Plots of the flow structure and isobars, for different geometries of channel and for different values of vorticity on entrance, are also presented.

  8. Resonant vibrations of a submerged beam

    NASA Astrophysics Data System (ADS)

    Achenbach, J. D.; Qu, J.

    1986-03-01

    Forced vibration of a simply supported submerged beam of circular cross section is investigated by the use of two mathematical methods. In the first approach the problem formulation is reduced to a singular integro-differential equation for the transverse deflection. In the second approach the method of matched asymptotic expansions is employed. The integro-differential equation is solved numerically, to yield an exact solution for the frequency response. Subsequent use of a representation integral yields the radiated far field acoustic pressure. The exact results for the beam deflection are compared with approximate results that are available in the literature. Next, a matched asymptotic expansion is worked out by constructing "inner" and "outer" expansions for frequencies near and not near resonance frequencies, respectively. The two expansions are matched in an appropriate manner to yield a uniformly valid solution. The leading term of the matched asymptotic solution is compared with exact numerical results.

  9. Applying the Zel'dovich approximation to general relativity

    NASA Astrophysics Data System (ADS)

    Croudace, K. M.; Parry, J.; Salopek, D. S.; Stewart, J. M.

    1994-03-01

    Starting from general relativity, we give a systematic derivation of the Zel'dovich approximation describing the nonlinear evolution of collisionless dust. We begin by evolving dust along world lines, and we demonstrate that the Szekeres line element is an exact but apparently unstable solution of the evolution equations describing pancake collapse. Next, we solve the Einstein field equations by employing Hamilton-Jacobi techniques and a spatial gradient expansion. We give a prescription for evolving a primordial or 'seed' metric up to the formation of pancakes, and demonstrate its validity by rederiving the Szekeres solution approximately at third order and exactly at fifth order in spatial gradients. Finally we show that the range of validity of the expansion can be improved quite significantly if one notes that the 3-metric must have nonnegative eigenvalues. With this improvement the exact Szekeres solution is obtained after only one iteration.

  10. On a new class of completely integrable nonlinear wave equations. I. Infinitely many conservation laws

    NASA Astrophysics Data System (ADS)

    Nutku, Y.

    1985-06-01

    We point out a class of nonlinear wave equations which admit infinitely many conserved quantities. These equations are characterized by a pair of exact one-forms. The implication that they are closed gives rise to equations, the characteristics and Riemann invariants of which are readily obtained. The construction of the conservation laws requires the solution of a linear second-order equation which can be reduced to canonical form using the Riemann invariants. The hodograph transformation results in a similar linear equation. We discuss also the symplectic structure and Bäcklund transformations associated with these equations.

  11. Exact Exchange calculations for periodic systems: a real space approach

    NASA Astrophysics Data System (ADS)

    Natan, Amir; Marom, Noa; Makmal, Adi; Kronik, Leeor; Kuemmel, Stephan

    2011-03-01

    We present a real-space method for exact-exchange Kohn-Sham calculations of periodic systems. The method is based on self-consistent solutions of the optimized effective potential (OEP) equation on a three-dimensional non-orthogonal grid, using norm conserving pseudopotentials. These solutions can be either exact, using the S-iteration approach, or approximate, using the Krieger, Li, and Iafrate (KLI) approach. We demonstrate, using a variety of systems, the importance of singularity corrections and use of appropriate pseudopotentials.

  12. Model reduction for stochastic chemical systems with abundant species.

    PubMed

    Smith, Stephen; Cianci, Claudia; Grima, Ramon

    2015-12-07

    Biochemical processes typically involve many chemical species, some in abundance and some in low molecule numbers. We first identify the rate constant limits under which the concentrations of a given set of species will tend to infinity (the abundant species) while the concentrations of all other species remains constant (the non-abundant species). Subsequently, we prove that, in this limit, the fluctuations in the molecule numbers of non-abundant species are accurately described by a hybrid stochastic description consisting of a chemical master equation coupled to deterministic rate equations. This is a reduced description when compared to the conventional chemical master equation which describes the fluctuations in both abundant and non-abundant species. We show that the reduced master equation can be solved exactly for a number of biochemical networks involving gene expression and enzyme catalysis, whose conventional chemical master equation description is analytically impenetrable. We use the linear noise approximation to obtain approximate expressions for the difference between the variance of fluctuations in the non-abundant species as predicted by the hybrid approach and by the conventional chemical master equation. Furthermore, we show that surprisingly, irrespective of any separation in the mean molecule numbers of various species, the conventional and hybrid master equations exactly agree for a class of chemical systems.

  13. Exact solutions and conservation laws of the system of two-dimensional viscous Burgers equations

    NASA Astrophysics Data System (ADS)

    Abdulwahhab, Muhammad Alim

    2016-10-01

    Fluid turbulence is one of the phenomena that has been studied extensively for many decades. Due to its huge practical importance in fluid dynamics, various models have been developed to capture both the indispensable physical quality and the mathematical structure of turbulent fluid flow. Among the prominent equations used for gaining in-depth insight of fluid turbulence is the two-dimensional Burgers equations. Its solutions have been studied by researchers through various methods, most of which are numerical. Being a simplified form of the two-dimensional Navier-Stokes equations and its wide range of applicability in various fields of science and engineering, development of computationally efficient methods for the solution of the two-dimensional Burgers equations is still an active field of research. In this study, Lie symmetry method is used to perform detailed analysis on the system of two-dimensional Burgers equations. Optimal system of one-dimensional subalgebras up to conjugacy is derived and used to obtain distinct exact solutions. These solutions not only help in understanding the physical effects of the model problem but also, can serve as benchmarks for constructing algorithms and validation of numerical solutions of the system of Burgers equations under consideration at finite Reynolds numbers. Independent and nontrivial conserved vectors are also constructed.

  14. Analytical drafting curves provide exact equations for plotted data

    NASA Technical Reports Server (NTRS)

    Stewart, R. B.

    1967-01-01

    Analytical drafting curves provide explicit mathematical expressions for any numerical data that appears in the form of graphical plots. The curves each have a reference coordinate axis system indicated on the curve as well as the mathematical equation from which the curve was generated.

  15. On a class of exact solutions of the equations of motion of a viscous fluid

    NASA Technical Reports Server (NTRS)

    Yatseyev, V I

    1953-01-01

    The general solution is obtained of the equations of motion of a viscous fluid in which the velocity field is inversely proportional to the distance from a certain point. Some particular cases of such motion are investigated.

  16. Delay chemical master equation: direct and closed-form solutions

    PubMed Central

    Leier, Andre; Marquez-Lago, Tatiana T.

    2015-01-01

    The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived. PMID:26345616

  17. Delay chemical master equation: direct and closed-form solutions.

    PubMed

    Leier, Andre; Marquez-Lago, Tatiana T

    2015-07-08

    The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.

  18. Enskog theory for polydisperse granular mixtures. I. Navier-Stokes order transport.

    PubMed

    Garzó, Vicente; Dufty, James W; Hrenya, Christine M

    2007-09-01

    A hydrodynamic description for an s -component mixture of inelastic, smooth hard disks (two dimensions) or spheres (three dimensions) is derived based on the revised Enskog theory for the single-particle velocity distribution functions. In this first part of the two-part series, the macroscopic balance equations for mass, momentum, and energy are derived. Constitutive equations are calculated from exact expressions for the fluxes by a Chapman-Enskog expansion carried out to first order in spatial gradients, thereby resulting in a Navier-Stokes order theory. Within this context of small gradients, the theory is applicable to a wide range of restitution coefficients and densities. The resulting integral-differential equations for the zeroth- and first-order approximations of the distribution functions are given in exact form. An approximate solution to these equations is required for practical purposes in order to cast the constitutive quantities as algebraic functions of the macroscopic variables; this task is described in the companion paper.

  19. A numerical study of nonlinear waves in a transcritical flow of stratified fluid past an obstacle

    NASA Astrophysics Data System (ADS)

    Hanazaki, Hideshi

    1992-10-01

    A numerical study of the flow of stratified fluid past an obstacle in a horizontal channel is described. Upstream advancing of waves near critically (resonance) appears in the case of ordinary two-layer flow, in which case the flow is described well by the solution of the forced extended Korteweg-de Vries (KdV) equation which has a cubic nonlinear term. It is shown theoretically that the upstream waves in the general two-layer flow cannot be well described by the forced KdV equation except when the wave amplitude is very small. The critical-level flow is also governed by the forced extended KdV equation. However, because of the smallness of the coefficient of the quadratic nonlinear term, the bore cannot propagate upstream at exact resonance. The results for the linearly stratified Boussinesq flow show good agreement with the solution of the Grimshaw and Yi (1991) equation, at least for exact resonance.

  20. Anisotropic hydrodynamics for conformal Gubser flow

    NASA Astrophysics Data System (ADS)

    Nopoush, Mohammad; Ryblewski, Radoslaw; Strickland, Michael

    2015-02-01

    We derive the equations of motion for a system undergoing boost-invariant longitudinal and azimuthally symmetric transverse "Gubser flow" using leading-order anisotropic hydrodynamics. This is accomplished by assuming that the one-particle distribution function is ellipsoidally symmetric in the momenta conjugate to the de Sitter coordinates used to parametrize the Gubser flow. We then demonstrate that the S O (3 )q symmetry in de Sitter space further constrains the anisotropy tensor to be of spheroidal form. The resulting system of two coupled ordinary differential equations for the de Sitter-space momentum scale and anisotropy parameter are solved numerically and compared to a recently obtained exact solution of the relaxation-time-approximation Boltzmann equation subject to the same flow. We show that anisotropic hydrodynamics describes the spatiotemporal evolution of the system better than all currently known dissipative hydrodynamics approaches. In addition, we prove that anisotropic hydrodynamics gives the exact solution of the relaxation-time approximation Boltzmann equation in the ideal, η /s →0 , and free-streaming, η /s →∞, limits.

  1. Solution of two-body relativistic bound state equations with confining plus Coulomb interactions

    NASA Technical Reports Server (NTRS)

    Maung, Khin Maung; Kahana, David E.; Norbury, John W.

    1992-01-01

    Studies of meson spectroscopy have often employed a nonrelativistic Coulomb plus Linear Confining potential in position space. However, because the quarks in mesons move at an appreciable fraction of the speed of light, it is necessary to use a relativistic treatment of the bound state problem. Such a treatment is most easily carried out in momentum space. However, the position space Linear and Coulomb potentials lead to singular kernels in momentum space. Using a subtraction procedure we show how to remove these singularities exactly and thereby solve the Schroedinger equation in momentum space for all partial waves. Furthermore, we generalize the Linear and Coulomb potentials to relativistic kernels in four dimensional momentum space. Again we use a subtraction procedure to remove the relativistic singularities exactly for all partial waves. This enables us to solve three dimensional reductions of the Bethe-Salpeter equation. We solve six such equations for Coulomb plus Confining interactions for all partial waves.

  2. A critical evaluation of two-equation models for near wall turbulence

    NASA Technical Reports Server (NTRS)

    Speziale, Charles G.; Anderson, E. Clay; Abid, Ridha

    1990-01-01

    A basic theoretical and computational study of two-equation models for near-wall turbulent flows was conducted. Two major problems established for the K-epsilon model are discussed, the lack of natural boundary conditions for the dissipation rate and the appearance of higher-order correlations in the balance of terms for the dissipation rate at the wall. The K-omega equation is shown to have two problems also: an exact viscous term is missing, and the destruction of the dissipation term is not properly damped near the wall. A new K-tau model (where tau = 1/omega is the turbulent time scale) was developed by inclusion of the exact viscous term, and by introduction of new wall damping functions with improved asymptotic behavior. A preliminary test of the new model yields improved predictions for the flat-plate turbulent boundary layer.

  3. Stellar Equilibrium in Semiclassical Gravity.

    PubMed

    Carballo-Rubio, Raúl

    2018-02-09

    The phenomenon of quantum vacuum polarization in the presence of a gravitational field is well understood and is expected to have a physical reality, but studies of its backreaction on the dynamics of spacetime are practically nonexistent outside of the specific context of homogeneous cosmologies. Building on previous results of quantum field theory in curved spacetimes, in this Letter we first derive the semiclassical equations of stellar equilibrium in the s-wave Polyakov approximation. It is highlighted that incorporating the polarization of the quantum vacuum leads to a generalization of the classical Tolman-Oppenheimer-Volkoff equation. Despite the complexity of the resulting field equations, it is possible to find exact solutions. Aside from being the first known exact solutions that describe relativistic stars including the nonperturbative backreaction of semiclassical effects, these are identified as a nontrivial combination of the black star and gravastar proposals.

  4. Exact renormalization group equations: an introductory review

    NASA Astrophysics Data System (ADS)

    Bagnuls, C.; Bervillier, C.

    2001-07-01

    We critically review the use of the exact renormalization group equations (ERGE) in the framework of the scalar theory. We lay emphasis on the existence of different versions of the ERGE and on an approximation method to solve it: the derivative expansion. The leading order of this expansion appears as an excellent textbook example to underline the nonperturbative features of the Wilson renormalization group theory. We limit ourselves to the consideration of the scalar field (this is why it is an introductory review) but the reader will find (at the end of the review) a set of references to existing studies on more complex systems.

  5. 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.

  6. An exact solution for ideal dam-break floods on steep slopes

    USGS Publications Warehouse

    Ancey, C.; Iverson, R.M.; Rentschler, M.; Denlinger, R.P.

    2008-01-01

    The shallow-water equations are used to model the flow resulting from the sudden release of a finite volume of frictionless, incompressible fluid down a uniform slope of arbitrary inclination. The hodograph transformation and Riemann's method make it possible to transform the governing equations into a linear system and then deduce an exact analytical solution expressed in terms of readily evaluated integrals. Although the solution treats an idealized case never strictly realized in nature, it is uniquely well-suited for testing the robustness and accuracy of numerical models used to model shallow-water flows on steep slopes. Copyright 2008 by the American Geophysical Union.

  7. Exactly solvable Schrödinger equation with double-well potential for hydrogen bond

    NASA Astrophysics Data System (ADS)

    Sitnitsky, A. E.

    2017-05-01

    We construct a double-well potential for which the Schrödinger equation can be exactly solved via reducing to the confluent Heun's one. Thus the wave function is expressed via the confluent Heun's function. The latter is tabulated in Maple so that the obtained solution is easily treated. The potential is infinite at the boundaries of the final interval that makes it to be highly suitable for modeling hydrogen bonds (both ordinary and low-barrier ones). We exemplify theoretical results by detailed treating the hydrogen bond in KHCO3 and show their good agreement with literature experimental data.

  8. Exact and approximate stochastic simulation of intracellular calcium dynamics.

    PubMed

    Wieder, Nicolas; Fink, Rainer H A; Wegner, Frederic von

    2011-01-01

    In simulations of chemical systems, the main task is to find an exact or approximate solution of the chemical master equation (CME) that satisfies certain constraints with respect to computation time and accuracy. While Brownian motion simulations of single molecules are often too time consuming to represent the mesoscopic level, the classical Gillespie algorithm is a stochastically exact algorithm that provides satisfying results in the representation of calcium microdomains. Gillespie's algorithm can be approximated via the tau-leap method and the chemical Langevin equation (CLE). Both methods lead to a substantial acceleration in computation time and a relatively small decrease in accuracy. Elimination of the noise terms leads to the classical, deterministic reaction rate equations (RRE). For complex multiscale systems, hybrid simulations are increasingly proposed to combine the advantages of stochastic and deterministic algorithms. An often used exemplary cell type in this context are striated muscle cells (e.g., cardiac and skeletal muscle cells). The properties of these cells are well described and they express many common calcium-dependent signaling pathways. The purpose of the present paper is to provide an overview of the aforementioned simulation approaches and their mutual relationships in the spectrum ranging from stochastic to deterministic algorithms.

  9. BPS equations and non-trivial compactifications

    NASA Astrophysics Data System (ADS)

    Tyukov, Alexander; Warner, Nicholas P.

    2018-05-01

    We consider the problem of finding exact, eleven-dimensional, BPS supergravity solutions in which the compactification involves a non-trivial Calabi-Yau manifold, Y , as opposed to simply a T 6. Since there are no explicitly-known metrics on non-trivial, compact Calabi-Yau manifolds, we use a non-compact "local model" and take the compactification manifold to be Y={M}_{GH}× {T}^2 , where ℳGH is a hyper-Kähler, Gibbons-Hawking ALE space. We focus on backgrounds with three electric charges in five dimensions and find exact families of solutions to the BPS equations that have the same four supersymmetries as the three-charge black hole. Our exact solution to the BPS system requires that the Calabi-Yau manifold be fibered over the space-time using compensators on Y . The role of the compensators is to ensure smoothness of the eleven-dimensional metric when the moduli of Y depend on the space-time. The Maxwell field Ansatz also implicitly involves the compensators through the frames of the fibration. We examine the equations of motion and discuss the brane distributions on generic internal manifolds that do not have enough symmetry to allow smearing.

  10. 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.

  11. A well-balanced finite volume scheme for the Euler equations with gravitation. The exact preservation of hydrostatic equilibrium with arbitrary entropy stratification

    NASA Astrophysics Data System (ADS)

    Käppeli, R.; Mishra, S.

    2016-03-01

    Context. Many problems in astrophysics feature flows which are close to hydrostatic equilibrium. However, standard numerical schemes for compressible hydrodynamics may be deficient in approximating this stationary state, where the pressure gradient is nearly balanced by gravitational forces. Aims: We aim to develop a second-order well-balanced scheme for the Euler equations. The scheme is designed to mimic a discrete version of the hydrostatic balance. It therefore can resolve a discrete hydrostatic equilibrium exactly (up to machine precision) and propagate perturbations, on top of this equilibrium, very accurately. Methods: A local second-order hydrostatic equilibrium preserving pressure reconstruction is developed. Combined with a standard central gravitational source term discretization and numerical fluxes that resolve stationary contact discontinuities exactly, the well-balanced property is achieved. Results: The resulting well-balanced scheme is robust and simple enough to be very easily implemented within any existing computer code that solves time explicitly or implicitly the compressible hydrodynamics equations. We demonstrate the performance of the well-balanced scheme for several astrophysically relevant applications: wave propagation in stellar atmospheres, a toy model for core-collapse supernovae, convection in carbon shell burning, and a realistic proto-neutron star.

  12. Exact solutions for the static bending of Euler-Bernoulli beams using Eringen's two-phase local/nonlocal model

    NASA Astrophysics Data System (ADS)

    Wang, Y. B.; Zhu, X. W.; Dai, H. H.

    2016-08-01

    Though widely used in modelling nano- and micro- structures, Eringen's differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen's two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings are considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein has a consistent softening effect. Comparisons are also made with existing analytical and numerical results, which further shows the advantages of the analytical results obtained. Additionally, it seems that the once controversial nonlocal bar problem in the literature is well resolved by the reduction method.

  13. Thermal conductivity and phonon transport properties of silicon using perturbation theory and the environment-dependent interatomic potential

    NASA Astrophysics Data System (ADS)

    Pascual-Gutiérrez, José A.; Murthy, Jayathi Y.; Viskanta, Raymond

    2009-09-01

    Silicon thermal conductivities are obtained from the solution of the linearized phonon Boltzmann transport equation without the use of any parameter-fitting. Perturbation theory is used to compute the strength of three-phonon and isotope scattering mechanisms. Matrix elements based on Fermi's golden rule are computed exactly without assuming either average or mode-dependent Grüeisen parameters, and with no underlying assumptions of crystal isotropy. The environment-dependent interatomic potential is employed to describe the interatomic force constants and the perturbing Hamiltonians. A detailed methodology to accurately find three-phonon processes satisfying energy- and momentum-conservation rules is also described. Bulk silicon thermal conductivity values are computed across a range of temperatures and shown to match experimental data very well. It is found that about two-thirds of the heat transport in bulk silicon may be attributed to transverse acoustic modes. Effective relaxation times and mean free paths are computed in order to provide a more complete picture of the detailed transport mechanisms and for use with carrier transport models based on the Boltzmann transport equation.

  14. Improvements to the George/Castillo Boundary Layer Theory

    NASA Astrophysics Data System (ADS)

    Wosnik, Martin; George, William K.; Castillo, Luciano

    2000-11-01

    George and Castillo (1997)(George WK and Castillo L (1997) Appl.Mech.Rev.), 50, 12/1, 689-729. presented a new theory for Zero Pressure Gradient Turbulent Boundary Layers based on an application of Near-Asymptotics to scaling laws derived from equilibrium similarity to the Reynolds-averaged equations. The resulting overlap velocity profiles retained a dependence on local Reynolds number, the parameters for which had to satisfy the following constraint equation: ln \\varepsilon fracdγd ln δ^+ = fracdln [C_o/C_i] d ln δ^+ where γ is the power exponent, Co and Ci are the coefficients in inner and outer variables respectively. GC considered only the first term in an asymptotic expansion of the exact solution, but higher order terms can be considered with no increase in the number of unknowns. The improved theory is tested against new experimental Zero Pressure Gradient Turbulent Boundary Layer data of Smith (1994), Oesterlund (1999) and Johansson and Castillo (2000). For the friction law, the first order term is sufficient, but for Co and γ the higher order terms improve the fit to the velocity profiles significantly.

  15. Learning the inverse kinetics of an octopus-like manipulator in three-dimensional space.

    PubMed

    Giorelli, M; Renda, F; Calisti, M; Arienti, A; Ferri, G; Laschi, C

    2015-05-13

    This work addresses the inverse kinematics problem of a bioinspired octopus-like manipulator moving in three-dimensional space. The bioinspired manipulator has a conical soft structure that confers the ability of twirling around objects as a real octopus arm does. Despite the simple design, the soft conical shape manipulator driven by cables is described by nonlinear differential equations, which are difficult to solve analytically. Since exact solutions of the equations are not available, the Jacobian matrix cannot be calculated analytically and the classical iterative methods cannot be used. To overcome the intrinsic problems of methods based on the Jacobian matrix, this paper proposes a neural network learning the inverse kinematics of a soft octopus-like manipulator driven by cables. After the learning phase, a feed-forward neural network is able to represent the relation between manipulator tip positions and forces applied to the cables. Experimental results show that a desired tip position can be achieved in a short time, since heavy computations are avoided, with a degree of accuracy of 8% relative average error with respect to the total arm length.

  16. Quantitative kinetic theory of active matter

    NASA Astrophysics Data System (ADS)

    Ihle, Thomas; Chou, Yen-Liang

    2014-03-01

    Models of self-driven agents similar to the Vicsek model [Phys. Rev. Lett. 75 (1995) 1226] are studied by means of kinetic theory. In these models, particles try to align their travel directions with the average direction of their neighbours. At strong alignment a globally ordered state of collective motion forms. An Enskog-like kinetic theory is derived from the exact Chapman-Kolmogorov equation in phase space using Boltzmann's mean-field approximation of molecular chaos. The kinetic equation is solved numerically by a nonlocal Lattice-Boltzmann-like algorithm. Steep soliton-like waves are observed that lead to an abrupt jump of the global order parameter if the noise level is changed. The shape of the wave is shown to follow a novel scaling law and to quantitatively agree within 3 % with agent-based simulations at large particle speeds. This provides a mean-field mechanism to change the second-order character of the flocking transition to first order. Diagrammatic techniques are used to investigate small particle speeds, where the mean-field assumption of Molecular Chaos is invalid and where correlation effects need to be included.

  17. A Gaussian wave packet phase-space representation of quantum canonical statistics

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

    Coughtrie, David J.; Tew, David P.

    2015-07-28

    We present a mapping of quantum canonical statistical averages onto a phase-space average over thawed Gaussian wave-packet (GWP) parameters, which is exact for harmonic systems at all temperatures. The mapping invokes an effective potential surface, experienced by the wave packets, and a temperature-dependent phase-space integrand, to correctly transition from the GWP average at low temperature to classical statistics at high temperature. Numerical tests on weakly and strongly anharmonic model systems demonstrate that thermal averages of the system energy and geometric properties are accurate to within 1% of the exact quantum values at all temperatures.

  18. Advanced propeller noise prediction in the time domain

    NASA Technical Reports Server (NTRS)

    Farassat, F.; Dunn, M. H.; Spence, P. L.

    1992-01-01

    The time domain code ASSPIN gives acousticians a powerful technique of advanced propeller noise prediction. Except for nonlinear effects, the code uses exact solutions of the Ffowcs Williams-Hawkings equation with exact blade geometry and kinematics. By including nonaxial inflow, periodic loading noise, and adaptive time steps to accelerate computer execution, the development of this code becomes complete.

  19. Exact Analytical Solutions for Elastodynamic Impact

    DTIC Science & Technology

    2015-11-30

    corroborated by derivation of exact discrete solutions from recursive equations for the impact problems. 15. SUBJECT TERMS One-dimensional impact; Elastic...wave propagation; Laplace transform; Floor function; Discrete solutions 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT UU 18...impact Elastic wave propagation Laplace transform Floor function Discrete solutionsWe consider the one-dimensional impact problem in which a semi

  20. Isogeometric Divergence-conforming B-splines for the Steady Navier-Stokes Equations

    DTIC Science & Technology

    2012-04-01

    discretizations produce pointwise divergence-free velocity elds and hence exactly satisfy mass conservation. Consequently, discrete variational formulations...cretizations produce pointwise divergence-free velocity fields and hence exactly satisfy mass conservation. Consequently, discrete variational ... variational formulation. Using a combination of an advective for- mulation, SUPG, PSPG, and grad-div stabilization, provably convergent numerical methods

  1. Bayesian Semiparametric Structural Equation Models with Latent Variables

    ERIC Educational Resources Information Center

    Yang, Mingan; Dunson, David B.

    2010-01-01

    Structural equation models (SEMs) with latent variables are widely useful for sparse covariance structure modeling and for inferring relationships among latent variables. Bayesian SEMs are appealing in allowing for the incorporation of prior information and in providing exact posterior distributions of unknowns, including the latent variables. In…

  2. On application of asymmetric Kan-like exact equilibria to the Earth magnetotail modeling

    NASA Astrophysics Data System (ADS)

    Korovinskiy, Daniil B.; Kubyshkina, Darya I.; Semenov, Vladimir S.; Kubyshkina, Marina V.; Erkaev, Nikolai V.; Kiehas, Stefan A.

    2018-04-01

    A specific class of solutions of the Vlasov-Maxwell equations, developed by means of generalization of the well-known Harris-Fadeev-Kan-Manankova family of exact two-dimensional equilibria, is studied. The examined model reproduces the current sheet bending and shifting in the vertical plane, arising from the Earth dipole tilting and the solar wind nonradial propagation. The generalized model allows magnetic configurations with equatorial magnetic fields decreasing in a tailward direction as slow as 1/x, contrary to the original Kan model (1/x3); magnetic configurations with a single X point are also available. The analytical solution is compared with the empirical T96 model in terms of the magnetic flux tube volume. It is found that parameters of the analytical model may be adjusted to fit a wide range of averaged magnetotail configurations. The best agreement between analytical and empirical models is obtained for the midtail at distances beyond 10-15 RE at high levels of magnetospheric activity. The essential model parameters (current sheet scale, current density) are compared to Cluster data of magnetotail crossings. The best match of parameters is found for single-peaked current sheets with medium values of number density, proton temperature and drift velocity.

  3. Some new solutions for the Derrida-Lebowitz-Speer-Spohn equation

    NASA Astrophysics Data System (ADS)

    Ramírez, J.; Romero, J. L.; Tracinà, R.

    2013-09-01

    The well-known Derrida-Lebowitz-Speer-Spohn equation is investigated. By using specific ansätze and the classical symmetries of the equation, several families of new exact solutions have been found. In particular, there appear traveling waves that include compactons and soliton-compactons. Some other solutions conserve the mass and exhibit diffusion and convection processes from an instantaneous source and localized peakons.

  4. New variational principles for locating periodic orbits of differential equations.

    PubMed

    Boghosian, Bruce M; Fazendeiro, Luis M; Lätt, Jonas; Tang, Hui; Coveney, Peter V

    2011-06-13

    We present new methods for the determination of periodic orbits of general dynamical systems. Iterative algorithms for finding solutions by these methods, for both the exact continuum case, and for approximate discrete representations suitable for numerical implementation, are discussed. Finally, we describe our approach to the computation of unstable periodic orbits of the driven Navier-Stokes equations, simulated using the lattice Boltzmann equation.

  5. Exact and Approximate Solutions for the Decades-Old Michaelis-Menten Equation: Progress-Curve Analysis through Integrated Rate Equations

    ERIC Educational Resources Information Center

    Golicnik, 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"[subscript M]) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to…

  6. How hairpin vortices emerge from exact invariant solutions

    NASA Astrophysics Data System (ADS)

    Schneider, Tobias M.; Farano, Mirko; de Palma, Pietro; Robinet, Jean-Christoph; Cherubini, Stefania

    2017-11-01

    Hairpin vortices are among the most commonly observed flow structures in wall-bounded shear flows. However, within the dynamical system approach to turbulence, those structures have not yet been described. They are not captured by known exact invariant solutions of the Navier-Stokes equations nor have other state-space structures supporting hairpins been identified. We show that hairpin structures are observed along an optimally growing trajectory leaving a well known exact traveling wave solution of plane Poiseuille flow. The perturbation triggering hairpins does not correspond to an unstable mode of the exact traveling wave but lies in the stable manifold where non-normality causes strong transient amplification.

  7. Collision statistics, thermodynamics, and transport coefficients of hard hyperspheres in three, four, and five dimensions

    NASA Astrophysics Data System (ADS)

    Lue, L.

    2005-01-01

    The collision statistics of hard hyperspheres are investigated. An exact, analytical formula is developed for the distribution of speeds of a sphere on collision, which is shown to be related to the average time between collisions for a sphere with a particular velocity. In addition, the relationship between the collision rate and the compressibility factor is generalized to arbitrary dimensions. Molecular dynamics simulations are performed for d=3, 4, and 5 dimensional hard-hypersphere fluids. From these simulations, the equation of state of these systems, the self-diffusion coefficient, the shear viscosity, and the thermal conductivity are determined as a function of density. Various aspects of the collision statistics and their dependence on the density and dimensionality of the system are also studied.

  8. Identification of the population density of a species model with nonlocal diffusion and nonlinear reaction

    NASA Astrophysics Data System (ADS)

    Tuan, Nguyen Huy; Van Au, Vo; Khoa, Vo Anh; Lesnic, Daniel

    2017-05-01

    The identification of the population density of a logistic equation backwards in time associated with nonlocal diffusion and nonlinear reaction, motivated by biology and ecology fields, is investigated. The diffusion depends on an integral average of the population density whilst the reaction term is a global or local Lipschitz function of the population density. After discussing the ill-posedness of the problem, we apply the quasi-reversibility method to construct stable approximation problems. It is shown that the regularized solutions stemming from such method not only depend continuously on the final data, but also strongly converge to the exact solution in L 2-norm. New error estimates together with stability results are obtained. Furthermore, numerical examples are provided to illustrate the theoretical results.

  9. Numerical method based on the lattice Boltzmann model for the Fisher equation.

    PubMed

    Yan, Guangwu; Zhang, Jianying; Dong, Yinfeng

    2008-06-01

    In this paper, a lattice Boltzmann model for the Fisher equation is proposed. First, the Chapman-Enskog expansion and the multiscale time expansion are used to describe higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. Second, the modified partial differential equation of the Fisher equation with the higher-order truncation error is obtained. Third, comparison between numerical results of the lattice Boltzmann models and exact solution is given. The numerical results agree well with the classical ones.

  10. Whitham modulation theory for the two-dimensional Benjamin-Ono equation.

    PubMed

    Ablowitz, Mark; Biondini, Gino; Wang, Qiao

    2017-09-01

    Whitham modulation theory for the two-dimensional Benjamin-Ono (2DBO) equation is presented. A system of five quasilinear first-order partial differential equations is derived. The system describes modulations of the traveling wave solutions of the 2DBO equation. These equations are transformed to a singularity-free hydrodynamic-like system referred to here as the 2DBO-Whitham system. Exact reductions of this system are discussed, the formulation of initial value problems is considered, and the system is used to study the transverse stability of traveling wave solutions of the 2DBO equation.

  11. Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation

    PubMed Central

    Wang, Gang wei; Xu, Tian zhou; Feng, Tao

    2014-01-01

    In this paper, using the Lie group analysis method, we study the invariance properties of the time fractional fifth-order KdV equation. A systematic research to derive Lie point symmetries to time fractional fifth-order KdV equation is performed. In the sense of point symmetry, all of the vector fields and the symmetry reductions of the fractional fifth-order KdV equation are obtained. At last, by virtue of the sub-equation method, some exact solutions to the fractional fifth-order KdV equation are provided. PMID:24523885

  12. The Kadomtsev{endash}Petviashvili equation as a source of integrable model equations

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

    Maccari, A.

    1996-12-01

    A new integrable and nonlinear partial differential equation (PDE) in 2+1 dimensions is obtained, by an asymptotically exact reduction method based on Fourier expansion and spatiotemporal rescaling, from the Kadomtsev{endash}Petviashvili equation. The integrability property is explicitly demonstrated, by exhibiting the corresponding Lax pair, that is obtained by applying the reduction technique to the Lax pair of the Kadomtsev{endash}Petviashvili equation. This model equation is likely to be of applicative relevance, because it may be considered a consistent approximation of a large class of nonlinear evolution PDEs. {copyright} {ital 1996 American Institute of Physics.}

  13. Second-order closure PBL model with new third-order moments: Comparison with LES data

    NASA Technical Reports Server (NTRS)

    Canuto, V. M.; Minotti, F.; Ronchi, C.; Ypma, R. M.; Zeman, O.

    1994-01-01

    This paper contains two parts. In the first part, a new set of diagnostic equations is derived for the third-order moments for a buoyancy-driven flow, by exact inversion of the prognostic equations for the third-order moment equations in the stationary case. The third-order moments exhibit a universal structure: they all are a linear combination of the derivatives of all the second-order moments, bar-w(exp 2), bar-w theta, bar-theta(exp 2), and bar-q(exp 2). Each term of the sum contains a turbulent diffusivity D(sub t), which also exhibits a universal structure of the form D(sub t) = a nu(sub t) + b bar-w theta. Since the sign of the convective flux changes depending on stable or unstable stratification, D(sub t) varies according to the type of stratification. Here nu(sub t) approximately equal to wl (l is a mixing length and w is an rms velocity) represents the 'mechanical' part, while the 'buoyancy' part is represented by the convective flux bar-w theta. The quantities a and b are functions of the variable N(sub tau)(exp 2), where N(exp 2) = g alpha derivative of Theta with respect to z and tau is the turbulence time scale. The new expressions for the third-order moments generalize those of Zeman and Lumley, which were subsequently adopted by Sun and Ogura, Chen and Cotton, and Finger and Schmidt in their treatments of the convective boundary layer. In the second part, the new expressions for the third-order moments are used to solve the ensemble average equations describing a purely convective boundary laye r heated from below at a constant rate. The computed second- and third-order moments are then compared with the corresponding Large Eddy Simulation (LES) results, most of which are obtained by running a new LES code, and part of which are taken from published results. The ensemble average results compare favorably with the LES data.

  14. Investigating inhomogeneous Szekeres models and their applications to precision cosmology

    NASA Astrophysics Data System (ADS)

    Peel, Austin Chandler

    Exact solutions of Einstein's field equations that can describe the evolution of complex structures in the universe provide complementary frameworks to standard perturbation theory in which to analyze cosmological and astrophysical phenomena. The flexibility and generality of the inhomogeneous and anisotropic Szekeres metric make it the best known exact solution to explore nonlinearities in the universe. We study applications of Szekeres models to precision cosmology, focusing on the influence of inhomogeneities in two primary contexts---the growth rate of cosmic structures and biases in distance determinations to remote sources. We first define and derive evolution equations for a Szekeres density contrast, which quantifies exact deviations from a smooth background cosmology. Solving these equations and comparing to the usual perturbative approach, we find that for models with the same matter content, the Szekeres growth rate is larger through the matter-dominated cosmic era. Including a cosmological constant, we consider exact global perturbations, as well as the evolution of a single extended structure surrounded by an almost homogeneous background. For the former, we use growth data to obtain a best fit Szekeres model and find that it can fit the data as well as the standard Lambda-Cold Dark Matter (LCDM) cosmological model but with different cosmological parameters. Next, to study effects of inhomogeneities on distance measures, we build an exact relativistic Swiss-cheese model of the universe, where a large number of non-symmetric and randomly placed Szekeres structures are embedded within a LCDM background. Solving the full relativistic propagation equations, light beams are traced through the model, where they traverse the inhomogeneous structures in a way that mimics the paths of real light beams in the universe. For beams crossing a single structure, their magnification or demagnification reflects primarily the net density encountered along the path. Despite nontrivial evolution and density distributions of the structures, the effect of tidal shearing on the beams remains small. Finally, we study source magnification probability distributions for various redshifts, finding a limitation of the models in that the distributions do not consistently resemble those of gravitational lensing analyses in cosmological simulations.

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

    Manjunath, Naren; Samajdar, Rhine; Jain, Sudhir R., E-mail: srjain@barc.gov.in

    Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in Samajdar and Jain (2014). The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards.

  16. Some exact solutions of (2+1)-dimensional Yang-Mills equations with the Chern-Simons term

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

    Oh, C. H.; Sia, L. C.; Teh, R.

    1989-07-15

    Two /ital Ansa/$/ital uml/---/ital tze/ for the gauge field potential are given so that the(2+1)-dimensional Yang-Mills equations with the Chern-Simons termcan be solved in terms of the modified Bessel functions and the ellipticfunction respectively.

  17. Antigravity: Spin-gravity coupling in action

    NASA Astrophysics Data System (ADS)

    Plyatsko, Roman; Fenyk, Mykola

    2016-08-01

    The typical motions of a spinning test particle in Schwarzschild's background which show the strong repulsive action of the highly relativistic spin-gravity coupling are considered using the exact Mathisson-Papapetrou equations. An approximated approach to choice solutions of these equations which describe motions of the particle's proper center of mass is developed.

  18. Revealing Numerical Solutions of a Differential Equation

    ERIC Educational Resources Information Center

    Glaister, P.

    2006-01-01

    In this article, the author considers a student exercise that involves determining the exact and numerical solutions of a particular differential equation. He shows how a typical student solution is at variance with a numerical solution, suggesting that the numerical solution is incorrect. However, further investigation shows that this numerical…

  19. Beyond Euler's Method: Implicit Finite Differences in an Introductory ODE Course

    ERIC Educational Resources Information Center

    Kull, Trent C.

    2011-01-01

    A typical introductory course in ordinary differential equations (ODEs) exposes students to exact solution methods. However, many differential equations must be approximated with numerical methods. Textbooks commonly include explicit methods such as Euler's and Improved Euler's. Implicit methods are typically introduced in more advanced courses…

  20. Slip and barodiffusion phenomena in slow flows of a gas mixture

    NASA Astrophysics Data System (ADS)

    Zhdanov, V. M.

    2017-03-01

    The slip and barodiffusion problems for the slow flows of a gas mixture are investigated on the basis of the linearized moment equations following from the Boltzmann equation. We restrict ourselves to the set of the third-order moment equations and state two general relations (resembling conservation equations) for the moments of the distribution function similar to the conditions used by Loyalka [S. K. Loyalka, Phys. Fluids 14, 2291 (1971), 10.1063/1.1693331] in his approximation method (the modified Maxwell method). The expressions for the macroscopic velocities of the gas mixture species, the partial viscous stress tensors, and the reduced heat fluxes for the stationary slow flow of a gas mixture in the semi-infinite space over a plane wall are obtained as a result of the exact solution of the linearized moment equations in the 10- and 13-moment approximations. The general expression for the slip velocity and the simple and accurate expressions for the viscous, thermal, diffusion slip, and baroslip coefficients, which are given in terms of the basic transport coefficients, are derived by using the modified Maxwell method. The solutions of moment equations are also used for investigation of the flow and diffusion of a gas mixture in a channel formed by two infinite parallel plates. A fundamental result is that the barodiffusion factor in the cross-section-averaged expression for the diffusion flux contains contributions associated with the viscous transfer of momentum in the gas mixture and the effect of the Knudsen layer. Our study revealed that the barodiffusion factor is equal to the diffusion slip coefficient (correct to the opposite sign). This result is consistent with the Onsager's reciprocity relations for kinetic coefficients following from nonequilibrium thermodynamics of the discontinuous systems.

  1. Stable boundary conditions and difference schemes for Navier-Stokes equations

    NASA Technical Reports Server (NTRS)

    Dutt, P.

    1985-01-01

    The Navier-Stokes equations can be viewed as an incompletely elliptic perturbation of the Euler equations. By using the entropy function for the Euler equations as a measure of energy for the Navier-Stokes equations, it was possible to obtain nonlinear energy estimates for the mixed initial boundary value problem. These estimates are used to derive boundary conditions which guarantee L2 boundedness even when the Reynolds number tends to infinity. Finally, a new difference scheme for modelling the Navier-Stokes equations in multidimensions for which it is possible to obtain discrete energy estimates exactly analogous to those we obtained for the differential equation was proposed.

  2. a Numerical Comparison of Langrange and Kane's Methods of AN Arm Segment

    NASA Astrophysics Data System (ADS)

    Rambely, Azmin Sham; Halim, Norhafiza Ab.; Ahmad, Rokiah Rozita

    A 2-D model of a two-link kinematic chain is developed using two dynamics equations of motion, namely Kane's and Lagrange Methods. The dynamics equations are reduced to first order differential equation and solved using modified Euler and fourth order Runge Kutta to approximate the shoulder and elbow joint angles during a smash performance in badminton. Results showed that Runge-Kutta produced a better and exact approximation than that of modified Euler and both dynamic equations produced better absolute errors.

  3. Exact soliton of (2 + 1)-dimensional fractional Schrödinger equation

    NASA Astrophysics Data System (ADS)

    Rizvi, S. T. R.; Ali, K.; Bashir, S.; Younis, M.; Ashraf, R.; Ahmad, M. O.

    2017-07-01

    The nonlinear fractional Schrödinger equation is the basic equation of fractional quantum mechanics introduced by Nick Laskin in 2002. We apply three tools to solve this mathematical-physical model. First, we find the solitary wave solutions including the trigonometric traveling wave solutions, bell and kink shape solitons using the F-expansion and Improve F-expansion method. We also obtain the soliton solution, singular soliton solutions, rational function solution and elliptic integral function solutions, with the help of the extended trial equation method.

  4. Exact RG flow equations and quantum gravity

    NASA Astrophysics Data System (ADS)

    de Alwis, S. P.

    2018-03-01

    We discuss the different forms of the functional RG equation and their relation to each other. In particular we suggest a generalized background field version that is close in spirit to the Polchinski equation as an alternative to the Wetterich equation to study Weinberg's asymptotic safety program for defining quantum gravity, and argue that the former is better suited for this purpose. Using the heat kernel expansion and proper time regularization we find evidence in support of this program in agreement with previous work.

  5. 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.

  6. An exact solution for the solidification of a liquid slab of binary mixture

    NASA Technical Reports Server (NTRS)

    Antar, B. N.; Collins, F. G.; Aumalia, A. E.

    1986-01-01

    The time dependent temperature and concentration profiles of a one dimensional finite slab of a binary liquid alloy is investigated during solidification. The governing equations are reduced to a set of coupled, nonlinear initial value problems using the method outlined by Meyer. Two methods will be used to solve these equations. The first method uses a Runge-Kutta-Fehlberg integrator to solve the equations numerically. The second method comprises of finding closed form solutions of the equations.

  7. Recent Selected Papers of Northwestern Polytechnical University in Two Parts. Part I. 1979.

    DTIC Science & Technology

    1981-08-20

    pressure coefficient is calculated by the exact Bernoulli equation. Two numerical examples are included, and the results agree fairly well with known... Bernoulli equation is applied to cal- culate the pressure coefficient: ft. 2 2, Lq32) In the above expression, all the derivatives are calculated by...0.14: R Olt(6) The real part on the unit circle in ecuation (2) is used. Making use of equations (5) and (6), both sides of equation (2) are expanded

  8. Fluid dynamics of out of equilibrium boost invariant plasmas

    NASA Astrophysics Data System (ADS)

    Blaizot, Jean-Paul; Yan, Li

    2018-05-01

    By solving a simple kinetic equation, in the relaxation time approximation, and for a particular set of moments of the distribution function, we establish a set of equations which, on the one hand, capture exactly the dynamics of the kinetic equation, and, on the other hand, coincide with the hierarchy of equations of viscous hydrodynamics, to arbitrary order in the viscous corrections. This correspondence sheds light on the underlying mechanism responsible for the apparent success of hydrodynamics in regimes that are far from local equilibrium.

  9. Vertical distribution of ozone: a new method of determination using satellite measurements.

    PubMed

    Aruga, T; Igarashi, T

    1976-01-01

    A new method to determine the vertical distribution of atmospheric ozone over a wide range from the spectral measurement of backscattered solar uv radiation is proposed. Equations for the diffuse reflection in an inhomogeneous atmosphere are introduced, and some theoretical approximations are discussed. An inversion equation is formulated in such a way that the change of radiance at each wavelength, caused by the minute relative increment of ozone density at each altitude, is obtained exactly. The equation is solved by an iterative procedure using the weight function obtained in this work. The results of computer simulation indicate that the ozone distribution from the mesopause to the tropopause can be determined, and that although it is impossible to suggest exactly the complicated profile with fine structure, the smoothed ozone distribution and the total content can be determined with almost the same accuracy as the accuracies of measurement and theoretical calculation of the spectral intensity.

  10. Anomalous diffusion associated with nonlinear fractional derivative fokker-planck-like equation: exact time-dependent solutions

    PubMed

    Bologna; Tsallis; Grigolini

    2000-08-01

    We consider the d=1 nonlinear Fokker-Planck-like equation with fractional derivatives ( partial differential/ partial differentialt)P(x,t)=D( partial differential(gamma)/ partial differentialx(gamma))[P(x,t)](nu). Exact time-dependent solutions are found for nu=(2-gamma)/(1+gamma)(-infinity

  11. Partial differential equation-based localization of a monopole source from a circular array.

    PubMed

    Ando, Shigeru; Nara, Takaaki; Levy, Tsukassa

    2013-10-01

    Wave source localization from a sensor array has long been the most active research topics in both theory and application. In this paper, an explicit and time-domain inversion method for the direction and distance of a monopole source from a circular array is proposed. The approach is based on a mathematical technique, the weighted integral method, for signal/source parameter estimation. It begins with an exact form of the source-constraint partial differential equation that describes the unilateral propagation of wide-band waves from a single source, and leads to exact algebraic equations that include circular Fourier coefficients (phase mode measurements) as their coefficients. From them, nearly closed-form, single-shot and multishot algorithms are obtained that is suitable for use with band-pass/differential filter banks. Numerical evaluation and several experimental results obtained using a 16-element circular microphone array are presented to verify the validity of the proposed method.

  12. Preliminary study of the effects of a reversible chemical reaction on gas bubble dissolution. [for space glass refining

    NASA Technical Reports Server (NTRS)

    Weinberg, M. C.

    1982-01-01

    A preliminary investigation is carried out of the effects of a reversible chemical reaction on the dissolution of an isolated, stationary gas bubble in a glass melt. The exact governing equations for the model system are formulated and analyzed. The approximate quasi-steady-state version of these equations is solved analytically, and a calculation is made of bubble dissolution rates. The results are then compared with numerical solutions obtained from the finite difference form of the exact governing equations. It is pointed out that in the microgravity condition of space, the buoyant rise of a gas bubble in a glass melt will be negligible on the time scale of most experiments. For this reason, a determination of the behavior of a stationary gas bubble in a melt is relevant for an understanding of glass refining in space.

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

    Vieira, H.S., E-mail: horacio.santana.vieira@hotmail.com; Centro de Ciências, Tecnologia e Saúde, Universidade Estadual da Paraíba, CEP 58233-000, Araruna, PB; Bezerra, V.B., E-mail: valdir@fisica.ufpb.br

    Charged massive scalar fields are considered in the gravitational and electromagnetic field produced by a dyonic black hole with a cosmic string along its axis of symmetry. Exact solutions of both angular and radial parts of the covariant Klein–Gordon equation in this background are obtained, and are given in terms of the confluent Heun functions. The role of the presence of the cosmic string in these solutions is showed up. From the radial solution, we obtain the exact wave solutions near the exterior horizon of the black hole, and discuss the Hawking radiation spectrum and the energy flux. -- Highlights:more » •A cosmic string is introduced along the axis of symmetry of the dyonic black hole. •The covariant Klein–Gordon equation for a charged massive scalar field in this background is analyzed. •Both angular and radial parts are transformed to a confluent Heun equation. •The resulting Hawking radiation spectrum and the energy flux are obtained.« less

  14. Modulation instability, Fermi-Pasta-Ulam recurrence, rogue waves, nonlinear phase shift, and exact solutions of the Ablowitz-Ladik equation.

    PubMed

    Akhmediev, Nail; Ankiewicz, Adrian

    2011-04-01

    We study modulation instability (MI) of the discrete constant-background wave of the Ablowitz-Ladik (A-L) equation. We derive exact solutions of the A-L equation which are nonlinear continuations of MI at longer times. These periodic solutions comprise a family of two-parameter solutions with an arbitrary background field and a frequency of initial perturbation. The solutions are recurrent, since they return the field state to the original constant background solution after the process of nonlinear evolution has passed. These solutions can be considered as a complete resolution of the Fermi-Pasta-Ulam paradox for the A-L system. One remarkable consequence of the recurrent evolution is the nonlinear phase shift gained by the constant background wave after the process. A particular case of this family is the rational solution of the first-order or fundamental rogue wave.

  15. Uni-directional optical pulses, temporal propagation, and spatial and temporal dispersion

    NASA Astrophysics Data System (ADS)

    Kinsler, P.

    2018-02-01

    I derive a temporally propagated uni-directional optical pulse equation valid in the few cycle limit. Temporal propagation is advantageous because it naturally preserves causality, unlike the competing spatially propagated models. The exact coupled bi-directional equations that this approach generates can be efficiently approximated down to a uni-directional form in cases where an optical pulse changes little over one optical cycle. They also permit a direct term-to-term comparison of the exact bi-directional theory with its corresponding approximate uni-directional theory. Notably, temporal propagation handles dispersion in a different way, and this difference serves to highlight existing approximations inherent in spatially propagated treatments of dispersion. Accordingly, I emphasise the need for future work in clarifying the limitations of the dispersion conversion required by these types of approaches; since the only alternative in the few cycle limit may be to resort to the much more computationally intensive full Maxwell equation solvers.

  16. A spectral mimetic least-squares method

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

    Bochev, Pavel; Gerritsma, Marc

    We present a spectral mimetic least-squares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a first-order system for two scalar and two vector variables shifts material properties from the differential equations to a pair of constitutive relations. We also use this system to motivate a new least-squares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the four-field least-squares functional by spectral spaces compatible with the differential operators leads to a least-squares method in which the differential equations are alsomore » satisfied exactly. Additionally, the latter are reduced to purely topological relationships for the degrees of freedom that can be satisfied without reference to basis functions. Furthermore, numerical experiments confirm the spectral accuracy of the method and its local conservation.« less

  17. A spectral mimetic least-squares method

    DOE PAGES

    Bochev, Pavel; Gerritsma, Marc

    2014-09-01

    We present a spectral mimetic least-squares method for a model diffusion–reaction problem, which preserves key conservation properties of the continuum problem. Casting the model problem into a first-order system for two scalar and two vector variables shifts material properties from the differential equations to a pair of constitutive relations. We also use this system to motivate a new least-squares functional involving all four fields and show that its minimizer satisfies the differential equations exactly. Discretization of the four-field least-squares functional by spectral spaces compatible with the differential operators leads to a least-squares method in which the differential equations are alsomore » satisfied exactly. Additionally, the latter are reduced to purely topological relationships for the degrees of freedom that can be satisfied without reference to basis functions. Furthermore, numerical experiments confirm the spectral accuracy of the method and its local conservation.« less

  18. The SMM Model as a Boundary Value Problem Using the Discrete Diffusion Equation

    NASA Technical Reports Server (NTRS)

    Campbell, Joel

    2007-01-01

    A generalized single step stepwise mutation model (SMM) is developed that takes into account an arbitrary initial state to a certain partial difference equation. This is solved in both the approximate continuum limit and the more exact discrete form. A time evolution model is developed for Y DNA or mtDNA that takes into account the reflective boundary modeling minimum microsatellite length and the original difference equation. A comparison is made between the more widely known continuum Gaussian model and a discrete model, which is based on modified Bessel functions of the first kind. A correction is made to the SMM model for the probability that two individuals are related that takes into account a reflecting boundary modeling minimum microsatellite length. This method is generalized to take into account the general n-step model and exact solutions are found. A new model is proposed for the step distribution.

  19. Modeling of Electromagnetic Scattering by Discrete and Discretely Heterogeneous Random Media by Using Numerically Exact Solutions of the Maxwell Equations

    NASA Technical Reports Server (NTRS)

    Dlugach, Janna M.; Mishchenko, Michael I.

    2017-01-01

    In this paper, we discuss some aspects of numerical modeling of electromagnetic scattering by discrete random medium by using numerically exact solutions of the macroscopic Maxwell equations. Typical examples of such media are clouds of interstellar dust, clouds of interplanetary dust in the Solar system, dusty atmospheres of comets, particulate planetary rings, clouds in planetary atmospheres, aerosol particles with numerous inclusions and so on. Our study is based on the results of extensive computations of different characteristics of electromagnetic scattering obtained by using the superposition T-matrix method which represents a direct computer solver of the macroscopic Maxwell equations for an arbitrary multisphere configuration. As a result, in particular, we clarify the range of applicability of the low-density theories of radiative transfer and coherent backscattering as well as of widely used effective-medium approximations.

  20. Induced drag ideal efficiency factor of arbitrary lateral-vertical wing forms

    NASA Technical Reports Server (NTRS)

    Deyoung, J.

    1980-01-01

    A relatively simple equation is presented for estimating the induced drag ideal efficiency factor e for arbitrary cross sectional wing forms. This equation is based on eight basic but varied wing configurations which have exact solutions. The e function which relates the basic wings is developed statistically and is a continuous function of configuration geometry. The basic wing configurations include boxwings shaped as a rectangle, ellipse, and diamond; the V-wing; end-plate wing; 90 degree cruciform; circle dumbbell; and biplane. Example applications of the e equations are made to many wing forms such as wings with struts which form partial span rectangle dumbbell wings; bowtie, cruciform, winglet, and fan wings; and multiwings. Derivations are presented in the appendices of exact closed form solutions found of e for the V-wing and 90 degree cruciform wing and for an asymptotic solution for multiwings.

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