Integral equation study of soft-repulsive dimeric fluids
NASA Astrophysics Data System (ADS)
Munaò, Gianmarco; Saija, Franz
2017-03-01
We study fluid structure and water-like anomalies of a system constituted by dimeric particles interacting via a purely repulsive core-softened potential by means of integral equation theories. In our model, dimers interact through a repulsive pair potential of inverse-power form with a softened repulsion strength. By employing the Ornstein–Zernike approach and the reference interaction site model (RISM) theory, we study the behavior of water-like anomalies upon progressively increasing the elongation λ of the dimers from the monomeric case (λ =0 ) to the tangent configuration (λ =1 ). For each value of the elongation we consider two different values of the interaction potential, corresponding to one and two length scales, with the aim to provide a comprehensive description of the possible fluid scenarios of this model. Our theoretical results are systematically compared with already existing or newly generated Monte Carlo data: we find that theories and simulations agree in providing the picture of a fluid exhibiting density and structural anomalies for low values of λ and for both the two values of the interaction potential. Integral equation theories give accurate predictions for pressure and radial distribution functions, whereas the temperatures where anomalies occur are underestimated. Upon increasing the elongation, the RISM theory still predicts the existence of anomalies; the latter are no longer observed in simulations, since their development is likely precluded by the onset of crystallization. We discuss our results in terms of the reliability of integral equation theories in predicting the existence of water-like anomalies in core-softened fluids.
Integral equation study of soft-repulsive dimeric fluids.
Munaò, Gianmarco; Saija, Franz
2017-03-22
We study fluid structure and water-like anomalies of a system constituted by dimeric particles interacting via a purely repulsive core-softened potential by means of integral equation theories. In our model, dimers interact through a repulsive pair potential of inverse-power form with a softened repulsion strength. By employing the Ornstein-Zernike approach and the reference interaction site model (RISM) theory, we study the behavior of water-like anomalies upon progressively increasing the elongation λ of the dimers from the monomeric case ([Formula: see text]) to the tangent configuration ([Formula: see text]). For each value of the elongation we consider two different values of the interaction potential, corresponding to one and two length scales, with the aim to provide a comprehensive description of the possible fluid scenarios of this model. Our theoretical results are systematically compared with already existing or newly generated Monte Carlo data: we find that theories and simulations agree in providing the picture of a fluid exhibiting density and structural anomalies for low values of λ and for both the two values of the interaction potential. Integral equation theories give accurate predictions for pressure and radial distribution functions, whereas the temperatures where anomalies occur are underestimated. Upon increasing the elongation, the RISM theory still predicts the existence of anomalies; the latter are no longer observed in simulations, since their development is likely precluded by the onset of crystallization. We discuss our results in terms of the reliability of integral equation theories in predicting the existence of water-like anomalies in core-softened fluids.
Fluctuations in a ferrofluid monolayer: an integral equation study.
Luo, Liang; Klapp, Sabine H L
2009-07-21
Using integral equation theory in the reference hypernetted chain (RHNC) approximation we investigate the structure and phase behavior of a monolayer of dipolar spheres. The dipole orientations of the particles fluctuate within the plane. The resulting angle dependence of the correlation functions is treated via an expansion in two-dimensional rotational invariants. For homogeneous, isotropic states the RHNC correlation functions turn out to be in good agreement with Monte Carlo simulation data. We then use the RHNC theory combined with a stability (fluctuation) analysis to identify precursors of the low-temperature behavior. As expected, the fluctuations point to pair and cluster formation in the range of low and moderate densities. At high densities, there is no clear indication for a ferroelectric transition, contrary to what is found in three-dimensional dipolar fluids. The stability analysis rather indicates an alignment of chains supplemented by local crystal-like order.
Properties of the two-dimensional heterogeneous Lennard-Jones dimers: An integral equation study
NASA Astrophysics Data System (ADS)
Urbic, Tomaz
2016-11-01
Structural and thermodynamic properties of a planar heterogeneous soft dumbbell fluid are examined using Monte Carlo simulations and integral equation theory. Lennard-Jones particles of different sizes are the building blocks of the dimers. The site-site integral equation theory in two dimensions is used to calculate the site-site radial distribution functions and the thermodynamic properties. Obtained results are compared to Monte Carlo simulation data. The critical parameters for selected types of dimers were also estimated and the influence of the Lennard-Jones parameters was studied. We have also tested the correctness of the site-site integral equation theory using different closures.
Properties of the two-dimensional heterogeneous Lennard-Jones dimers: An integral equation study.
Urbic, Tomaz
2016-11-21
Structural and thermodynamic properties of a planar heterogeneous soft dumbbell fluid are examined using Monte Carlo simulations and integral equation theory. Lennard-Jones particles of different sizes are the building blocks of the dimers. The site-site integral equation theory in two dimensions is used to calculate the site-site radial distribution functions and the thermodynamic properties. Obtained results are compared to Monte Carlo simulation data. The critical parameters for selected types of dimers were also estimated and the influence of the Lennard-Jones parameters was studied. We have also tested the correctness of the site-site integral equation theory using different closures.
Properties of the Lennard-Jones dimeric fluid in two dimensions: an integral equation study.
Urbic, Tomaz; Dias, Cristiano L
2014-03-07
The thermodynamic and structural properties of the planar soft-sites dumbbell fluid are examined by Monte Carlo simulations and integral equation theory. The dimers are built of two Lennard-Jones segments. Site-site integral equation theory in two dimensions is used to calculate the site-site radial distribution functions for a range of elongations and densities and the results are compared with Monte Carlo simulations. The critical parameters for selected types of dimers were also estimated. We analyze the influence of the bond length on critical point as well as tested correctness of site-site integral equation theory with different closures. The integral equations can be used to predict the phase diagram of dimers whose molecular parameters are known.
Properties of the Lennard-Jones dimeric fluid in two dimensions: An integral equation study
Urbic, Tomaz; Dias, Cristiano L.
2014-03-07
The thermodynamic and structural properties of the planar soft-sites dumbbell fluid are examined by Monte Carlo simulations and integral equation theory. The dimers are built of two Lennard-Jones segments. Site-site integral equation theory in two dimensions is used to calculate the site-site radial distribution functions for a range of elongations and densities and the results are compared with Monte Carlo simulations. The critical parameters for selected types of dimers were also estimated. We analyze the influence of the bond length on critical point as well as tested correctness of site-site integral equation theory with different closures. The integral equations can be used to predict the phase diagram of dimers whose molecular parameters are known.
The application of the integral equation theory to study the hydrophobic interaction
Mohorič, Tomaž; Urbic, Tomaz; Hribar-Lee, Barbara
2014-01-01
The Wertheim's integral equation theory was tested against newly obtained Monte Carlo computer simulations to describe the potential of mean force between two hydrophobic particles. An excellent agreement was obtained between the theoretical and simulation results. Further, the Wertheim's integral equation theory with polymer Percus-Yevick closure qualitatively correctly (with respect to the experimental data) describes the solvation structure under conditions where the simulation results are difficult to obtain with good enough accuracy. PMID:24437891
Range, Gabriel M; Klapp, Sabine H L
2006-03-21
Using the reference hypernetted chain (RHNC) integral equation theory and an accompanying stability analysis we investigate the structural and phase behaviors of model bidisperse ferrocolloids based on correlations of the homogeneous isotropic high-temperature phase. Our model consists of two species of dipolar hard spheres (DHSs) which dipole moments are proportional to the particle volume. At small packing fractions our results indicate the onset of chain formation, where the (more strongly coupled) A species behaves essentially as a one-component DHS fluid in a background of B particles. At high packing fractions, on the other hand, the RHNC theory indicates the appearance of isotropic-to-ferromagnetic transitions (volume ratios close to one) and demixing transitions (smaller volume ratios). However, contrary with the related case of monodisperse DHS mixtures previously studied by us [Phys. Rev. E 70, 031201 (2004)], none of the present bidisperse systems exhibit demixing within the isotropic phase, rather we observe coupled ferromagnetic/demixing phase transitions.
Some properties for a new integrable soliton equation
NASA Astrophysics Data System (ADS)
Pan, Chaohong; Ling, Liming
2015-02-01
In this paper, we introduce an integrable soliton equation and present its Lax pair and bi-Hamiltonian structure. We demonstrate that this integrable equation possesses special kink waves. The relationship between the integrable soliton equation and Gardner's equation is established by a reciprocal transformation. Our study extends previous research through a comparison between the soliton equation and its generalized version.
Integration rules for scattering equations
NASA Astrophysics Data System (ADS)
Baadsgaard, Christian; Bjerrum-Bohr, N. E. J.; Bourjaily, Jacob L.; Damgaard, Poul H.
2015-09-01
As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints fo any Möbius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.
Do, D D; Nicholson, D; Fan, Chunyan
2011-12-06
We present equations to calculate the differential and integral enthalpy changes of adsorption for their use in Monte Carlo simulation. Adsorption of a system of N molecules, subject to an external potential energy, is viewed as one of transferring these molecules from a reference gas phase (state 1) to the adsorption system (state 2) at the same temperature and equilibrium pressure (same chemical potential). The excess amount adsorbed is the difference between N and the hypothetical amount of gas occupying the accessible volume of the system at the same density as the reference gas. The enthalpy change is a state function, which is defined as the difference between the enthalpies of state 2 and state 1, and the isosteric heat is defined as the negative of the derivative of this enthalpy change with respect to the excess amount of adsorption. It is suitable to determine how the system behaves for a differential increment in the excess phase adsorbed under subcritical conditions. For supercritical conditions, use of the integral enthalpy of adsorption per particle is recommended since the isosteric heat becomes infinite at the maximum excess concentration. With these unambiguous definitions we derive equations which are applicable for a general case of adsorption and demonstrate how they can be used in a Monte Carlo simulation. We apply the new equations to argon adsorption at various temperatures on a graphite surface to illustrate the need to use the correct equation to describe isosteric heat of adsorption.
Integral equation study of particle confinement effects in a polymer/particle mixture
Henderson, D; Trokhymchuk, A; Kalyuzhnyi, Y; Gee, R; Lacevic, N
2007-05-09
Integral equation theory techniques are applied to evaluate the structuring of the polymer when large solid particles are embedded into a bulk polymer melt. The formalism presented here is applied to obtain an insight into the filler particle aggregation tendency. We find that with the employed polymer-particle interaction model it is very unlikely that the particles will aggregate. We believe that in such a system aggregation and clustering can occur when the filler particles are dressed by tightly bound polymer layers.
Freezing lines of colloidal Yukawa spheres. I. A Rogers-Young integral equation study.
Gapinski, Jacek; Nägele, Gerhard; Patkowski, Adam
2012-01-14
Using the Rogers-Young (RY) integral equation scheme for the static structure factor combined with the one-phase Hansen-Verlet (HV) freezing rule, we study the equilibrium structure and two-parameter freezing lines of colloidal particles with Yukawa-type pair interactions representing charge-stabilized silica spheres suspended in dimethylformamide (DMF). Results are presented for a vast range of concentrations, salinities and effective charges covering particles with masked excluded-volume interactions. The freezing lines were obtained for the low-charge and high-charge solutions of the static structure factor, for various two-parameter sets of experimentally accessible system parameters. All RY-HV based freezing lines can be mapped on a universal fluid-solid coexistence line in good agreement with computer simulation predictions. The RY-HV calculations extend the freezing lines obtained in earlier simulations to a broader parameter range. The experimentally observed fluid-bcc-fluid reentrant transition of charged silica spheres in DMF can be explained using the freezing lines obtained in this work.
Freezing lines of colloidal Yukawa spheres. I. A Rogers-Young integral equation study
NASA Astrophysics Data System (ADS)
Gapinski, Jacek; Nägele, Gerhard; Patkowski, Adam
2012-01-01
Using the Rogers-Young (RY) integral equation scheme for the static structure factor combined with the one-phase Hansen-Verlet (HV) freezing rule, we study the equilibrium structure and two-parameter freezing lines of colloidal particles with Yukawa-type pair interactions representing charge-stabilized silica spheres suspended in dimethylformamide (DMF). Results are presented for a vast range of concentrations, salinities and effective charges covering particles with masked excluded-volume interactions. The freezing lines were obtained for the low-charge and high-charge solutions of the static structure factor, for various two-parameter sets of experimentally accessible system parameters. All RY-HV based freezing lines can be mapped on a universal fluid-solid coexistence line in good agreement with computer simulation predictions. The RY-HV calculations extend the freezing lines obtained in earlier simulations to a broader parameter range. The experimentally observed fluid-bcc-fluid reentrant transition of charged silica spheres in DMF can be explained using the freezing lines obtained in this work.
Studies of the accuracy of time integration methods for reaction-diffusion equations
NASA Astrophysics Data System (ADS)
Ropp, David L.; Shadid, John N.; Ober, Curtis C.
2004-03-01
In this study we present numerical experiments of time integration methods applied to systems of reaction-diffusion equations. Our main interest is in evaluating the relative accuracy and asymptotic order of accuracy of the methods on problems which exhibit an approximate balance between the competing component time scales. Nearly balanced systems can produce a significant coupling of the physical mechanisms and introduce a slow dynamical time scale of interest. These problems provide a challenging test for this evaluation and tend to reveal subtle differences between the various methods. The methods we consider include first- and second-order semi-implicit, fully implicit, and operator-splitting techniques. The test problems include a prototype propagating nonlinear reaction-diffusion wave, a non-equilibrium radiation-diffusion system, a Brusselator chemical dynamics system and a blow-up example. In this evaluation we demonstrate a "split personality" for the operator-splitting methods that we consider. While operator-splitting methods often obtain very good accuracy, they can also manifest a serious degradation in accuracy due to stability problems.
Invariant imbedding and a matrix integral equation of neuronal networks.
NASA Technical Reports Server (NTRS)
Kalaba, R.; Ruspini, E. H.
1971-01-01
A matrix Fredholm integral equation of neuronal networks is transformed into a Cauchy system suited for numerical and analytical studies. A special case is discussed, and a connection with the classical renewal integral equation of stochastic point processes is presented.
Study of time-accurate integration of the variable-density Navier-Stokes equations
NASA Astrophysics Data System (ADS)
Lu, Xiaoyi; Pantano, Carlos
2015-11-01
We present several theoretical elements that affect time-consistent integration of the low-Mach number approximation of variable-density Navier-Stokes equations. The goal is for velocity, pressure, density, and scalars to achieve uniform order of accuracy, consistent with the time integrator being used. We show examples of second-order (using Crank-Nicolson and Adams-Bashforth) and third-order (using additive semi-implicit Runge-Kutta) uniform convergence with the proposed conceptual framework. Furthermore, the consistent approach can be extended to other time integrators. In addition, the method is formulated using approximate/incomplete factorization methods for easy incorporation in existing solvers. One of the observed benefits of the proposed approach is improved stability, even for large density difference, in comparison with other existing formulations. A linearized stability analysis is also carried out for some test problems to better understand the behavior of the approach. This work was supported in part by the Department of Energy, National Nuclear Security Administration, under award no. DE-NA0002382 and the California Institute of Technology.
Explicit integration of Friedmann's equation with nonlinear equations of state
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong E-mail: gwg1@damtp.cam.ac.uk
2015-05-01
In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
Painleve Chains for the Study of Integrable Higher Order Differential Equations.
1986-12-18
evolution equations , 1,2,3,4, 5 has become of special interest to theoretical physicists. Such equations possess a special type of elementary solution taking...diverse areas of physics including fluid dynamics, ferromagnetism, quantum optics, and crystal dislocations. Solution of important evolution equations ...and the most important evolution equations including the Burgers, Korteweg-de Vries ( KdV ), modified KdV , and Boussinesq equations . The present paper
Algorithms For Integrating Nonlinear Differential Equations
NASA Technical Reports Server (NTRS)
Freed, A. D.; Walker, K. P.
1994-01-01
Improved algorithms developed for use in numerical integration of systems of nonhomogenous, nonlinear, first-order, ordinary differential equations. In comparison with integration algorithms, these algorithms offer greater stability and accuracy. Several asymptotically correct, thereby enabling retention of stability and accuracy when large increments of independent variable used. Accuracies attainable demonstrated by applying them to systems of nonlinear, first-order, differential equations that arise in study of viscoplastic behavior, spread of acquired immune-deficiency syndrome (AIDS) virus and predator/prey populations.
PREFACE: Symmetries and Integrability of Difference Equations
NASA Astrophysics Data System (ADS)
Doliwa, Adam; Korhonen, Risto; Lafortune, Stéphane
2007-10-01
The notion of integrability was first introduced in the 19th century in the context of classical mechanics with the definition of Liouville integrability for Hamiltonian flows. Since then, several notions of integrability have been introduced for partial and ordinary differential equations. Closely related to integrability theory is the symmetry analysis of nonlinear evolution equations. Symmetry analysis takes advantage of the Lie group structure of a given equation to study its properties. Together, integrability theory and symmetry analysis provide the main method by which nonlinear evolution equations can be solved explicitly. Difference equations (DE), like differential equations, are important in numerous fields of science and have a wide variety of applications in such areas as mathematical physics, computer visualization, numerical analysis, mathematical biology, economics, combinatorics, and quantum field theory. It is thus crucial to develop tools to study and solve DEs. While the theory of symmetry and integrability for differential equations is now largely well-established, this is not yet the case for discrete equations. Although over recent years there has been significant progress in the development of a complete analytic theory of difference equations, further tools are still needed to fully understand, for instance, the symmetries, asymptotics and the singularity structure of difference equations. The series of SIDE meetings on Symmetries and Integrability of Difference Equations started in 1994. Its goal is to provide a platform for an international and interdisciplinary communication for researchers working in areas associated with integrable discrete systems, such as classical and quantum physics, computer science and numerical analysis, mathematical biology and economics, discrete geometry and combinatorics, theory of special functions, etc. The previous SIDE meetings took place in Estérel near Montréal, Canada (1994), at the University of
On third order integrable vector Hamiltonian equations
NASA Astrophysics Data System (ADS)
Meshkov, A. G.; Sokolov, V. V.
2017-03-01
A complete list of third order vector Hamiltonian equations with the Hamiltonian operator Dx having an infinite series of higher conservation laws is presented. A new vector integrable equation on the sphere is found.
Lax integrable nonlinear partial difference equations
NASA Astrophysics Data System (ADS)
Sahadevan, R.; Nagavigneshwari, G.
2015-03-01
A systematic investigation to derive nonlinear lattice equations governed by partial difference equations admitting specific Lax representation is presented. Further whether or not the identified lattice equations possess other characteristics of integrability namely Consistency Around the Cube (CAC) property and linearizability through a global transformation is analyzed. Also it is presented that how to derive higher order ordinary difference equations or mappings from the obtained lattice equations through periodic reduction and investigated whether they are measure preserving or linearizable and admit sufficient number of integrals leading to their integrability.
Feasibility study of the numerical integration of shell equations using the field method
NASA Technical Reports Server (NTRS)
Cohen, G. A.
1973-01-01
The field method is developed for arbitrary open branch domains subjected to general linear boundary conditions. Although closed branches are within the scope of the method, they are not treated here. The numerical feasibility of the method has been demonstrated by implementing it in a computer program for the linear static analysis of open branch shells of revolution under asymmetric loads. For such problems the field method eliminates the well-known numerical problem of long subintervals associated with the rapid growth of extraneous solutions. Also, the method appears to execute significantly faster than other numerical integration methods.
Holographic integral equations and walking technicolor
Alvares, Raul; Evans, Nick; Gebauer, Astrid; Weatherill, George James
2010-01-15
We study chiral symmetry breaking in the holographic D3-D7 system in a simple model with an arbitrary running coupling. We derive equations for the D7 embedding and show there is a light pion. In particular we present simple integral equations, involving just the running coupling and the quark self-energy, for the quark condensate and the pion decay constant. We compare these to the Pagels-Stokar or constituent quark model equivalent. We discuss the implications for walking technicolor theories. We also perform a similar analysis in the four-dimensional field theory whose dual is the nonsupersymmetric D3-D5 system and propose that it represents a walking theory in which the quark condensate has dimension 2+{radical}(3)
Linearization properties, first integrals, nonlocal transformation for heat transfer equation
NASA Astrophysics Data System (ADS)
Orhan, Özlem; Özer, Teoman
2016-08-01
We examine first integrals and linearization methods of the second-order ordinary differential equation which is called fin equation in this study. Fin is heat exchange surfaces which are used widely in industry. We analyze symmetry classification with respect to different choices of thermal conductivity and heat transfer coefficient functions of fin equation. Finally, we apply nonlocal transformation to fin equation and examine the results for different functions.
NASA Astrophysics Data System (ADS)
Toporkov, Jakov V.
A numerical study of electromagnetic scattering by one-dimensional perfectly conducting randomly rough surfaces with an ocean-like Pierson-Moskowitz spectrum is presented. Simulations are based on solving the Magnetic Field Integral Equation (MFIE) using the numerical technique called the Method of Ordered Multiple Interactions (MOMI). The study focuses on the application and validation of this integral equation-based technique to scattering at low grazing angles and considers other aspects of numerical simulations crucial to obtaining correct results in the demanding low grazing angle regime. It was found that when the MFIE propagator matrix is used with zeros on its diagonal (as has often been the practice) the results appear to show an unexpected sensitivity to the sampling interval. This sensitivity is especially pronounced in the case of horizontal polarization and at low grazing angles. We show---both numerically and analytically---that the problem lies not with the particular numerical technique used (MOMI) but rather with how the MFIE is discretized. It is demonstrated that the inclusion of so-called "curvature terms" (terms that arise from a correct discretization procedure and are proportional to the second surface derivative) in the diagonal of the propagator matrix eliminates the problem completely. A criterion for the choice of the sampling interval used in discretizing the MFIE based on both electromagnetic wavelength and the surface spectral cutoff is established. The influence of the surface spectral cutoff value on the results of scattering simulations is investigated and a recommendation for the choice of this spectral cutoff for numerical simulation purposes is developed. Also studied is the applicability of the tapered incident field at low grazing incidence angles. It is found that when a Gaussian-like taper with fixed beam waist is used there is a characteristic pattern (anomalous jump) in the calculated average backscattered cross section at
A SYMPLECTIC INTEGRATOR FOR HILL'S EQUATIONS
Quinn, Thomas; Barnes, Rory; Perrine, Randall P.; Richardson, Derek C.
2010-02-15
Hill's equations are an approximation that is useful in a number of areas of astrophysics including planetary rings and planetesimal disks. We derive a symplectic method for integrating Hill's equations based on a generalized leapfrog. This method is implemented in the parallel N-body code, PKDGRAV, and tested on some simple orbits. The method demonstrates a lack of secular changes in orbital elements, making it a very useful technique for integrating Hill's equations over many dynamical times. Furthermore, the method allows for efficient collision searching using linear extrapolation of particle positions.
Integral equations for flows in wind tunnels
NASA Technical Reports Server (NTRS)
Fromme, J. A.; Golberg, M. A.
1979-01-01
This paper surveys recent work on the use of integral equations for the calculation of wind tunnel interference. Due to the large number of possible physical situations, the discussion is limited to two-dimensional subsonic and transonic flows. In the subsonic case, the governing boundary value problems are shown to reduce to a class of Cauchy singular equations generalizing the classical airfoil equation. The theory and numerical solution are developed in some detail. For transonic flows nonlinear singular equations result, and a brief discussion of the work of Kraft and Kraft and Lo on their numerical solution is given. Some typical numerical results are presented and directions for future research are indicated.
Variational integrators for nonvariational partial differential equations
NASA Astrophysics Data System (ADS)
Kraus, Michael; Maj, Omar
2015-08-01
Variational integrators for Lagrangian dynamical systems provide a systematic way to derive geometric numerical methods. These methods preserve a discrete multisymplectic form as well as momenta associated to symmetries of the Lagrangian via Noether's theorem. An inevitable prerequisite for the derivation of variational integrators is the existence of a variational formulation for the considered problem. Even though for a large class of systems this requirement is fulfilled, there are many interesting examples which do not belong to this class, e.g., equations of advection-diffusion type frequently encountered in fluid dynamics or plasma physics. On the other hand, it is always possible to embed an arbitrary dynamical system into a larger Lagrangian system using the method of formal (or adjoint) Lagrangians. We investigate the application of the variational integrator method to formal Lagrangians, and thereby extend the application domain of variational integrators to include potentially all dynamical systems. The theory is supported by physically relevant examples, such as the advection equation and the vorticity equation, and numerically verified. Remarkably, the integrator for the vorticity equation combines Arakawa's discretisation of the Poisson brackets with a symplectic time stepping scheme in a fully covariant way such that the discrete energy is exactly preserved. In the presentation of the results, we try to make the geometric framework of variational integrators accessible to non specialists.
Solution of a system of dual integral equations.
NASA Technical Reports Server (NTRS)
Buell, J.; Kagiwada, H.; Kalaba, R.; Ruspini, E.; Zagustin, E.
1972-01-01
The solution of a presented system of differential equations with initial values is shown to satisfy a system of dual integral equations of a type appearing in the study of axisymmetric problems of potential theory. Of practical interest are possible applications in biomechanics, particularly, for the case of trauma due to impact.
NASA Astrophysics Data System (ADS)
Caplan, Ronald Meyer
We numerically study the dynamics and interactions of vortex rings in the nonlinear Schrodinger equation (NLSE). Single ring dynamics for both bright and dark vortex rings are explored including their traverse velocity, stability, and perturbations resulting in quadrupole oscillations. Multi-ring dynamics of dark vortex rings are investigated, including scattering and merging of two colliding rings, leapfrogging interactions of co-traveling rings, as well as co-moving steady-state multi-ring ensembles. Simulations of choreographed multi-ring setups are also performed, leading to intriguing interaction dynamics. Due to the inherent lack of a close form solution for vortex rings and the dimensionality where they live, efficient numerical methods to integrate the NLSE have to be developed in order to perform the extensive number of required simulations. To facilitate this, compact high-order numerical schemes for the spatial derivatives are developed which include a new semi-compact modulus-squared Dirichlet boundary condition. The schemes are combined with a fourth-order Runge-Kutta time-stepping scheme in order to keep the overall method fully explicit. To ensure efficient use of the schemes, a stability analysis is performed to find bounds on the largest usable time step-size as a function of the spatial step-size. The numerical methods are implemented into codes which are run on NVIDIA graphic processing unit (GPU) parallel architectures. The codes running on the GPU are shown to be many times faster than their serial counterparts. The codes are developed with future usability in mind, and therefore are written to interface with MATLAB utilizing custom GPU-enabled C codes with a MEX-compiler interface. Reproducibility of results is achieved by combining the codes into a code package called NLSEmagic which is freely distributed on a dedicated website.
Exact Solutions and Conservation Laws for a New Integrable Equation
Gandarias, M. L.; Bruzon, M. S.
2010-09-30
In this work we study a generalization of an integrable equation proposed by Qiao and Liu from the point of view of the theory of symmetry reductions in partial differential equations. Among the solutions we obtain a travelling wave with decaying velocity and a smooth soliton solution. We determine the subclass of these equations which are quasi-self-adjoint and we get a nontrivial conservation law.
Integral kinetic equation in dechanneling problem
NASA Astrophysics Data System (ADS)
Ryabov, V.
1989-11-01
A version of dechanneling theory, based on using an integral kinetic equation in both the phase and transverse energy space, is described. It is derived from the binary collision model and it takes into account consistently the thermal multiple and single scattering of axial and planar channeled particles. The connection between the method developed and that of Oshiyama and of Gartner is discussed.
Polynomial solutions of nonlinear integral equations
NASA Astrophysics Data System (ADS)
Dominici, Diego
2009-05-01
We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of Bender and Ben-Naim (2007 J. Phys. A: Math. Theor. 40 F9, 2008 J. Nonlinear Math. Phys. 15 (Suppl. 3) 73). We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel polynomials.
An integrable coupled short pulse equation
NASA Astrophysics Data System (ADS)
Feng, Bao-Feng
2012-03-01
An integrable coupled short pulse (CSP) equation is proposed for the propagation of ultra-short pulses in optical fibers. Based on two sets of bilinear equations to a two-dimensional Toda lattice linked by a Bäcklund transformation, and an appropriate hodograph transformation, the proposed CSP equation is derived. Meanwhile, its N-soliton solutions are given by the Casorati determinant in a parametric form. The properties of one- and two-soliton solutions are investigated in detail. Same as the short pulse equation, the two-soliton solution turns out to be a breather type if the wave numbers are complex conjugate. We also illustrate an example of soliton-breather interaction.
Lectures on differential equations for Feynman integrals
NASA Astrophysics Data System (ADS)
Henn, Johannes M.
2015-04-01
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations (DE). These lectures give a review of these developments, while not assuming any prior knowledge of the subject. After an introduction to DE for Feynman integrals, we point out how they can be simplified using algorithms available in the mathematical literature. We discuss how this is related to a recent conjecture for a canonical form of the equations. We also discuss a complementary approach that is based on properties of the space-time loop integrands, and explain how the ideas of leading singularities and d-log representations can be used to find an optimal basis for the DE. Finally, as an application of these ideas we show how single-scale integrals can be bootstrapped using the Drinfeld associator of a DE.
Huš, Matej; Munaò, Gianmarco; Urbic, Tomaz
2014-01-01
Thermodynamic and structural properties of a coarse-grained model of methanol are examined by Monte Carlo simulations and reference interaction site model (RISM) integral equation theory. Methanol particles are described as dimers formed from an apolar Lennard-Jones sphere, mimicking the methyl group, and a sphere with a core-softened potential as the hydroxyl group. Different closure approximations of the RISM theory are compared and discussed. The liquid structure of methanol is investigated by calculating site-site radial distribution functions and static structure factors for a wide range of temperatures and densities. Results obtained show a good agreement between RISM and Monte Carlo simulations. The phase behavior of methanol is investigated by employing different thermodynamic routes for the calculation of the RISM free energy, drawing gas-liquid coexistence curves that match the simulation data. Preliminary indications for a putative second critical point between two different liquid phases of methanol are also discussed. PMID:25362323
Huš, Matej; Urbic, Tomaz; Munaò, Gianmarco
2014-10-28
Thermodynamic and structural properties of a coarse-grained model of methanol are examined by Monte Carlo simulations and reference interaction site model (RISM) integral equation theory. Methanol particles are described as dimers formed from an apolar Lennard-Jones sphere, mimicking the methyl group, and a sphere with a core-softened potential as the hydroxyl group. Different closure approximations of the RISM theory are compared and discussed. The liquid structure of methanol is investigated by calculating site-site radial distribution functions and static structure factors for a wide range of temperatures and densities. Results obtained show a good agreement between RISM and Monte Carlo simulations. The phase behavior of methanol is investigated by employing different thermodynamic routes for the calculation of the RISM free energy, drawing gas-liquid coexistence curves that match the simulation data. Preliminary indications for a putative second critical point between two different liquid phases of methanol are also discussed.
On some new forms of lattice integrable equations
NASA Astrophysics Data System (ADS)
Babalic, Corina N.; Carstea, Adrian S.
2014-05-01
Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon — type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattice equations is studied and it is shown that all of them, including the higher order ones, can be integrated to Quispel-Roberts-Thomson (QRT) mappings.
Isogeometric Analysis of Boundary Integral Equations
2015-04-21
problems governed by Laplace’s equation. It is shown that the smoothness of geometric parametriza- tions central to computer-aided design can be exploited...smoothness of geometric parametriza- tions central to computer-aided design can be exploited for regularizing integral operators. As a result, one...functions coincide with those used for geometric parametrizations in computer aided design (CAD). Thus, in contrast to conventional finite element
Integrals and integral equations in linearized wing theory
NASA Technical Reports Server (NTRS)
Lomax, Harvard; Heaslet, Max A; Fuller, Franklyn B
1951-01-01
The formulas of subsonic and supersonic wing theory for source, doublet, and vortex distributions are reviewed and a systematic presentation is provided which relates these distributions to the pressure and to the vertical induced velocity in the plane of the wing. It is shown that care must be used in treating the singularities involved in the analysis and that the order of integration is not always reversible. Concepts suggested by the irreversibility of order of integration are shown to be useful in the inversion of singular integral equations when operational techniques are used. A number of examples are given to illustrate the methods presented, attention being directed to supersonic flight speed.
Integrability of the Gross Pitaevskii equation with Feshbach resonance management
NASA Astrophysics Data System (ADS)
Zhao, Dun; Luo, Hong-Gang; Chai, Hua-Yue
2008-08-01
In this Letter we study the integrability of a class of Gross-Pitaevskii equations managed by Feshbach resonance in an expulsive parabolic external potential. By using WTC test, we find a condition under which the Gross-Pitaevskii equation is completely integrable. Under the present model, this integrability condition is completely consistent with that proposed by Serkin, Hasegawa, and Belyaeva [V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102]. Furthermore, this integrability can also be explicitly shown by a transformation, which can convert the Gross-Pitaevskii equation into the well-known standard nonlinear Schrödinger equation. By this transformation, each exact solution of the standard nonlinear Schrödinger equation can be converted into that of the Gross-Pitaevskii equation, which builds a systematical connection between the canonical solitons and the so-called nonautonomous ones. The finding of this transformation has a significant contribution to understanding the essential properties of the nonautonomous solitons and the dynamics of the Bose-Einstein condensates by using the Feshbach resonance technique.
Alternative field representations and integral equations for modeling inhomogeneous dielectrics
NASA Technical Reports Server (NTRS)
Volakis, John L.
1992-01-01
New volume and volume-surface integral equations are presented for modeling inhomogeneous dielectric regions. The presented integral equations result in more efficient numerical implementations and should, therefore, be useful in a variety of electromagnetic applications.
On integrability aspects of the supersymmetric sine-Gordon equation
NASA Astrophysics Data System (ADS)
Bertrand, S.
2017-04-01
In this paper we study certain integrability properties of the supersymmetric sine-Gordon equation. We construct Lax pairs with their zero-curvature representations which are equivalent to the supersymmetric sine-Gordon equation. From the fermionic linear spectral problem, we derive coupled sets of super Riccati equations and the auto-Bäcklund transformation of the supersymmetric sine-Gordon equation. In addition, a detailed description of the associated Darboux transformation is presented and non-trivial super multisoliton solutions are constructed. These integrability properties allow us to provide new explicit geometric characterizations of the bosonic supersymmetric version of the Sym–Tafel formula for the immersion of surfaces in a Lie superalgebra. These characterizations are expressed only in terms of the independent bosonic and fermionic variables.
Integrability test for discrete equations via generalized symmetries
Levi, D.; Yamilov, R. I.
2010-12-23
In this article we present some integrability conditions for partial difference equations obtained using the formal symmetries approach. We apply them to find integrable partial difference equations contained in a class of equations obtained by the multiple scale analysis of the general multilinear dispersive difference equation defined on the square.
Distribution theory for Schrödinger’s integral equation
Lange, Rutger-Jan
2015-12-15
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger’s equation. This paper, in contrast, investigates the integral form of Schrödinger’s equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger’s integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger’s differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov’s [J. Math. Anal. Appl. 201(1), 297–323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger’s integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger’s differential equation. Third, we derive boundary conditions for “super-singular” potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger’s integral equation is a viable tool for studying singular interactions in quantum mechanics.
Distribution theory for Schrödinger's integral equation
NASA Astrophysics Data System (ADS)
Lange, Rutger-Jan
2015-12-01
Much of the literature on point interactions in quantum mechanics has focused on the differential form of Schrödinger's equation. This paper, in contrast, investigates the integral form of Schrödinger's equation. While both forms are known to be equivalent for smooth potentials, this is not true for distributional potentials. Here, we assume that the potential is given by a distribution defined on the space of discontinuous test functions. First, by using Schrödinger's integral equation, we confirm a seminal result by Kurasov, which was originally obtained in the context of Schrödinger's differential equation. This hints at a possible deeper connection between both forms of the equation. We also sketch a generalisation of Kurasov's [J. Math. Anal. Appl. 201(1), 297-323 (1996)] result to hypersurfaces. Second, we derive a new closed-form solution to Schrödinger's integral equation with a delta prime potential. This potential has attracted considerable attention, including some controversy. Interestingly, the derived propagator satisfies boundary conditions that were previously derived using Schrödinger's differential equation. Third, we derive boundary conditions for "super-singular" potentials given by higher-order derivatives of the delta potential. These boundary conditions cannot be incorporated into the normal framework of self-adjoint extensions. We show that the boundary conditions depend on the energy of the solution and that probability is conserved. This paper thereby confirms several seminal results and derives some new ones. In sum, it shows that Schrödinger's integral equation is a viable tool for studying singular interactions in quantum mechanics.
Characterizations of linear Volterra integral equations with nonnegative kernels
NASA Astrophysics Data System (ADS)
Naito, Toshiki; Shin, Jong Son; Murakami, Satoru; Ngoc, Pham Huu Anh
2007-11-01
We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.
Maximal regularity for perturbed integral equations on periodic Lebesgue spaces
NASA Astrophysics Data System (ADS)
Lizama, Carlos; Poblete, Verónica
2008-12-01
We characterize the maximal regularity of periodic solutions for an additive perturbed integral equation with infinite delay in the vector-valued Lebesgue spaces. Our method is based on operator-valued Fourier multipliers. We also study resonances, characterizing the existence of solutions in terms of a compatibility condition on the forcing term.
An integral equation arising in two group neutron transport theory
NASA Astrophysics Data System (ADS)
Cassell, J. S.; Williams, M. M. R.
2003-07-01
An integral equation describing the fuel distribution necessary to maintain a flat flux in a nuclear reactor in two group transport theory is reduced to the solution of a singular integral equation. The formalism developed enables the physical aspects of the problem to be better understood and its relationship with the corresponding diffusion theory model is highlighted. The integral equation is solved by reducing it to a non-singular Fredholm equation which is then evaluated numerically.
Master equations and the theory of stochastic path integrals.
Weber, Markus F; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers-Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman-Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a 'generating functional', which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a 'forward' and a 'backward' path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered from them. Upon
Master equations and the theory of stochastic path integrals
NASA Astrophysics Data System (ADS)
Weber, Markus F.; Frey, Erwin
2017-04-01
This review provides a pedagogic and self-contained introduction to master equations and to their representation by path integrals. Since the 1930s, master equations have served as a fundamental tool to understand the role of fluctuations in complex biological, chemical, and physical systems. Despite their simple appearance, analyses of master equations most often rely on low-noise approximations such as the Kramers–Moyal or the system size expansion, or require ad-hoc closure schemes for the derivation of low-order moment equations. We focus on numerical and analytical methods going beyond the low-noise limit and provide a unified framework for the study of master equations. After deriving the forward and backward master equations from the Chapman–Kolmogorov equation, we show how the two master equations can be cast into either of four linear partial differential equations (PDEs). Three of these PDEs are discussed in detail. The first PDE governs the time evolution of a generalized probability generating function whose basis depends on the stochastic process under consideration. Spectral methods, WKB approximations, and a variational approach have been proposed for the analysis of the PDE. The second PDE is novel and is obeyed by a distribution that is marginalized over an initial state. It proves useful for the computation of mean extinction times. The third PDE describes the time evolution of a ‘generating functional’, which generalizes the so-called Poisson representation. Subsequently, the solutions of the PDEs are expressed in terms of two path integrals: a ‘forward’ and a ‘backward’ path integral. Combined with inverse transformations, one obtains two distinct path integral representations of the conditional probability distribution solving the master equations. We exemplify both path integrals in analysing elementary chemical reactions. Moreover, we show how a well-known path integral representation of averaged observables can be recovered
Evaluating impedances in a Sacherer integral equation
Zhang, S.Y.; Weng, W.T.
1994-08-01
In Sacherer integral equation, the beam line density is expanded on the phase deviation {phi}, generating a Hankel spectrum, rather than on the time, which generates a Fourier spectrum. This is a natural choice to deal with the particle evolution in phase space, it however causes complications whenever the impedance corresponding to the spectrum has to be evaluated. In this article, the line density expansion on {phi} is shown to be equivalent to a beam time modulation under an acceptable condition. Therefore for a Hankel spectrum, a number of sidebands, and the corresponding impedance as well, will be involved. For wideband resonators, it is shown that the original Sacherer solution is adequate. For narrowband resonators, the solution had been compromised, therefore a modification may be needed.
NASA Astrophysics Data System (ADS)
Levi, Decio; Olver, Peter; Thomova, Zora; Winternitz, Pavel
2009-11-01
The concept of integrability was introduced in classical mechanics in the 19th century for finite dimensional continuous Hamiltonian systems. It was extended to certain classes of nonlinear differential equations in the second half of the 20th century with the discovery of the inverse scattering transform and the birth of soliton theory. Also at the end of the 19th century Lie group theory was invented as a powerful tool for obtaining exact analytical solutions of large classes of differential equations. Together, Lie group theory and integrability theory in its most general sense provide the main tools for solving nonlinear differential equations. Like differential equations, difference equations play an important role in physics and other sciences. They occur very naturally in the description of phenomena that are genuinely discrete. Indeed, they may actually be more fundamental than differential equations if space-time is actually discrete at very short distances. On the other hand, even when treating continuous phenomena described by differential equations it is very often necessary to resort to numerical methods. This involves a discretization of the differential equation, i.e. a replacement of the differential equation by a difference one. Given the well developed and understood techniques of symmetry and integrability for differential equations a natural question to ask is whether it is possible to develop similar techniques for difference equations. The aim is, on one hand, to obtain powerful methods for solving `integrable' difference equations and to establish practical integrability criteria, telling us when the methods are applicable. On the other hand, Lie group methods can be adapted to solve difference equations analytically. Finally, integrability and symmetry methods can be combined with numerical methods to obtain improved numerical solutions of differential equations. The origin of the SIDE meetings goes back to the early 1990s and the first
Bezerra, Rui M F; Pinto, Paula A; Fraga, Irene; Dias, Albino A
2016-03-01
To determine initial velocities of enzyme catalyzed reactions without theoretical errors it is necessary to consider the use of the integrated Michaelis-Menten equation. When the reaction product is an inhibitor, this approach is particularly important. Nevertheless, kinetic studies usually involved the evaluation of other inhibitors beyond the reaction product. The occurrence of these situations emphasizes the importance of extending the integrated Michaelis-Menten equation, assuming the simultaneous presence of more than one inhibitor because reaction product is always present. This methodology is illustrated with the reaction catalyzed by alkaline phosphatase inhibited by phosphate (reaction product, inhibitor 1) and urea (inhibitor 2). The approach is explained in a step by step manner using an Excel spreadsheet (available as a template in Appendix). Curve fitting by nonlinear regression was performed with the Solver add-in (Microsoft Office Excel). Discrimination of the kinetic models was carried out based on Akaike information criterion. This work presents a methodology that can be used to develop an automated process, to discriminate in real time the inhibition type and kinetic constants as data (product vs. time) are achieved by the spectrophotometer.
Calculation of transonic flows using an extended integral equation method
NASA Technical Reports Server (NTRS)
Nixon, D.
1976-01-01
An extended integral equation method for transonic flows is developed. In the extended integral equation method velocities in the flow field are calculated in addition to values on the aerofoil surface, in contrast with the less accurate 'standard' integral equation method in which only surface velocities are calculated. The results obtained for aerofoils in subcritical flow and in supercritical flow when shock waves are present compare satisfactorily with the results of recent finite difference methods.
Strongly Nonlinear Integral Equations of Hammerstein Type
Browder, Felix E.
1975-01-01
This paper studies the solution of the nonlinear Hammerstein equation u(x) + ʃ k(x,y)f[y,u(y)]μ(dy) = h(x) in the singular case, i.e., where the linear operator K with kernel k(x,y) is not defined for all the range of the nonlinear mapping F given by Fu(y) = f[y,u(y)] over the whole class X of functions u which are potential solutions of the equation. An existence theorem is derived under relatively minimal assumptions upon k and f, namely that (Ku,u) ≥ 0, that K maps L1 into L1loc and is compact from L1 [unk] L∞ into L1loc, that f(y,s) has the same sign as s for ǀsǀ ≥ R, and that for each constant r > 0, ǀf(y,s)ǀ ≤ gr(y) for ǀsǀ ≤ r where g is bounded and summable. The proof is obtained by combining a priori bounds, a truncation procedure, and a convergence argument using the Dunford-Pettis theorem. PMID:16578727
Numerical integration of asymptotic solutions of ordinary differential equations
NASA Technical Reports Server (NTRS)
Thurston, Gaylen A.
1989-01-01
Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration.
Phase-integral solution of the radial Dirac equation
Linnaeus, Staffan
2010-03-15
A phase-integral (WKB) solution of the radial Dirac equation is constructed, retaining perfect symmetry between the two components of the wave function and introducing no singularities except at the classical transition points. The potential is allowed to be the time component of a four-vector, a Lorentz scalar, a pseudoscalar, or any combination of these. The key point in the construction is the transformation from two coupled first-order equations constituting the radial Dirac equation to a single second-order Schroedinger-type equation. This transformation can be carried out in infinitely many ways, giving rise to different second-order equations but with the same spectrum. A unique transformation is found that produces a particularly simple second-order equation and correspondingly simple and well-behaved phase-integral solutions. The resulting phase-integral formulas are applied to unbound and bound states of the Coulomb potential. For bound states, the exact energy levels are reproduced.
Renaissance Learning Equating Study. Report
ERIC Educational Resources Information Center
Sewell, Julie; Sainsbury, Marian; Pyle, Katie; Keogh, Nikki; Styles, Ben
2007-01-01
An equating study was carried out in autumn 2006 by the National Foundation for Educational Research (NFER) on behalf of Renaissance Learning, to provide validation evidence for the use of the Renaissance Star Reading and Star Mathematics tests in English schools. The study investigated the correlation between the Star tests and established tests.…
Exponential Methods for the Time Integration of Schroedinger Equation
Cano, B.; Gonzalez-Pachon, A.
2010-09-30
We consider exponential methods of second order in time in order to integrate the cubic nonlinear Schroedinger equation. We are interested in taking profit of the special structure of this equation. Therefore, we look at symmetry, symplecticity and approximation of invariants of the proposed methods. That will allow to integrate till long times with reasonable accuracy. Computational efficiency is also our aim. Therefore, we make numerical computations in order to compare the methods considered and so as to conclude that explicit Lawson schemes projected on the norm of the solution are an efficient tool to integrate this equation.
Stability of negative solitary waves for an integrable modified Camassa-Holm equation
Yin Jiuli; Tian Lixin; Fan Xinghua
2010-05-15
In this paper, we prove that the modified Camassa-Holm equation is Painleve integrable. We also study the orbital stability problem of negative solitary waves for this integrable equation. It is shown that the negative solitary waves are stable for arbitrary wave speed of propagation.
Integrable fourth-order difference equations
NASA Astrophysics Data System (ADS)
Maheswari, C. Uma; Sahadevan, R.
2010-06-01
In this paper an attempt is made to find four-dimensional analogs of two-dimensional Quispel, Roberts and Thompson mappings and identified four distinct cases have been identified. The obtained mappings are measure preserving. The integrability of the isolated mappings is examined by constructing a sufficient number of integrals and their symplectic structure wherever possible.
Integrability of the Wong Equations in the Class of Linear Integrals of Motion
NASA Astrophysics Data System (ADS)
Magazev, A. A.
2016-04-01
The Wong equations, which describe the motion of a classical charged particle with isospin in an external gauge field, are considered. The structure of the Lie algebra of the linear integrals of motion of these equations is investigated. An algebraic condition for integrability of the Wong equations is formulated. Some examples are considered.
Integrable semi-discretization of a multi-component short pulse equation
NASA Astrophysics Data System (ADS)
Feng, Bao-Feng; Maruno, Ken-ichi; Ohta, Yasuhiro
2015-04-01
In the present paper, we mainly study the integrable semi-discretization of a multi-component short pulse equation. First, we briefly review the bilinear equations for a multi-component short pulse equation proposed by Matsuno [J. Math. Phys. 52, 123702 (2011)] and reaffirm its N-soliton solution in terms of pfaffians. Then by using a Bäcklund transformation of the bilinear equations and defining a discrete hodograph (reciprocal) transformation, an integrable semi-discrete multi-component short pulse equation is constructed. Meanwhile, its N-soliton solution in terms of pfaffians is also proved.
On polynomial integrability of the Euler equations on so(4)
NASA Astrophysics Data System (ADS)
Llibre, Jaume; Yu, Jiang; Zhang, Xiang
2015-10-01
In this paper we prove that the Euler equations on the Lie algebra so(4) with a diagonal quadratic Hamiltonian either satisfy the Manakov condition, or have at most four functionally independent polynomial first integrals.
Numerical integration of ordinary differential equations of various orders
NASA Technical Reports Server (NTRS)
Gear, C. W.
1969-01-01
Report describes techniques for the numerical integration of differential equations of various orders. Modified multistep predictor-corrector methods for general initial-value problems are discussed and new methods are introduced.
Stability of drift waves with the integral eigenmode equation
Chen, L.; Ke, F.J.; Xu, M.J.; Tsai, S.T.; Lee, Y.C.; Antonsen, T.M. Jr.
1981-11-01
An analytical theory on the stability properties of drift-wave eigenmodes in a slab plasma with finite magnetic shear is presented. The corresponding eigenmode equation is the integral equation first given by Coppi, Rosenbluth, and Sagdeev (1967) and rederived here, in a relatively simpler fashion, via the gyrokinetic equation. It is then proved that the universal drift-wave eigenmodes remain absolutely stable and finite electron temperature gradients do not alter the stability.
On the solution of integral equations with strongly singular kernels
NASA Technical Reports Server (NTRS)
Kaya, A. C.; Erdogan, F.
1987-01-01
Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m, m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup-m, terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.
On the solution of integral equations with strongly singular kernels
NASA Technical Reports Server (NTRS)
Kaya, A. C.; Erdogan, F.
1986-01-01
Some useful formulas are developed to evaluate integrals having a singularity of the form (t-x) sup-m ,m greater than or equal 1. Interpreting the integrals with strong singularities in Hadamard sense, the results are used to obtain approximate solutions of singular integral equations. A mixed boundary value problem from the theory of elasticity is considered as an example. Particularly for integral equations where the kernel contains, in addition to the dominant term (t-x) sup -m , terms which become unbounded at the end points, the present technique appears to be extremely effective to obtain rapidly converging numerical results.
Nonlinear partial differential equations: Integrability, geometry and related topics
NASA Astrophysics Data System (ADS)
Krasil'shchik, Joseph; Rubtsov, Volodya
2017-03-01
Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.
Monograph - The Numerical Integration of Ordinary Differential Equations.
ERIC Educational Resources Information Center
Hull, T. E.
The materials presented in this monograph are intended to be included in a course on ordinary differential equations at the upper division level in a college mathematics program. These materials provide an introduction to the numerical integration of ordinary differential equations, and they can be used to supplement a regular text on this…
Integral equation for small perturbations of irrotational flows
NASA Technical Reports Server (NTRS)
Haviland, J. K.
1973-01-01
An investigation is conducted concerning the validity of analytical methods which are based on deriving an integral equation, taking into account small perturbations in the case of a nonuniform but irrotational flow. The results obtained apply to a wide Mach number range, but are restricted to small amplitude motions and to nonviscous flows. It is shown that the integral equation relating the unknown velocity potential to the known normal flow velocity can be derived from the appropriate Green's identity.
NASA Technical Reports Server (NTRS)
Sloss, J. M.; Kranzler, S. K.
1972-01-01
The equivalence of a considered integral equation form with an infinite system of linear equations is proved, and the localization of the eigenvalues of the infinite system is expressed. Error estimates are derived, and the problems of finding upper bounds and lower bounds for the eigenvalues are solved simultaneously.
NASA Astrophysics Data System (ADS)
Choudhury, A. Ghose; Guha, Partha; Khanra, Barun
2009-10-01
The Darboux integrability method is particularly useful to determine first integrals of nonplanar autonomous systems of ordinary differential equations, whose associated vector fields are polynomials. In particular, we obtain first integrals for a variant of the generalized Raychaudhuri equation, which has appeared in string inspired modern cosmology.
a Simple Method to Construct Integrable Coupling System for the MKdV Equation Hierarchy
NASA Astrophysics Data System (ADS)
Yu, Fajun
In this paper, we will extend Ma's method to construct the integrable couplings of soliton equation hierarchy with the Kronecker product and two-nilpotent matrix. A direct application to the MKdV spectral problem leads to a novel integrable coupling system of soliton equation hierarchy. It is shown that the study of integrable couplings using the Kronecker product is an efficient and straightforward method.
Liu, Ai-Lin; Zhou, Ting; He, Feng-Yun; Xu, Jing-Juan; Lu, Yu; Chen, Hong-Yuan; Xia, Xing-Hua
2006-06-01
We firstly transformed the traditional Michaelis-Menten equation into an off-line form which can be used for evaluating the Michaelis-Menten constant after the enzymatic reaction. For experimental estimation of the kinetics of enzymatic reactions, we have developed a facile and effective method by integrating an enzyme microreactor into direct-printing polymer microchips. Strong nonspecific adsorption of proteins was utilized to effectively immobilize enzymes onto the microchannel wall, forming the integrated on-column enzyme microreactor in a microchip. The properties of the integrated enzyme microreactor were evaluated by using the enzymatic reaction of glucose oxidase (GOx) with its substrate glucose as a model system. The reaction product, hydrogen peroxide, was electrochemically (EC) analyzed using a Pt microelectrode. The data for enzyme kinetics using our off-line form of the Michaelis-Menten equation was obtained (K(m) = 2.64 mM), which is much smaller than that reported in solution (K(m) = 6.0 mM). Due to the hydrophobic property and the native mesoscopic structure of the poly(ethylene terephthalate) film, the immobilized enzyme in the microreactor shows good stability and bioactivity under the flowing conditions.
Master integrals for splitting functions from differential equations in QCD
NASA Astrophysics Data System (ADS)
Gituliar, Oleksandr
2016-02-01
A method for calculating phase-space master integrals for the decay process 1 → n masslesspartonsinQCDusingintegration-by-partsanddifferentialequationstechniques is discussed. The method is based on the appropriate choice of the basis for master integrals which leads to significant simplification of differential equations. We describe an algorithm how to construct the desirable basis, so that the resulting system of differential equations can be recursively solved in terms of (G) HPLs as a series in the dimensional regulator ɛ to any order. We demonstrate its power by calculating master integrals for the NLO time-like splitting functions and discuss future applications of the proposed method at the NNLO precision.
Nonzero solutions of nonlinear integral equations modeling infectious disease
Williams, L.R.; Leggett, R.W.
1982-01-01
Sufficient conditions to insure the existence of periodic solutions to the nonlinear integral equation, x(t) = ..integral../sup t//sub t-tau/f(s,x(s))ds, are given in terms of simple product and product integral inequalities. The equation can be interpreted as a model for the spread of infectious diseases (e.g., gonorrhea or any of the rhinovirus viruses) if x(t) is the proportion of infectives at time t and f(t,x(t)) is the proportion of new infectives per unit time.
NASA Astrophysics Data System (ADS)
Dimitriu, G.; Satco, B.
2016-10-01
Motivated by the fact that bounded variation (often discontinuous) functions frequently appear when studying integral equations that describe physical phenomena, we focus on the existence of bounded variation solutions for Urysohn integral measure driven equations. Due to numerous applications of Urysohn integral equations in various domains, problems of this kind have been extensively studied in literature, under more restrictive assumptions. Our approach concerns the framework of Kurzweil-Stieltjes integration, which allows the occurrence of high oscillatory features on the right hand side of the equation. A discussion about interesting consequences of our main result (given by particular cases of the measure driving the equation) is presented. Finally, we show the generality of our results by investigating two examples of impulsive type problems (from both theoretical and numerical perspective) and giving an application in electronics industry concerning polarization properties of ferroelectric materials.
NASA Astrophysics Data System (ADS)
Cui-Cui, Liao; Jin-Chao, Cui; Jiu-Zhen, Liang; Xiao-Hua, Ding
2016-01-01
In this paper, we propose a variational integrator for nonlinear Schrödinger equations with variable coefficients. It is shown that our variational integrator is naturally multi-symplectic. The discrete multi-symplectic structure of the integrator is presented by a multi-symplectic form formula that can be derived from the discrete Lagrangian boundary function. As two examples of nonlinear Schrödinger equations with variable coefficients, cubic nonlinear Schrödinger equations and Gross-Pitaevskii equations are extensively studied by the proposed integrator. Our numerical simulations demonstrate that the integrator is capable of preserving the mass, momentum, and energy conservation during time evolutions. Convergence tests are presented to verify that our integrator has second-order accuracy both in time and space. Project supported by the National Natural Science Foundation of China (Grant No. 11401259) and the Fundamental Research Funds for the Central Universities, China (Grant No. JUSRR11407).
The LEM exponential integrator for advection-diffusion-reaction equations
NASA Astrophysics Data System (ADS)
Caliari, Marco; Vianello, Marco; Bergamaschi, Luca
2007-12-01
We implement a second-order exponential integrator for semidiscretized advection-diffusion-reaction equations, obtained by coupling exponential-like Euler and Midpoint integrators, and computing the relevant matrix exponentials by polynomial interpolation at Leja points. Numerical tests on 2D models discretized in space by finite differences or finite elements, show that the Leja-Euler-Midpoint (LEM) exponential integrator can be up to 5 times faster than a classical second-order implicit solver.
Canonical algorithms for numerical integration of charged particle motion equations
NASA Astrophysics Data System (ADS)
Efimov, I. N.; Morozov, E. A.; Morozova, A. R.
2017-02-01
A technique for numerically integrating the equation of charged particle motion in a magnetic field is considered. It is based on the canonical transformations of the phase space in Hamiltonian mechanics. The canonical transformations make the integration process stable against counting error accumulation. The integration algorithms contain a minimum possible amount of arithmetics and can be used to design accelerators and devices of electron and ion optics.
Integrability and structural stability of solutions to the Ginzburg-Landau equation
NASA Technical Reports Server (NTRS)
Keefe, Laurence R.
1986-01-01
The integrability of the Ginzburg-Landau equation is studied to investigate if the existence of chaotic solutions found numerically could have been predicted a priori. The equation is shown not to possess the Painleveproperty, except for a special case of the coefficients that corresponds to the integrable, nonlinear Schroedinger (NLS) equation. Regarding the Ginzburg-Landau equation as a dissipative perturbation of the NLS, numerical experiments show all but one of a family of two-tori solutions, possessed by the NLS under particular conditions, to disappear under real perturbations to the NLS coefficients of O(10 to the -6th).
de Munck, J C
1992-09-01
A method is presented to compute the potential distribution on the surface of a homogeneous isolated conductor of arbitrary shape. The method is based on an approximation of a boundary integral equation as a set linear algebraic equations. The potential is described as a piecewise linear or quadratic function. The matrix elements of the discretized equation are expressed as analytical formulas.
Linear Multistep Methods for Integrating Reversible Differential Equations
NASA Astrophysics Data System (ADS)
Evans, N. Wyn; Tremaine, Scott
1999-10-01
This paper studies multistep methods for the integration of reversible dynamical systems, with particular emphasis on the planar Kepler problem. It has previously been shown by Cano & Sanz-Serna that reversible linear multisteps for first-order differential equations are generally unstable. Here we report on a subset of these methods-the zero-growth methods-that evade these instabilities. We provide an algorithm for identifying these rare methods. We find and study all zero-growth, reversible multisteps with six or fewer steps. This select group includes two well-known second-order multisteps (the trapezoidal and explicit midpoint methods), as well as three new fourth-order multisteps-one of which is explicit. Variable time steps can be readily implemented without spoiling the reversibility. Tests on Keplerian orbits show that these new reversible multisteps work well on orbits with low or moderate eccentricity, although at least 100 steps per radian are required for stability.
Integrable nonlocal nonlinear Schrödinger equation.
Ablowitz, Mark J; Musslimani, Ziad H
2013-02-08
A new integrable nonlocal nonlinear Schrödinger equation is introduced. It possesses a Lax pair and an infinite number of conservation laws and is PT symmetric. The inverse scattering transform and scattering data with suitable symmetries are discussed. A method to find pure soliton solutions is given. An explicit breathing one soliton solution is found. Key properties are discussed and contrasted with the classical nonlinear Schrödinger equation.
Discrete Painlevé equations: an integrability paradigm
NASA Astrophysics Data System (ADS)
Grammaticos, B.; Ramani, A.
2014-03-01
In this paper we present a review of results on discrete Painlevé equations. We begin with an introduction which serves as a refresher on the continuous Painlevé equations. Next, in the first, main part of the paper, we introduce the discrete Painlevé equations, the various methods for their derivation, and their properties as well as their classification scheme. Along the way we present a brief summary of the two major discrete integrability detectors and of Quispel-Roberts-Thompson mapping, which plays a primordial role in the derivation of discrete Painlevé equations. The second part of the paper is more technical and focuses on the presentation of new results on what are called asymmetric discrete Painlevé equations.
An integral equation solution for multistage turbomachinery design calculations
NASA Technical Reports Server (NTRS)
Mcfarland, Eric R.
1993-01-01
A method was developed to calculate flows in multistage turbomachinery. The method is an extension of quasi-three-dimensional blade-to-blade solution methods. Governing equations for steady compressible inviscid flow are linearized by introducing approximations. The linearized flow equations are solved using integral equation techniques. The flows through both stationary and rotating blade rows are determined in a single calculation. Multiple bodies can be modelled for each blade row, so that arbitrary blade counts can be analyzed. The method's benefits are its speed and versatility.
Five-wave classical scattering matrix and integrable equations
NASA Astrophysics Data System (ADS)
Zakharov, V. E.; Odesskii, A. V.; Cisternino, M.; Onorato, M.
2014-07-01
We study the five-wave classical scattering matrix for nonlinear and dispersive Hamiltonian equations with a nonlinearity of the type u∂u/∂x. Our aim is to find the most general nontrivial form of the dispersion relation ω(k) for which the five-wave interaction scattering matrix is identically zero on the resonance manifold. As could be expected, the matrix in one dimension is zero for the Korteweg-de Vries equation, the Benjamin-Ono equation, and the intermediate long-wave equation. In two dimensions, we find a new equation that satisfies our requirement.
The new integrable symplectic map and the symmetry of integrable nonlinear lattice equation
NASA Astrophysics Data System (ADS)
Dong, Huanhe; Zhang, Yong; Zhang, Xiaoen
2016-07-01
A discrete matrix spectral problem is presented and the hierarchy of discrete integrable systems is derived. Their Hamiltonian structures are established. As to the discrete integrable system, nonlinearization of the spatial parts of the Lax pairs and the adjoint Lax pairs generate a new integrable symplectic map. Based on the theory, a new integrable symplectic map and a family of finite-dimension completely integrable systems are given. Especially, two explicit equations are obtained under the Bargmann constraint. Finally, the symmetry of the discrete equation is provided according to the recursion operator and the seed symmetry. Although the solutions of the discrete equations have been gained by many methods, there are few articles that solving the discrete equation via the symmetry. So the solution of the discrete lattice equation is obtained through the symmetry theory.
Plasmonic properties of metal nanoislands: surface integral equations approach
NASA Astrophysics Data System (ADS)
Scherbak, S. A.; Lipovskii, A. A.
2016-08-01
The surface integral equations method is used to analyse the surface plasmon resonance position in a metal island film formed by non-interacting axisymmetrical prolate/oblate hemispheroids placed on a dielectric substrate. The approach is verified via the comparison of results obtained for a hemisphere on a substrate with the ones obtained using the multipole expansion method. The preference of the integral equations method is in obtaining a simple final analytical expression for a particle polarizability in which any dielectric function of a metal can be substituted. Such simple formulae for the hemispherical particle on the substrate and calculated dependences of the hemispheroid resonant wavelength on its aspect ratio are presented.
A spectral boundary integral equation method for the 2-D Helmholtz equation
NASA Technical Reports Server (NTRS)
Hu, Fang Q.
1994-01-01
In this paper, we present a new numerical formulation of solving the boundary integral equations reformulated from the Helmholtz equation. The boundaries of the problems are assumed to be smooth closed contours. The solution on the boundary is treated as a periodic function, which is in turn approximated by a truncated Fourier series. A Fourier collocation method is followed in which the boundary integral equation is transformed into a system of algebraic equations. It is shown that in order to achieve spectral accuracy for the numerical formulation, the nonsmoothness of the integral kernels, associated with the Helmholtz equation, must be carefully removed. The emphasis of the paper is on investigating the essential elements of removing the nonsmoothness of the integral kernels in the spectral implementation. The present method is robust for a general boundary contour. Aspects of efficient implementation of the method using FFT are also discussed. A numerical example of wave scattering is given in which the exponential accuracy of the present numerical method is demonstrated.
Comparison of four stable numerical methods for Abel's integral equation
NASA Technical Reports Server (NTRS)
Murio, Diego A.; Mejia, Carlos E.
1991-01-01
The 3-D image reconstruction from cone-beam projections in computerized tomography leads naturally, in the case of radial symmetry, to the study of Abel-type integral equations. If the experimental information is obtained from measured data, on a discrete set of points, special methods are needed in order to restore continuity with respect to the data. A new combined Regularized-Adjoint-Conjugate Gradient algorithm, together with two different implementations of the Mollification Method (one based on a data filtering technique and the other on the mollification of the kernal function) and a regularization by truncation method (initially proposed for 2-D ray sample schemes and more recently extended to 3-D cone-beam image reconstruction) are extensively tested and compared for accuracy and numerical stability as functions of the level of noise in the data.
Kedziora, D J; Ankiewicz, A; Chowdury, A; Akhmediev, N
2015-10-01
We present an infinite nonlinear Schrödinger equation hierarchy of integrable equations, together with the recurrence relations defining it. To demonstrate integrability, we present the Lax pairs for the whole hierarchy, specify its Darboux transformations and provide several examples of solutions. These resulting wavefunctions are given in exact analytical form. We then show that the Lax pair and Darboux transformation formalisms still apply in this scheme when the coefficients in the hierarchy depend on the propagation variable (e.g., time). This extension thus allows for the construction of complicated solutions within a greatly diversified domain of generalised nonlinear systems.
Galerkin Boundary Integral Analysis for the 3D Helmholtz Equation
Swager, Melissa; Gray, Leonard J; Nintcheu Fata, Sylvain
2010-01-01
A linear element Galerkin boundary integral analysis for the three-dimensional Helmholtz equation is presented. The emphasis is on solving acoustic scattering by an open (crack) surface, and to this end both a dual equation formulation and a symmetric hypersingular formulation have been developed. All singular integrals are defined and evaluated via a boundary limit process, facilitating the evaluation of the (finite) hypersingular Galerkin integral. This limit process is also the basis for the algorithm for post-processing of the surface gradient. The analytic integrations required by the limit process are carried out by employing a Taylor series expansion for the exponential factor in the Helmholtz fundamental solutions. For the open surface, the implementations are validated by comparing the numerical results obtained by using the two different methods.
Volume integrals of ellipsoids associated with the inhomogeneous Helmholtz equation
NASA Technical Reports Server (NTRS)
Fu, L. S.; Mura, T.
1982-01-01
Problems of wave phenomena in the fields of acoustics, electromagnetics and elasticity are often reduced to an integration of the inhomogeneous Helmholtz equation. Results are presented for volume integrals associated with the inhomogeneous Helmholtz equation, for an ellipsoidal region. By using appropriate Taylor series expansions and the multinomial theorem, these volume integrals are obtained in series form for regions r greater than r-prime and r less than r-prime, where r and r-prime are the distances from the origin to the point of observation and the source. Derivatives of these integrals are easily evaluated. When the wavenumber approaches zero the results reduce directly to the potentials of ellipsoids of variable densities.
On the numeric integration of dynamic attitude equations
NASA Technical Reports Server (NTRS)
Crouch, P. E.; Yan, Y.; Grossman, Robert
1992-01-01
We describe new types of numerical integration algorithms developed by the authors. The main aim of the algorithms is to numerically integrate differential equations which evolve on geometric objects, such as the rotation group. The algorithms provide iterates which lie on the prescribed geometric object, either exactly, or to some prescribed accuracy, independent of the order of the algorithm. This paper describes applications of these algorithms to the evolution of the attitude of a rigid body.
NASA Astrophysics Data System (ADS)
Zink, M.
The integral equation method in the form of the electric field integral equation for wire grid models provides the current distribution on the surface of structures under study. Characteristic parameters such as the input impedance and the radiation diagram are obtained in this fashion. These parameters are determined for a dipole in free space, a monopole over a circular ground plane, and a torus antenna. Good results are obtained for the far field and the variables related to it.
On the Implementation of 3D Galerkin Boundary Integral Equations
Nintcheu Fata, Sylvain; Gray, Leonard J
2010-01-01
In this article, a reverse contribution technique is proposed to accelerate the construction of the dense influence matrices associated with a Galerkin approximation of singular and hypersingular boundary integral equations of mixed-type in potential theory. In addition, a general-purpose sparse preconditioner for boundary element methods has also been developed to successfully deal with ill-conditioned linear systems arising from the discretization of mixed boundary-value problems on non-smooth surfaces. The proposed preconditioner, which originates from the precorrected-FFT method, is sparse, easy to generate and apply in a Krylov subspace iterative solution of discretized boundary integral equations. Moreover, an approximate inverse of the preconditioner is implicitly built by employing an incomplete LU factorization. Numerical experiments involving mixed boundary-value problems for the Laplace equation are included to illustrate the performance and validity of the proposed techniques.
The Cauchy problem for the integrable Novikov equation
NASA Astrophysics Data System (ADS)
Yan, Wei; Li, Yongsheng; Zhang, Yimin
In this paper we consider the Cauchy problem for the integrable Novikov equation. By using the Littlewood-Paley decomposition and nonhomogeneous Besov spaces, we prove that the Cauchy problem for the integrable Novikov equation is locally well-posed in the Besov space Bp,rs with 1⩽p,r⩽+∞ and s>max{1+1/p,3/2}. In particular, when u0∈Bp,rs∩H1 with 1⩽p,r⩽+∞ and s>max{1+1/p,3/2}, for all t∈[0,T], we have that ‖u(t)‖=‖u0‖. We also prove that the local well-posedness of the Cauchy problem for the Novikov equation fails in B2,∞3/2.
Efficient Integration of Quantum Mechanical Wave Equations by Unitary Transforms
Bauke, Heiko; Keitel, Christoph H.
2009-08-13
The integration of time dependent quantum mechanical wave equations is a fundamental problem in computational physics and computational chemistry. The energy and momentum spectrum of a wave function imposes fundamental limits on the performance of numerical algorithms for this problem. We demonstrate how unitary transforms can help to surmount these limitations.
Equation Free Projective Integration and its Applicability for Simulating Plasma
NASA Astrophysics Data System (ADS)
Jemella, B.; Shay, M. A.; Drake, J. F.; Dorland, W.
2004-12-01
We examine a novel simulation scheme called equation free projective integration1 which has the potential to allow global simulations of plasmas while still including the global effects of microscale physics. These simulation codes would be ideal for such multiscale problems as the Earth's magnetosphere, tokamaks, and the solar corona. In this method, the global plasma variables stepped forward in time are not time-integrated directly using dynamical differential equations, hence the name "equation free." Instead, these variables are represented on a microgrid using a kinetic simulation. This microsimulation is integrated forward long enough to determine the time derivatives of the global plasma variables, which are then used to integrate forward the global variables with much larger time steps. We are exploring the feasibility of applying this scheme to simulate plasma, and we will present the results of exploratory test problems including the development of 1-D shocks and magnetic reconnection. 1 I. G. Kevrekidis et. al., ``Equation-free multiscale computation: Enabling microscopic simulators to perform system-level tasks,'' arXiv:physics/0209043.
Equation free projective integration and its applicability for simulating plasma
NASA Astrophysics Data System (ADS)
Shay, Michael A.; Drake, James F.; Dorland, William; Swisdak, Marc
2004-11-01
We examine a novel simulation scheme called equation free projective integration^1 which has the potential to allow global simulations of plasmas while still including the global effects of microscale physics. These simulation codes would be ideal for such multiscale problems as tokamaks, the Earth's magnetosphere, and the solar corona. In this method, the global plasma variables stepped forward in time are not time-integrated directly using dynamical differential equations, hence the name ``equation free.'' Instead, these variables are represented on a microgrid using a kinetic simulation. This microsimulation is integrated forward long enough to determine the time derivatives of the global plasma variables, which are then used to integrate forward the global variables with much larger time steps. We are exploring the feasibility of applying this scheme to simulate plasma, and we will present the results of exploratory test problems including the development of 1-D shocks and magnetic reconnection. ^1 I. G. Kevrekidis et. al., ``Equation-free multiscale computation: Enabling microscopic simulators to perform system-level tasks,'' arXiv:physics/0209043.
Application of boundary integral equations to elastoplastic problems
NASA Technical Reports Server (NTRS)
Mendelson, A.; Albers, L. U.
1975-01-01
The application of boundary integral equations to elastoplastic problems is reviewed. Details of the analysis as applied to torsion problems and to plane problems is discussed. Results are presented for the elastoplastic torsion of a square cross section bar and for the plane problem of notched beams. A comparison of different formulations as well as comparisons with experimental results are presented.
Integral Equations and the Bound-State Problem.
ERIC Educational Resources Information Center
Bagchi, B.; Seyler, R. G.
1980-01-01
An integral equation for the s-wave bound-state solution is derived and then solved for a square-well potential. It is shown that the scattering solutions continue to exist at negative energies, and when evaluated at the energy of a bound state these solutions do reduce to the bound-state solution.
Integral equations for the microstructures of supercritical fluids
Lee, L.L.; Cochran, H.D.
1993-11-01
Molecular interactions and molecular distributions are at the heart of the supercritical behavior of fluid mixtures. The distributions, i.e. structure, can be obtained through any of the three routes: (1) scattering experiments, (2) Monte Carlo or molecular dynamics simulation, and (3) integral equations that govern the relation between the molecular interactions u(r) and the probability distributions g{sub ij}(r). Most integral equations are based on the Ornstein-Zernike relation connecting the total correlation to the direct correlation. The OZ relation requires a {open_quotes}closure{close_quotes} equation to be solvable. Thus the Percus-Yevick, hypernetted chain, and mean spherical approximations have been proposed. The authors outline the numerical methods of solution for these integral equations, including the Picard, Labik-Gillan, and Baxter methods. Solution of these equations yields the solvent-solute, solvent-solvent, and solute-solute pair correlation functions (pcf`s). Interestingly, these pcf`s exhibit characteristical signatures for supercritical mixtures that are classified as {open_quotes}attractive{close_quotes} or {open_quotes}repulsive{close_quotes} in nature. Close to the critical locus, the pcf shows enhanced first neighbor peaks with concomitant long-range build-ups (sic attractive behavior) or reduced first peaks plus long-range depletion (sic repulsive behavior) of neighbors. For ternary mixtures with entrainers, there are synergistic effects between solvent and cosolvent, or solute and cosolute. These are also detectable on the distribution function level. The thermodynamic consequences are deciphered through the Kirkwood-Buff fluctuation integrals (G{sub ij}) and their matrix inverses: the direct correlation function integrals (DCFI`s). These quantities connect the correlation functions to the chemical potential derivatives (macroscopic variables) thus acting as {open_quotes}bridges{close_quotes} between the two Weltanschauungen.
Boundary regularized integral equation formulation of the Helmholtz equation in acoustics.
Sun, Qiang; Klaseboer, Evert; Khoo, Boo-Cheong; Chan, Derek Y C
2015-01-01
A boundary integral formulation for the solution of the Helmholtz equation is developed in which all traditional singular behaviour in the boundary integrals is removed analytically. The numerical precision of this approach is illustrated with calculation of the pressure field owing to radiating bodies in acoustic wave problems. This method facilitates the use of higher order surface elements to represent boundaries, resulting in a significant reduction in the problem size with improved precision. Problems with extreme geometric aspect ratios can also be handled without diminished precision. When combined with the CHIEF method, uniqueness of the solution of the exterior acoustic problem is assured without the need to solve hypersingular integrals.
NASA Astrophysics Data System (ADS)
Momani, Shaher; Ibrahim, Rabha W.
2008-03-01
In this paper, we study the existence of periodic solutions for a nonlinear integral equation of periodic functions involving Weyl-Riesz fractional integral operator under the mixed generalized Lipschitz, Carathéodory and monotonicity conditions. The fixed point theorems due to Dhage are the main tool in carrying out our proofs.
Hamiltonian time integrators for Vlasov-Maxwell equations
He, Yang; Xiao, Jianyuan; Zhang, Ruili; Liu, Jian; Qin, Hong; Sun, Yajuan
2015-12-15
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.
Hamiltonian time integrators for Vlasov-Maxwell equations
NASA Astrophysics Data System (ADS)
He, Yang; Qin, Hong; Sun, Yajuan; Xiao, Jianyuan; Zhang, Ruili; Liu, Jian
2015-12-01
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.
Singularity Preserving Numerical Methods for Boundary Integral Equations
NASA Technical Reports Server (NTRS)
Kaneko, Hideaki (Principal Investigator)
1996-01-01
In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract.
Goličnik, Marko
2011-04-15
Various explicit reformulations of time-dependent solutions for the classical two-step irreversible Michaelis-Menten enzyme reaction model have been described recently. In the current study, I present further improvements in terms of a generalized integrated form of the Michaelis-Menten equation for computation of substrate or product concentrations as functions of time for more real-world, enzyme-catalyzed reactions affected by the product. The explicit equations presented here can be considered as a simpler and useful alternative to the exact solution for the generalized integrated Michaelis-Menten equation when fitted to time course data using standard curve-fitting software.
Podesta, John J.
2012-08-15
The electric field generated by a time varying point charge in a three-dimensional, unbounded, spatially homogeneous plasma with a uniform background magnetic field and a uniform (static) flow velocity is studied in the electrostatic approximation which is often valid in the near field. For plasmas characterized by Maxwell distribution functions with isotropic temperatures, the linearized Vlasov-Poisson equations may be formulated in terms of an equivalent integral equation in the time domain. The kernel of the integral equation has a relatively simple mathematical form consisting of elementary functions such as exponential and trigonometric functions (sines and cosines), and contains no infinite sums of Bessel functions. Consequently, the integral equation is amenable to numerical solutions and may be useful for the study of the impulse response of magnetized plasmas and, more generally, the response to arbitrary waveforms.
Phase integral theory, coupled wave equations, and mode conversion.
Littlejohn, Robert G.; Flynn, William G.
1992-01-01
Phase integral or WKB theory is applied to multicomponent wave equations, i.e., wave equations in which the wave field is a vector, spinor, or tensor of some kind. Specific examples of physical interest often have special features that simplify their analysis, when compared with the general theory. The case of coupled channel equations in atomic or molecular scattering theory in the Born-Oppenheimer approximation is examined in this context. The problem of mode conversion, also called surface jumping or Landau-Zener-Stuckelberg transitions, is examined in the multidimensional case, and cast into normal form. The group theoretical principles of the normal form transformation are laid out, and shown to involve both the Lorentz group and the symplectic group.
Integral equation methods for vesicle electrohydrodynamics in three dimensions
NASA Astrophysics Data System (ADS)
Veerapaneni, Shravan
2016-12-01
In this paper, we develop a new boundary integral equation formulation that describes the coupled electro- and hydro-dynamics of a vesicle suspended in a viscous fluid and subjected to external flow and electric fields. The dynamics of the vesicle are characterized by a competition between the elastic, electric and viscous forces on its membrane. The classical Taylor-Melcher leaky-dielectric model is employed for the electric response of the vesicle and the Helfrich energy model combined with local inextensibility is employed for its elastic response. The coupled governing equations for the vesicle position and its transmembrane electric potential are solved using a numerical method that is spectrally accurate in space and first-order in time. The method uses a semi-implicit time-stepping scheme to overcome the numerical stiffness associated with the governing equations.
Solving conical diffraction grating problems with integral equations.
Goray, Leonid I; Schmidt, Gunther
2010-03-01
Off-plane scattering of time-harmonic plane waves by a plane diffraction grating with arbitrary conductivity and general surface profile is considered in a rigorous electromagnetic formulation. Integral equations for conical diffraction are obtained involving, besides the boundary integrals of the single and double layer potentials, singular integrals, the tangential derivative of single-layer potentials. We derive an explicit formula for the calculation of the absorption in conical diffraction. Some rules that are expedient for the numerical implementation of the theory are presented. The efficiencies and polarization angles compared with those obtained by Lifeng Li for transmission and reflection gratings are in a good agreement. The code developed and tested is found to be accurate and efficient for solving off-plane diffraction problems including high-conductive gratings, surfaces with edges, real profiles, and gratings working at short wavelengths.
ERIC Educational Resources Information Center
Field, J. H.
2011-01-01
It is shown how the time-dependent Schrodinger equation may be simply derived from the dynamical postulate of Feynman's path integral formulation of quantum mechanics and the Hamilton-Jacobi equation of classical mechanics. Schrodinger's own published derivations of quantum wave equations, the first of which was also based on the Hamilton-Jacobi…
NASA Technical Reports Server (NTRS)
Baker, A. J.; Soliman, M. O.
1978-01-01
A study of accuracy and convergence of linear functional finite element solution to linear parabolic and hyperbolic partial differential equations is presented. A variable-implicit integration procedure is employed for the resultant system of ordinary differential equations. Accuracy and convergence is compared for the consistent and two lumped assembly procedures for the identified initial-value matrix structure. Truncation error estimation is accomplished using Richardson extrapolation.
NASA Technical Reports Server (NTRS)
Hayes, E. F.; Kouri, D. J.
1971-01-01
Coupled integral equations are derived for the full scattering amplitudes for both reactive and nonreactive channels. The equations do not involve any partial wave expansion and are obtained using channel operators for reactive and nonreactive collisions. These coupled integral equations are similar in nature to equations derived for purely nonreactive collisions of structureless particles. Using numerical quadrature techniques, these equations may be reduced to simultaneous algebraic equations which may then be solved.
Exponential integrators for the incompressible Navier-Stokes equations.
Newman, Christopher K.
2004-07-01
We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with problem size. Exponential integrators, used typically to solve systems of ODEs, utilize matrix vector products of the exponential of the Jacobian on a vector. For large systems, this product can be approximated efficiently by Krylov subspace methods. However, in contrast to explicit methods, KBEIs are not restricted by the time step. While implicit methods require a solution of a linear system with the Jacobian, KBEIs only require matrix vector products of the Jacobian. Furthermore, these methods are based on linearization, so there is no non-linear system solve at each time step. Differential-algebraic equations (DAEs) are ordinary differential equations (ODEs) subject to algebraic constraints. The discretized NSE constitute a system of DAEs, where the incompressibility condition is the algebraic constraint. Exponential integrators can be extended to DAEs with linear constraints imposed via a projection onto the constraint manifold. This results in a projected ODE that is integrated by a KBEI. In this approach, the Krylov subspace satisfies the constraint, hence the solution at the advanced time step automatically satisfies the constraint as well. For the NSE, the projection onto the constraint is typically achieved by a projection induced by the L{sup 2} inner product. We examine this L{sup 2} projection and an H{sup 1} projection induced by the H{sup 1} semi-inner product. The H
Application of boundary integral equations to elastoplastic problems
NASA Technical Reports Server (NTRS)
Mendelson, A.; Albers, L. U.
1975-01-01
The application of the boundary integral equation method (BIE) to the elastoplastic torsion problem is considered. It is found that the BIE is very suitable for the elastoplastic analysis of the torsion of prismatic bars. A comparison of the BIE with the finite difference method shows savings for the BIE concerning the number of unknowns which have to be determined and also a much faster convergence rate. Attention is given to the problem of an edge-notched beam in pure bending, taking into account a biharmonic formulation and a displacement formulation.
Investigation of ODE integrators using interactive graphics. [Ordinary Differential Equations
NASA Technical Reports Server (NTRS)
Brown, R. L.
1978-01-01
Two FORTRAN programs using an interactive graphic terminal to generate accuracy and stability plots for given multistep ordinary differential equation (ODE) integrators are described. The first treats the fixed stepsize linear case with complex variable solutions, and generates plots to show accuracy and error response to step driving function of a numerical solution, as well as the linear stability region. The second generates an analog to the stability region for classes of non-linear ODE's as well as accuracy plots. Both systems can compute method coefficients from a simple specification of the method. Example plots are given.
Free energies from integral equation theories: enforcing path independence.
Kast, Stefan M
2003-04-01
A variational formalism is constructed for deriving the chemical potential and the Helmholtz free energy in various statistical-mechanical integral equation theories of fluids. Nonzero bridge functions extending the scope of the theories beyond the hypernetted chain approximation can be classified as to whether or not they imply path dependence of the free energy. Classes of bridge functions free of the path dependence problem are derived, based on which a route is devised toward direct computation of free energies from the simulation of a single state.
Numerical solution of nonlinear Hammerstein fuzzy functional integral equations
NASA Astrophysics Data System (ADS)
Enkov, Svetoslav; Georgieva, Atanaska; Nikolla, Renato
2016-12-01
In this work we investigate nonlinear Hammerstein fuzzy functional integral equation. Our aim is to provide an efficient iterative method of successive approximations by optimal quadrature formula for classes of fuzzy number-valued functions of Lipschitz type to approximate the solution. We prove the convergence of the method by Banach's fixed point theorem and investigate the numerical stability of the presented method with respect to the choice of the first iteration. Finally, illustrative numerical experiment demonstrate the accuracy and the convergence of the proposed method.
Phase-integral method for the radial Dirac equation
Linnæus, Staffan
2014-09-15
A phase-integral (WKB) solution of the radial Dirac equation is calculated up to the third order of approximation, retaining perfect symmetry between the two components of the wave function and introducing no singularities except at the zeroth-order transition points. The potential is allowed to be of scalar, vector, or tensor type, or any combination of these. The connection problem is investigated in detail. Explicit formulas are given for single-turning-point phase shifts and single-well energy levels.
NASA Astrophysics Data System (ADS)
Utama, Briandhika; Purqon, Acep
2016-08-01
Path Integral is a method to transform a function from its initial condition to final condition through multiplying its initial condition with the transition probability function, known as propagator. At the early development, several studies focused to apply this method for solving problems only in Quantum Mechanics. Nevertheless, Path Integral could also apply to other subjects with some modifications in the propagator function. In this study, we investigate the application of Path Integral method in financial derivatives, stock options. Black-Scholes Model (Nobel 1997) was a beginning anchor in Option Pricing study. Though this model did not successfully predict option price perfectly, especially because its sensitivity for the major changing on market, Black-Scholes Model still is a legitimate equation in pricing an option. The derivation of Black-Scholes has a high difficulty level because it is a stochastic partial differential equation. Black-Scholes equation has a similar principle with Path Integral, where in Black-Scholes the share's initial price is transformed to its final price. The Black-Scholes propagator function then derived by introducing a modified Lagrange based on Black-Scholes equation. Furthermore, we study the correlation between path integral analytical solution and Monte-Carlo numeric solution to find the similarity between this two methods.
Exact solutions for the fractional differential equations by using the first integral method
NASA Astrophysics Data System (ADS)
Aminikhah, Hossein; Sheikhani, A. Refahi; Rezazadeh, Hadi
2015-03-01
In this paper, we apply the first integral method to study the solutions of the nonlinear fractional modified Benjamin-Bona-Mahony equation, the nonlinear fractional modified Zakharov-Kuznetsov equation and the nonlinear fractional Whitham-Broer-Kaup-Like systems. This method is based on the ring theory of commutative algebra. The results obtained by the proposed method show that the approach is effective and general. This approach can also be applied to other nonlinear fractional differential equations, which are arising in the theory of solitons and other areas.
Numerical solution of a class of integral equations arising in two-dimensional aerodynamics
NASA Technical Reports Server (NTRS)
Fromme, J.; Golberg, M. A.
1978-01-01
We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This allows the problem to be studied in terms of an equivalent integral equation of the second kind. A convergent numerical algorithm for its solution is derived by using Galerkin's method. It is shown that this algorithm is numerically equivalent to Bland's collocation method, which is then used as the method of computation. Extensive numerical calculations are presented establishing the validity of the theory.
Gauge Drift in Numerical Integrations of the Lagrange Planetary Equations
NASA Astrophysics Data System (ADS)
Murison, M. A.; Efroimsky, M.
2003-08-01
Efroimsky (2002) and Newman & Efroimsky (2003) recognized that the Lagrange and Delaunay planetary equations of celestial mechanics may be generalized to allow transformations analogous to the familiar gauge transformations in electrodynamics. As usually presented, the Lagrange equations, which are derived by the method of variation of parameters (invented by Euler and Lagrange for this very purpose), assume the Lagrange constraint, whereby a certain combination of parameter time derivatives is arbitrarily equated to zero. This particular constraint ensures an osculating orbit that is unique. The transformation of the description, as given by the (time-varying) osculating elements, into that given by the Cartesian coordinates and velocities is invertible. Relaxing the constraint enables one to substitute instead an arbitrary gauge function. This breaks the uniqueness and invertibility between the orbit instantaneously described by the orbital elements and the position and velocity components (i.e., many different orbits, precessing at different rates, can at a given instant share the same physical position and physical velocity through space). However, the orbit described by the (varying) orbital elements obeying a different gauge is no longer osculating. In numerical calculations that integrate the traditional Lagrange and Delaunay equations, even starting off in a certain (say, Lagrange's) gauge, some fraction of the numerical errors will, nevertheless, diffuse into violation of the chosen constraint. This results in an unintended ``gauge drift''. Geometrically, numerical errors cause the trajectory in phase space to leave the gauge-defined submanifold to which the motion was constrained, so that it is then moving on a different submanifold. The method of Lagrange multipliers can be utilized to return the motion to the original submanifold (e.g., Nacozy 1971, Murison 1989). Alternatively, the accumulated gauge drift may be compensated by a gauge transformation
Discretization of the Induced-Charge Boundary Integral Equation
Bardhan, Jaydeep P.; Eisenberg, Robert S.; Gillespie, Dirk
2013-01-01
Boundary-element methods (BEM) for solving integral equations numerically have been used in many fields to compute the induced charges at dielectric boundaries. In this paper, we consider a more accurate implementation of BEM in the context of ions in aqueous solution near proteins, but our results are applicable more generally. The ions that modulate protein function are often within a few Angstroms of the protein, which leads to the significant accumulation of polarization charge at the protein/solvent interface. Computing the induced charge accurately and quickly poses a numerical challenge in solving a popular integral equation using BEM. In particular, the accuracy of simulations can depend strongly on seemingly minor details of how the entries of the BEM matrix are calculated. We demonstrate that when the dielectric interface is discretized into flat tiles, the qualocation method of Tausch, Wang, and White (IEEE. Trans. Comput.-Aided Des. 20:1398, 2001) to compute the BEM matrix elements is always more accurate than the traditional centroid collocation method. Qualocation is no more expensive to implement than collocation and can save significant computional time by reducing the number of boundary elements needed to discretize the dielectric interfaces. PMID:19658728
Existence of solutions to nonlinear Hammerstein integral equations and applications
NASA Astrophysics Data System (ADS)
Li, Fuyi; Li, Yuhua; Liang, Zhanping
2006-11-01
In this paper, we study the existence and multiplicity of solutions of the operator equation Kfu=u in the real Hilbert space L2(G). Under certain conditions on the linear operator K, we establish the conditions on f which are able to guarantee that the operator equation has at least one solution, a unique solution, and infinitely many solutions, respectively. The monotone operator principle and the critical point theory are employed to discuss this problem, respectively. In argument, quadratic root operator K1/2 and its properties play an important role. As an application, we investigate the existence and multiplicity of solutions to fourth-order boundary value problems for ordinary differential equations with two parameters, and give some new existence results of solutions.
Numerical method to solve Cauchy type singular integral equation with error bounds
NASA Astrophysics Data System (ADS)
Setia, Amit; Sharma, Vaishali; Liu, Yucheng
2017-01-01
Cauchy type singular integral equations with index zero naturally occur in the field of aerodynamics. Literature is very much developed for these equations and Chebyshevs polynomials are most frequently used to solve these integral equations. In this paper, a residual based Galerkins method has been proposed by using Legendre polynomial as basis functions to solve Cauchy singular integral equation of index zero. It converts the Cauchy singular integral equation into system of equations which can be easily solved. The test examples are given for illustration of proposed numerical method. Error bounds are derived as well as implemented in all the test examples.
ERIC Educational Resources Information Center
Alejandra, Almirón; Fernando, Bifano; Leonardo, Lupinacci
2015-01-01
Solving systems of equations at school, at least in Argentina, is usually a task that students are given as a series of techniques that "allow" them to find a solution. How to overcome educational obstacles that are generated from a fragmented approach of knowledge? What can DGS do, in particular the CAS environment? What epistemic and…
Integration of the Equations of Classical Electrode-Effect Theory with Aerosols
NASA Astrophysics Data System (ADS)
Kalinin, A. V.; Leont'ev, N. V.; Terent'ev, A. M.; Umnikov, E. D.
2016-04-01
This paper is devoted to an analytical study of the one-dimensional stationary system of equations for modeling of the electrode effect in the Earth's atmospheric layer with aerosols. New integrals of the system are derived. Using these integrals, the expressions for solutions of the system and estimates of the electrode layer's thickness as a function of the aerosol concentration are obtained for numerical parameters close to real.
ERIC Educational Resources Information Center
Marsh, Herbert W.; Muthen, Bengt; Asparouhov, Tihomir; Ludtke, Oliver; Robitzsch, Alexander; Morin, Alexandre J. S.; Trautwein, Ulrich
2009-01-01
This study is a methodological-substantive synergy, demonstrating the power and flexibility of exploratory structural equation modeling (ESEM) methods that integrate confirmatory and exploratory factor analyses (CFA and EFA), as applied to substantively important questions based on multidimentional students' evaluations of university teaching…
Chen, Ke
1996-12-31
We study various preconditioning techniques for the iterative solution of boundary integral equations, and aim to provide a theory for a class of sparse preconditioners. Two related ideas are explored here: singularity separation and inverse approximation. Our preliminary conclusion is that singularity separation based preconditioners perform better than approximate inverse based while it is desirable to have both features.
Integrated optics technology study
NASA Technical Reports Server (NTRS)
Chen, B.
1982-01-01
The materials and processes available for the fabrication of single mode integrated electrooptical components are described. Issues included in the study are: (1) host material and orientation, (2) waveguide formation, (3) optical loss mechanisms, (4) wavelength selection, (5) polarization effects and control, (6) laser to integrated optics coupling,(7) fiber optic waveguides to integrated optics coupling, (8) souces, (9) detectors. The best materials, technology and processes for fabrication of integrated optical components for communications and fiber gyro applications are recommended.
A bin integral method for solving the kinetic collection equation
NASA Astrophysics Data System (ADS)
Wang, Lian-Ping; Xue, Yan; Grabowski, Wojciech W.
2007-09-01
A new numerical method for solving the kinetic collection equation (KCE) is proposed, and its accuracy and convergence are investigated. The method, herein referred to as the bin integral method with Gauss quadrature (BIMGQ), makes use of two binwise moments, namely, the number and mass concentration in each bin. These two degrees of freedom define an extended linear representation of the number density distribution for each bin following Enukashvily (1980). Unlike previous moment-based methods in which the gain and loss integrals are evaluated for a target bin, the concept of source-bin pair interactions is used to transfer bin moments from source bins to target bins. Collection kernels are treated by bilinear interpolations. All binwise interaction integrals are then handled exactly by Gauss quadrature of various orders. In essence the method combines favorable features in previous spectral moment-based and bin-based pair-interaction (or flux) methods to greatly enhance the logic, consistency, and simplicity in the numerical method and its implementation. Quantitative measures are developed to rigorously examine the accuracy and convergence properties of BIMGQ for both the Golovin kernel and hydrodynamic kernels. It is shown that BIMGQ has a superior accuracy for the Golovin kernel and a monotonic convergence behavior for hydrodynamic kernels. Direct comparisons are also made with the method of Berry and Reinhardt (1974), the linear flux method of Bott (1998), and the linear discrete method of Simmel et al. (2002).
The reduced basis method for the electric field integral equation
Fares, M.; Hesthaven, J.S.; Maday, Y.; Stamm, B.
2011-06-20
We introduce the reduced basis method (RBM) as an efficient tool for parametrized scattering problems in computational electromagnetics for problems where field solutions are computed using a standard Boundary Element Method (BEM) for the parametrized electric field integral equation (EFIE). This combination enables an algorithmic cooperation which results in a two step procedure. The first step consists of a computationally intense assembling of the reduced basis, that needs to be effected only once. In the second step, we compute output functionals of the solution, such as the Radar Cross Section (RCS), independently of the dimension of the discretization space, for many different parameter values in a many-query context at very little cost. Parameters include the wavenumber, the angle of the incident plane wave and its polarization.
Yan, Zai You; Hung, Kin Chew; Zheng, Hui
2003-05-01
Regularization of the hypersingular integral in the normal derivative of the conventional Helmholtz integral equation through a double surface integral method or regularization relationship has been studied. By introducing the new concept of discretized operator matrix, evaluation of the double surface integrals is reduced to calculate the product of two discretized operator matrices. Such a treatment greatly improves the computational efficiency. As the number of frequencies to be computed increases, the computational cost of solving the composite Helmholtz integral equation is comparable to that of solving the conventional Helmholtz integral equation. In this paper, the detailed formulation of the proposed regularization method is presented. The computational efficiency and accuracy of the regularization method are demonstrated for a general class of acoustic radiation and scattering problems. The radiation of a pulsating sphere, an oscillating sphere, and a rigid sphere insonified by a plane acoustic wave are solved using the new method with curvilinear quadrilateral isoparametric elements. It is found that the numerical results rapidly converge to the corresponding analytical solutions as finer meshes are applied.
NASA Astrophysics Data System (ADS)
Keller, I. E.
2013-08-01
The equilibrium and compatibility equations for viscoplastic medium with an arbitrary material function relating the stress intensity to the strain rate intensity is considered. A general form of the function ensuring complete integrability of two-dimensional equations has been found. The obtained function has an N-shaped (spinodal) graph and in particular cases corresponds to a linearly viscous liquid and perfectly plastic solid. A change of the strain rate sensitivity sign corresponds to a change in the type of the system and passing over the discontinuity line in a solid. The obtained function provides decoupling of the operator in a pair of two-dimensional subspaces where the equations are exactly linearized. The results of this study allows us to extend the class of integrable problems to so-called "active materials" (or "materials with internal dynamics"), which have aroused considerable interest.
Liquid-vapor interfaces in XY -spin fluids: an inhomogeneous anisotropic integral-equation approach.
Omelyan, I P; Folk, R; Kovalenko, A; Fenz, W; Mryglod, I M
2009-01-01
An integral-equation approach is developed to study interfacial properties of anisotropic fluids with planar spins in the presence of an external magnetic field. The approach is based on the coupled set of the Lovett-Mou-Buff-Wertheim integro-differential equation for the inhomogeneous anisotropic one-particle density and the Ornstein-Zernike equation for the orientationally dependent two-particle correlation functions. Using the proposed inhomogeneous angle-harmonics expansion formalism we show that these integral equations can be reduced to a much simpler form similar to that inherent for a system of isotropic fluids. The interfacial orientationally dependent direct correlation function can be consistently constructed by means of a nonlinear interpolation via its values obtained in the coexisting anisotropic bulk phases. A soft mean spherical approximation is employed for the closure relation. This has allowed us to solve the complicated integral equations in the situation when both spatial inhomogeneity and orientational anisotropy are present simultaneously. The approach introduced is applied to an XY fluid model with ferromagnetic spin interactions. As a result, the density-orientation and magnetization profiles at the liquid-vapor interfaces are calculated in a wide range of temperatures up to subcritical regions. The influence of the external field on the microscopic structure of the interfaces and the surface tension is also analyzed in detail.
Boundary integral equation method for electromagnetic and elastic waves
NASA Astrophysics Data System (ADS)
Chen, Kun
In this thesis, the boundary integral equation method (BIEM) is studied and applied to electromagnetic and elastic wave problems. First of all, a spectral domain BIEM called the spectral domain approach is employed for full wave analysis of metal strip grating on grounded dielectric slab (MSG-GDS) and microstrips shielded with either perfect electric conductor (PEC) or perfect magnetic conductor (PMC) walls. The modal relations between these structures are revealed by exploring their symmetries. It is derived analytically and validated numerically that all the even and odd modes of the latter two (when they are mirror symmetric) find their correspondence in the modes of metal strip grating on grounded dielectric slab when the phase shift between adjacent two unit cells is 0 or pi. Extension to non-symmetric case is also made. Several factors, including frequency, grating period, slab thickness and strip width, are further investigated for their impacts on the effective permittivity of the dominant mode of PEC/PMC shielded microstrips. It is found that the PMC shielded microstrip generally has a larger wave number than the PEC shielded microstrip. Secondly, computational aspects of the layered medim doubly periodic Green's function (LMDPGF) in matrix-friendly formulation (MFF) are investigated. The MFF for doubly periodic structures in layered medium is derived, and the singularity of the periodic Green's function when the transverse wave number equals zero in this formulation is analytically extracted. A novel approach is proposed to calculate the LMDPGF, which makes delicate use of several techniques including factorization of the Green's function, generalized pencil of function (GPOF) method and high order Taylor expansion to derive the high order asymptotic expressions, which are then evaluated by newly derived fast convergent series. This approach exhibits robustness, high accuracy and fast and high order convergence; it also allows fast frequency sweep for
Inversion of airborne tensor VLF data using integral equations
NASA Astrophysics Data System (ADS)
Kamm, Jochen; Pedersen, Laust B.
2014-08-01
The Geological Survey of Sweden has been collecting airborne tensor very low frequency data (VLF) over several decades, covering large parts of the country. The data has been an invaluable source of information for identifying conductive structures that can among other things be related to water-filled fault zones, wet sediments that fill valleys or ore mineralizations. Because the method only uses two differently polarized plane waves of very similar frequency, vertical resolution is low and interpretation is in most cases limited to maps that are directly derived from the data. Occasionally, 2-D inversion is carried out along selected profiles. In this paper, we present for the first time a 3-D inversion for tensor VLF data in order to further increase the usefulness of the data set. The inversion is performed using a non-linear conjugate gradient scheme (Polak-Ribière) with an inexact line-search. The gradient is obtained by an algebraic adjoint method that requires one additional forward calculation involving the adjoint system matrix. The forward modelling is based on integral equations with an analytic formulation of the half-space Green's tensor. It avoids typically required Hankel transforms and is particularly amenable to singularity removal prior to the numerical integration over the volume elements. The system is solved iteratively, thus avoiding construction and storage of the dense system matrix. By using fast 3-D Fourier transforms on nested grids, subsequently farther away interactions are represented with less detail and therefore with less computational effort, enabling us to bridge the gap between the relatively short wavelengths of the fields (tens of metres) and the large model dimensions (several square kilometres). We find that the approximation of the fields can be off by several per cent, yet the transfer functions in the air are practically unaffected. We verify our code using synthetic calculations from well-established 2-D methods, and
Kleinert, H; Zatloukal, V
2013-11-01
The statistics of rare events, the so-called black-swan events, is governed by non-Gaussian distributions with heavy power-like tails. We calculate the Green functions of the associated Fokker-Planck equations and solve the related stochastic differential equations. We also discuss the subject in the framework of path integration.
Properties of Linear Integral Equations Related to the Six-Vertex Model with Disorder Parameter
NASA Astrophysics Data System (ADS)
Boos, Hermann; Göhmann, Frank
2011-10-01
One of the key steps in recent work on the correlation functions of the XXZ chain was to regularize the underlying six-vertex model by a disorder parameter α. For the regularized model it was shown that all static correlation functions are polynomials in only two functions. It was further shown that these two functions can be written as contour integrals involving the solutions of a certain type of linear and non-linear integral equations. The linear integral equations depend parametrically on α and generalize linear integral equations known from the study of the bulk thermodynamic properties of the model. In this note we consider the generalized dressed charge and a generalized magnetization density. We express the generalized dressed charge as a linear combination of two quotients of Q-functions, the solutions of Baxter's t-Q-equation. With this result we give a new proof of a lemma on the asymptotics of the generalized magnetization density as a function of the spectral parameter.
Modern integral equation techniques for quantum reactive scattering theory
Auerbach, Scott Michael
1993-11-01
Rigorous calculations of cross sections and rate constants for elementary gas phase chemical reactions are performed for comparison with experiment, to ensure that our picture of the chemical reaction is complete. We focus on the H/D+H_{2} → H_{2}/DH + H reaction, and use the time independent integral equation technique in quantum reactive scattering theory. We examine the sensitivity of H+H_{2} state resolved integral cross sections σ{sub v'j',vj}(E) for the transitions (v = 0,j = 0) to (v'} = 1,j' = 1,3), to the difference between the Liu-Siegbahn-Truhlar-Horowitz (LSTH) and double many body expansion (DMBE) ab initio potential energy surfaces (PES). This sensitivity analysis is performed to determine the origin of a large discrepancy between experimental cross sections with sharply peaked energy dependence and theoretical ones with smooth energy dependence. We find that the LSTH and DMBE PESs give virtually identical cross sections, which lends credence to the theoretical energy dependence.
Rapid Solution of Integral Equations of Scattering Theory in Two Dimensions.
1985-11-01
0184 1. Introduction One of standard approaches to numerical treatment of boundary value problems for elliptic partial differential equations (PDEs...Smirnov, E. B. Gliner, Differential Equations of Mathematical Physics, North-Holland, Amsterdam, 1964. [16] V. Rokhlin, Solution of Acoustic Scattering... Differential Equations , Computers and Mathematics with Applications, 11,No 7/8 (1985). [18] , Rapid Solution of Integral Equations of Classical Potential
NASA Technical Reports Server (NTRS)
Lakin, W. D.
1981-01-01
The use of integrating matrices in solving differential equations associated with rotating beam configurations is examined. In vibration problems, by expressing the equations of motion of the beam in matrix notation, utilizing the integrating matrix as an operator, and applying the boundary conditions, the spatial dependence is removed from the governing partial differential equations and the resulting ordinary differential equations can be cast into standard eigenvalue form. Integrating matrices are derived based on two dimensional rectangular grids with arbitrary grid spacings allowed in one direction. The derivation of higher dimensional integrating matrices is the initial step in the generalization of the integrating matrix methodology to vibration and stability problems involving plates and shells.
The Application of a Boundary Integral Equation Method to the Prediction of Ducted Fan Engine Noise
NASA Technical Reports Server (NTRS)
Dunn, M. H.; Tweed, J.; Farassat, F.
1999-01-01
The prediction of ducted fan engine noise using a boundary integral equation method (BIEM) is considered. Governing equations for the BIEM are based on linearized acoustics and describe the scattering of incident sound by a thin, finite-length cylindrical duct in the presence of a uniform axial inflow. A classical boundary value problem (BVP) is derived that includes an axisymmetric, locally reacting liner on the duct interior. Using potential theory, the BVP is recast as a system of hypersingular boundary integral equations with subsidiary conditions. We describe the integral equation derivation and solution procedure in detail. The development of the computationally efficient ducted fan noise prediction program TBIEM3D, which implements the BIEM, and its utility in conducting parametric noise reduction studies are discussed. Unlike prediction methods based on spinning mode eigenfunction expansions, the BIEM does not require the decomposition of the interior acoustic field into its radial and axial components which, for the liner case, avoids the solution of a difficult complex eigenvalue problem. Numerical spectral studies are presented to illustrate the nexus between the eigenfunction expansion representation and BIEM results. We demonstrate BIEM liner capability by examining radiation patterns for several cases of practical interest.
On integration of the first order differential equations in a finite terms
NASA Astrophysics Data System (ADS)
Malykh, M. D.
2017-01-01
There are several approaches to the description of the concept called briefly as integration of the first order differential equations in a finite terms or symbolical integration. In the report three of them are considered: 1.) finding of a rational integral (Beaune or Poincaré problem), 2.) integration by quadratures and 3.) integration when the general solution of given differential equation is an algebraical function of a constant (Painlevé problem). Their realizations in Sage are presented.
NASA Astrophysics Data System (ADS)
Miller, C. T.; Kees, C. E.
2002-12-01
Time integration methods that adapt in both the order of approximation and time step have been shown to provide efficient solutions for Richards' equation. In this work, we extend the same method of lines approach to solve a set of two-phase flow formulations and address some mass conservation issues from the previous work. We analyze these formulations and the nonlinear systems that result from applying the integration methods, placing particular emphasis on their index, range of applicability, and mass conservation characteristics. We conduct numerical experiments to study the behavior of the numerical models for three test problems. We demonstrate that higher order integration in time is more efficient than standard low-order methods for a variety of practical grids and integration tolerances, that the adaptive scheme successfully varies the step size in response to changing conditions, and that mass balance can be maintained efficiently using variable-order integration and an appropriately chosen numerical model formulation.
Hierarchies of nonlinear integrable equations and their symmetries in 2 + 1 dimensions
NASA Astrophysics Data System (ADS)
Cheng, Yi
1990-11-01
For a given nonlinear integrable equation in 2 + 1 dimensions, an approach is described to construct the hierarchies of equations and relevant Lie algebraic properties. The commutability and noncommutability of equations of the flow, their symmetries and mastersymmetries are then derived as direct results of these algebraic properties. The details for the modified Kadomtsev-Petviashvilli equation are shown as an example and the main results for the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Katera-Sawada equation are given.
ERIC Educational Resources Information Center
Collins, Myrtle T.; Greenberg, Marvin
1974-01-01
Article described a plan to develop integrated study through music activities. Students learned to become more independent learners while concentrating on more complex and creative activities. (Author/RK)
Geometric and Integral Equation Methods for Scattering in Layered Media
NASA Astrophysics Data System (ADS)
Wiskin, James Walter
This dissertation is an extension of the Stenger -Johnson-Borup sinc and Fast Fourier Transform (FFT) based integral equation imaging algorithms to the case of a layered ambient medium. This scenario has medical, geophysical and nondestructive testing applications. It is also a first step in the direction of incorporating a geometric point of view in forward and inverse scattering. The construction of layered Green's functions and concomitant inverse scattering algorithms for inhomogeneities residing within a layered medium whose layers are known a priori is carried out. Computer simulations and numerical experiments investigate the ill -posedness of inverse scattering in this context. Both 2 and 3D ambient media are considered and the relationship to the distorted wave Born approximation are discussed. Noise contamination and attenuation in both the layered background medium and the inhomogeneity are included for realism. Global minimization techniques based on homotopy are introduced and generalized. Concepts from Cartan/Kahler differential geometry play a natural role in understanding homotopy methods of global minimization. These minimization methods have application to biomolecular modelling as well as scattering. Exterior Differential Forms provide a natural vehicle for extending results determined here to include shear effects in fully elastic media. It is also shown that the methods developed here can be extended to ambient media with different types of known structure.
NASA Astrophysics Data System (ADS)
Aspon, Siti Zulaiha; Murid, Ali Hassan Mohamed; Rahmat, Hamisan
2014-07-01
This research is about computing the Green's functions on unbounded doubly connected regions by using the method of boundary integral equation. The method depends on solving an exterior Dirichlet problem. The Dirichlet problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the generalized Neumann kernel. The method for solving this integral equation is by using the Nyström method with trapezoidal rule to discretize it to a linear system. The linear system is then solved by the Gaussian elimination method. Mathematica plots of Green's functions for several test regions are also presented.
A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
The efficiency of several algorithms used for numerical integration of stiff ordinary differential equations was compared. The methods examined included two general purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes were applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code available for the integration of combustion kinetic rate equations. It is shown that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient then evaluating the temperature by integrating its time-derivative.
Third-order integrable difference equations generated by a pair of second-order equations
NASA Astrophysics Data System (ADS)
Matsukidaira, Junta; Takahashi, Daisuke
2006-02-01
We show that the third-order difference equations proposed by Hirota, Kimura and Yahagi are generated by a pair of second-order difference equations. In some cases, the pair of the second-order equations are equivalent to the Quispel-Robert-Thomson (QRT) system, but in the other cases, they are irrelevant to the QRT system. We also discuss an ultradiscretization of the equations.
Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.
Fokas, A S
2006-05-19
The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.
The Reduction of Ducted Fan Engine Noise Via a Boundary Integral Equation Method
NASA Technical Reports Server (NTRS)
Tweed, John
2000-01-01
Engineering studies for reducing ducted fan engine noise were conducted using the noise prediction code TBIEM3D. To conduct parametric noise reduction calculations, it was necessary to advance certain theoretical and computational aspects of the boundary integral equation method (BIEM) described in and implemented in TBIEM3D. Also, enhancements and upgrades to TBIEM3D were made for facilitating the code's use in this research and by the aeroacoustics engineering community.
Stable and fast semi-implicit integration of the stochastic Landau-Lifshitz equation.
Mentink, J H; Tretyakov, M V; Fasolino, A; Katsnelson, M I; Rasing, Th
2010-05-05
We propose new semi-implicit numerical methods for the integration of the stochastic Landau-Lifshitz equation with built-in angular momentum conservation. The performance of the proposed integrators is tested on the 1D Heisenberg chain. For this system, our schemes show better stability properties and allow us to use considerably larger time steps than standard explicit methods. At the same time, these semi-implicit schemes are also of comparable accuracy to and computationally much cheaper than the standard midpoint implicit method. The results are of key importance for atomistic spin dynamics simulations and the study of spin dynamics beyond the macro spin approximation.
Nonlinear Smoothing and the EM Algorithm for Positive Integral Equations of the First Kind
Eggermont, P. P. B.
1999-01-15
We study a modification of the EMS algorithm in which each step of the EMS algorithm is preceded by a nonlinear smoothing step of the form Nf-exp(S*log f) , where S is the smoothing operator of the EMS algorithm. In the context of positive integral equations (a la positron emission tomography) the resulting algorithm is related to a convex minimization problem which always admits a unique smooth solution, in contrast to the unmodified maximum likelihood setup. The new algorithm has slightly stronger monotonicity properties than the original EM algorithm. This suggests that the modified EMS algorithm is actually an EM algorithm for the modified problem. The existence of a smooth solution to the modified maximum likelihood problem and the monotonicity together imply the strong convergence of the new algorithm. We also present some simulation results for the integral equation of stereology, which suggests that the new algorithm behaves roughly like the EMS algorithm.
ERIC Educational Resources Information Center
Huang, Heng-Tsung Danny
2010-01-01
Thus far, few research studies have examined the practice of integrated speaking test tasks in the field of second/foreign language oral assessment. This dissertation utilized structural equation modeling (SEM) and qualitative techniques to explore the relationships among topical knowledge, anxiety, and integrated speaking test performance and to…
(2+1)-dimensional non-isospectral multi-component AKNS equations and its integrable couplings
Sun Yepeng
2010-03-08
(2+1)-dimensional non-isospectral multi-component AKNS equations are derived from an arbitrary order matrix spectral problem. As a reduction, (2+1)-dimensional non-isospectral multi-component Schroedinger equations are obtained. Moreover, new (2+1)-dimensional non-isospectral integrable couplings of the resulting AKNS equations are constructed by enlarging the associated matrix spectral problem.
Integrability Test and Travelling-Wave Solutions of Higher-Order Shallow- Water Type Equations
NASA Astrophysics Data System (ADS)
Maldonado, Mercedes; Molinero, María Celeste; Pickering, Andrew; Prada, Julia
2010-04-01
We apply the Weiss-Tabor-Carnevale (WTC) Painlevé test to members of a sequence of higher-order shallow-water type equations. We obtain the result that the equations considered are non-integrable, although compatibility conditions at real resonances are satisfied. We also construct travelling-wave solutions for these and related equations.
Soliton dynamics to the multi-component complex coupled integrable dispersionless equation
NASA Astrophysics Data System (ADS)
Xu, Zong-Wei; Yu, Guo-Fu; Zhu, Zuo-Nong
2016-11-01
The generalized coupled integrable dispersionless (CID) equation describes the current-fed string in a certain external magnetic field. In this paper, we propose a multi-component complex CID equation. The integrability of the multi-component complex equation is confirmed by constructing Lax pairs. One-soliton and two-soliton solutions are investigated to exhibit rich evolution properties. Especially, similar as the multi-component short pulse equation and the first negative AKNS equation, periodic interaction, parallel solitons, elastic and inelastic interaction, energy re-distribution happen between two solitons. Multi-soliton solutions are given in terms of Pfaffian expression by virtue of Hirota's bilinear method.
Lax Pairs and Integrability Conditions of Higher-Order Nonlinear Schrödinger Equations
NASA Astrophysics Data System (ADS)
Asad-uz-zaman, M.; Chachou Samet, H.; Khawaja, U. Al
2016-08-01
We derive the Lax pairs and integrability conditions of the nonlinear Schrödinger equation with higher-order terms, complex potentials, and time-dependent coefficients. Cubic and quintic nonlinearities together with derivative terms are considered. The Lax pairs and integrability conditions for some of the well-known nonlinear Schrödinger equations, including a new equation which was not considered previously in the literature, are then derived as special cases. We show most clearly with a similarity transformation that the higher-order terms restrict the integrability to linear potential in contrast with quadratic potential for the standard nonlinear Schrödinger equation.
Laplace transform approach for solving integral equations using computer algebra system
NASA Astrophysics Data System (ADS)
Paneva-Konovska, Jordanka; Nikolova, Yanka
2016-12-01
The Laplace transform method, along with Computer Algebra Systems (CAS) "Maple" v. 13, are extremely successfully applied for solving a class of integral equations with an arbitrary order, including fractional order integral equations. The combining of both powerful approaches allows students more quickly, enjoyable and thoroughly to master the material.
NASA Technical Reports Server (NTRS)
Walker, K. P.; Freed, A. D.
1991-01-01
New methods for integrating systems of stiff, nonlinear, first order, ordinary differential equations are developed by casting the differential equations into integral form. Nonlinear recursive relations are obtained that allow the solution to a system of equations at time t plus delta t to be obtained in terms of the solution at time t in explicit and implicit forms. Examples of accuracy obtained with the new technique are given by considering systems of nonlinear, first order equations which arise in the study of unified models of viscoplastic behaviors, the spread of the AIDS virus, and predator-prey populations. In general, the new implicit algorithm is unconditionally stable, and has a Jacobian of smaller dimension than that which is acquired by current implicit methods, such as the Euler backward difference algorithm; yet, it gives superior accuracy. The asymptotic explicit and implicit algorithms are suitable for solutions that are of the growing and decaying exponential kinds, respectively, whilst the implicit Euler-Maclaurin algorithm is superior when the solution oscillates, i.e., when there are regions in which both growing and decaying exponential solutions exist.
A path-integral Langevin equation treatment of low-temperature doped helium clusters
NASA Astrophysics Data System (ADS)
Ing, Christopher; Hinsen, Konrad; Yang, Jing; Zeng, Toby; Li, Hui; Roy, Pierre-Nicholas
2012-06-01
We present an implementation of path integral molecular dynamics for sampling low temperature properties of doped helium clusters using Langevin dynamics. The robustness of the path integral Langevin equation and white-noise Langevin equation [M. Ceriotti, M. Parrinello, T. E. Markland, and D. E. Manolopoulos, J. Chem. Phys. 133, 124104 (2010)], 10.1063/1.3489925 sampling methods are considered for those weakly bound systems with comparison to path integral Monte Carlo (PIMC) in terms of efficiency and accuracy. Using these techniques, convergence studies are performed to confirm the systematic error reduction introduced by increasing the number of discretization steps of the path integral. We comment on the structural and energetic evolution of HeN-CO2 clusters from N = 1 to 20. To quantify the importance of both rotations and exchange in our simulations, we present a chemical potential and calculated band origin shifts as a function of cluster size utilizing PIMC sampling that includes these effects. This work also serves to showcase the implementation of path integral simulation techniques within the molecular modelling toolkit [K. Hinsen, J. Comp. Chem. 21, 79 (2000)], 10.1002/(SICI)1096-987X(20000130)21:2<79::AID-JCC1>3.0.CO;2-B, an open-source molecular simulation package.
Thermodynamic consistency and integral equations for the liquid structure
NASA Astrophysics Data System (ADS)
Leys, F. E.; March, N. H.; Lamoen, D.
2002-12-01
Within an assumed pair potential framework, it has been generally accepted for a long time that far from the critical point the asymptotic form of the direct correlation function c(r) at large r is given by [-φ(r)/kBT]. Here φ(r) is the pair potential and kBT the thermal energy. Subsequently, Kumar, March, and Wasserman [Phys. Chem. Liquids 11, 271 (1982)] examined the condition for thermodynamic consistency between virial and compressibility equations of state. Their study, together with later work by Senatore, Rashid, and March [Phys. Chem. Liquids 16, 1 (1986)], resulted in a decomposition of c(r) into a potential part cp(r) given by Kumar et al. for all r and involving the pair function g(r) and its density derivative, plus a "collective" part cc(r), which must obey a simple sum rule to satisfy thermodynamic consistency. The more recent study of B. C. Eu and K. Rah [J. Chem. Phys. 3, 3327 (1999)] prompts us to bring their results into direct contact with the study of Kumar et al. The work of Eu and Rah gives a prominent place to the Mayer function f(r)=e(-[φ(r)/kBT]-1 which tends to -[φ(r)/kBT] as r→∞ for potentials tending to zero at infinity.
Thermodynamic consistency and integral equations for the liquid structure
NASA Astrophysics Data System (ADS)
Leys, F. E.; March, N. H.; Lamoen, D.
Within an assumed pair potential framework, it has been generally accepted for a long time that far from the critical point the asymptotic form of the direct correlation function c(r) at large r is given by [- ϕ(r)/kBT]. Here ϕ(r) is the pair potential and kBT the thermal energy. Subsequently, Kumar, March, and Wasserman [Phys. Chem. Liquids 11, 271 (1982)] examined the condition for thermodynamic consistency between virial and compressibility equations of state. Their study, together with later work by Senatore, Rashid, and March [Phys. Chem. Liquids 16, 1 (1986)], resulted in a decomposition of c(r) into a potential part cp(r) given by Kumar et al. for all r and involving the pair function g(r) and its density derivative, plus a "collective" part cc(r), which must obey a simple Sum rule to satisfy thermodynamic consistency. The more recent study of B. C. Eu and K. Rah [J. Chem. Phys. 3, 3327 (1999)] prompts us to bring their results into direct contact with the study of Kumar et al. The work of Eu and Rah gives a prominent place to the Mayer function f(r) = e(-[ϕ(r) / kBT]-1 which tends to -[ϕ(r)/kBT] as r → ∞ for potentials tending to zero at infinity.
Sung, Bong June; Yethiraj, Arun
2005-08-15
The conformational properties and static structure of freely jointed hard-sphere chains in matrices composed of stationary hard spheres are studied using Monte Carlo simulations and integral equation theory. The simulations show that the chain size is a nonmonotonic function of the matrix density when the matrix spheres are the same size as the monomers. When the matrix spheres are of the order of the chain size the chain size decreases monotonically with increasing matrix volume fraction. The simulations are used to test the replica-symmetric polymer reference interaction site model (RSP) integral equation theory. When the simulation results for the intramolecular correlation functions are input into the theory, the agreement between theoretical predictions and simulation results for the pair-correlation functions is quantitative only at the highest fluid volume fractions and for small matrix sphere sizes. The RSP theory is also implemented in a self-consistent fashion, i.e., the intramolecular and intermolecular correlation functions are calculated self-consistently by combining a field theory with the integral equations. The theory captures qualitative trends observed in the simulations, such as the nonmonotonic dependence of the chain size on media fraction.
Third order difference equations with two rational integrals
NASA Astrophysics Data System (ADS)
Sahadevan, R.; Maheswari, C. Uma
2009-10-01
A systematic investigation to derive three-dimensional analogs of two-dimensional Quispel, Roberts and Thompson (QRT) mappings is presented. The question of integrability of the obtained three-dimensional mappings with two independent integrals is also analyzed. It is also shown that there exist three-dimensional QRT maps with three n-dependent integrals.
NASA Astrophysics Data System (ADS)
Urai, Janos L.; Kukla, Peter A.
2015-04-01
The growing importance of salt in the energy, subsurface storage, and chemical and food industries also increases the challenges with prediction of geometries, kinematics, stress and transport in salt. This requires an approach, which integrates a broader range of knowledge than is traditionally available in the different scientific and engineering disciplines. We aim to provide a starting point for a more integrated understanding of salt, by presenting an overview of the state of the art in a wide range of salt-related topics, from (i) the formation and metamorphism of evaporites, (ii) rheology and transport properties, (iii) salt tectonics and basin evolution, (iv) internal structure of evaporites, (v) fluid flow through salt, to (vi) salt engineering. With selected case studies we show how integration of these domains of knowledge can bring better predictions of (i) sediment architecture and reservoir distribution, (ii) internal structure of salt for optimized drilling and better cavern design, (iii) reliable long-term predictions of deformations and fluid flow in subsurface storage. A fully integrated workflow is based on geomechanical models, which include all laboratory and natural observations and links macro- and micro-scale studies. We present emerging concepts for (i) the initiation dynamics of halokinesis, (ii) the rheology and deformation of the evaporites by brittle and ductile processes, (iii) the coupling of processes in evaporites and the under- and overburden, and (iv) the impact of the layered evaporite rheology on the structural evolution.
Review of Integrated Noise Model (INM) Equations and Processes
NASA Technical Reports Server (NTRS)
Shepherd, Kevin P. (Technical Monitor); Forsyth, David W.; Gulding, John; DiPardo, Joseph
2003-01-01
The FAA's Integrated Noise Model (INM) relies on the methods of the SAE AIR-1845 'Procedure for the Calculation of Airplane Noise in the Vicinity of Airports' issued in 1986. Simplifying assumptions for aerodynamics and noise calculation were made in the SAE standard and the INM based on the limited computing power commonly available then. The key objectives of this study are 1) to test some of those assumptions against Boeing source data, and 2) to automate the manufacturer's methods of data development to enable the maintenance of a consistent INM database over time. These new automated tools were used to generate INM database submissions for six airplane types :737-700 (CFM56-7 24K), 767-400ER (CF6-80C2BF), 777-300 (Trent 892), 717-200 (BR7 15), 757-300 (RR535E4B), and the 737-800 (CFM56-7 26K).
Dissolution process analysis using model-free Noyes-Whitney integral equation.
Hattori, Yusuke; Haruna, Yoshimasa; Otsuka, Makoto
2013-02-01
Drug dissolution process of solid dosages is theoretically described by Noyes-Whitney-Nernst equation. However, the analysis of the process is demonstrated assuming some models. Normally, the model-dependent methods are idealized and require some limitations. In this study, Noyes-Whitney integral equation was proposed and applied to represent the drug dissolution profiles of a solid formulation via the non-linear least squares (NLLS) method. The integral equation is a model-free formula involving the dissolution rate constant as a parameter. In the present study, several solid formulations were prepared via changing the blending time of magnesium stearate (MgSt) with theophylline monohydrate, α-lactose monohydrate, and crystalline cellulose. The formula could excellently represent the dissolution profile, and thereby the rate constant and specific surface area could be obtained by NLLS method. Since the long time blending coated the particle surface with MgSt, it was found that the water permeation was disturbed by its layer dissociating into disintegrant particles. In the end, the solid formulations were not disintegrated; however, the specific surface area gradually increased during the process of dissolution. The X-ray CT observation supported this result and demonstrated that the rough surface was dominant as compared to dissolution, and thus, specific surface area of the solid formulation gradually increased.
Bifurcations of traveling wave solutions for an integrable equation
Li Jibin; Qiao Zhijun
2010-04-15
This paper deals with the following equation m{sub t}=(1/2)(1/m{sup k}){sub xxx}-(1/2)(1/m{sup k}){sub x}, which is proposed by Z. J. Qiao [J. Math. Phys. 48, 082701 (2007)] and Qiao and Liu [Chaos, Solitons Fractals 41, 587 (2009)]. By adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wave solutions of the equation, it is shown that for different k, the equation may have infinitely many solitary wave solutions, periodic wave solutions, kink/antikink wave solutions, cusped solitary wave solutions, and breaking loop solutions. We discuss in a detail the cases of k=-2,-(1/2),(1/2),2, and parametric representations of all possible bounded traveling wave solutions are given in the different (c,g)-parameter regions.
NASA Technical Reports Server (NTRS)
Park, K. C.; Belvin, W. Keith
1990-01-01
A general form for the first-order representation of the continuous second-order linear structural-dynamics equations is introduced to derive a corresponding form of first-order continuous Kalman filtering equations. Time integration of the resulting equations is carried out via a set of linear multistep integration formulas. It is shown that a judicious combined selection of computational paths and the undetermined matrices introduced in the general form of the first-order linear structural systems leads to a class of second-order discrete Kalman filtering equations involving only symmetric sparse N x N solution matrices.
NASA Astrophysics Data System (ADS)
Ji, Jia-Liang; Zhu, Zuo-Nong
2017-01-01
Very recently, Ablowitz and Musslimani introduced a new integrable nonlocal nonlinear Schrödinger equation. In this paper, we investigate an integrable nonlocal modified Korteweg-de Vries equation (mKdV) which can be derived from the well-known AKNS system. We construct the Darboux transformation for the nonlocal mKdV equation. Using the Darboux transformation, we obtain its different kinds of exact solutions including soliton, kink, antikink, complexiton, rogue-wave solution, and nonlocalized solution with singularities. It is shown that these solutions possess new properties which are different from the ones for mKdV equation.
NASA Astrophysics Data System (ADS)
Delogu, V. Antonuccio
1984-11-01
The equilibrium magnetofluidostatic (MFS) problem is formulated as an extremum problem for the energy integral. The resulting equations, obtained from general theorems of the calculus of variations, are transformed into integral equations, and these, together with conditions specifying the position of the feet of the field line, form a system of integral equations. Previously known configurations, like the pinch, are easily found with this method, which allows the analysis of two dimensional configurations of general aspect. The intrinsic limits of this method and the fundamental aspects of the MFS problem for coronal loops are discussed.
NASA Astrophysics Data System (ADS)
Belov, A. A.; Igumnov, L. A.; Litvinchuk, S. Yu.; Metrikin, V. S.
2016-11-01
Two approaches (classical and nonclassical) of the boundary integral equation method for solving three-dimensional dynamical boundary value problems of elasticity, viscoelasticity, and poroelasticity are considered. The boundary integral equation model is used for porous materials. The Kelvin-Voigt model and the weakly singular hereditary Abel kernel are used to describe the viscoelastic properties. Both approaches permit solving the dynamic problems exactly not only in the isotropic but also in the anisotropic case. The boundary integral equation solution scheme is constructed on the basis of the boundary element technique. The numerical results obtained by the classical and nonclassical approaches are compared.
Daeva, S.G.; Setukha, A.V.
2015-03-10
A numerical method for solving a problem of diffraction of acoustic waves by system of solid and thin objects based on the reduction the problem to a boundary integral equation in which the integral is understood in the sense of finite Hadamard value is proposed. To solve this equation we applied piecewise constant approximations and collocation methods numerical scheme. The difference between the constructed scheme and earlier known is in obtaining approximate analytical expressions to appearing system of linear equations coefficients by separating the main part of the kernel integral operator. The proposed numerical scheme is tested on the solution of the model problem of diffraction of an acoustic wave by inelastic sphere.
Integration of CAS in the Didactics of Differential Equations.
ERIC Educational Resources Information Center
Balderas Puga, Angel
In this paper are described some features of the intensive use of math software, primarily DERIVE, in the context of modeling in an introductory university course in differential equations. Different aspects are detailed: changes in the curriculum that included not only course contents, but also the sequence of introduction to various topics and…
NASA Astrophysics Data System (ADS)
Enukashvily, Isaac M.
1980-11-01
An extension of Bleck' method and of the method of moments is developed for the numerical integration of the kinetic equation of coalescence and breakup of cloud droplets. The number density function nk(x,t) in each separate cloud droplet packet between droplet mass grid points (xk,xk+1) is represented by an expansion in orthogonal polynomials with a given weighting function wk(x,k). The expansion coefficients describe the deviations of nk(x,t) from wk(x,k). In this way droplet number concentrations, liquid water contents and other moments in each droplet packet are conserved, and the problem of solving the kinetic equation is replaced by one of solving a set of coupled differential equations for the moments of the number density function nk(x,t). Equations for these moments in each droplet packet are derived. The method is tested against existing solutions of the coalescence equation. Numerical results are obtained when Bleck's uniform distribution hypothesis for nk(x,t) and Golovin's asymptotic solution of the coalescence equation is chosen for the, weighting function wk(x, k). A comparison between numerical results computed by Bleck's method and by the method of this study is made. It is shown that for the correct computation of the coalescence and breakup interactions between cloud droplet packets it is very important that the, approximation of the nk(x,t) between grid points (xk,xk+1) satisfies the conservation conditions for the number concentration, liquid water content and other moments of the cloud droplet packets. If these conservation conditions are provided, even the quasi-linear approximation of the nk(x,t) in comparison with Berry's six-point interpolation will give reasonable results which are very close to the existing analytic solutions.
The Prediction of Ducted Fan Engine Noise Via a Boundary Integral Equation Method
NASA Technical Reports Server (NTRS)
Farassat, F.; Dunn, M. H.; Tweed, J.
1996-01-01
A computationally efficient Boundary Integral Equation Method (BIEM) for the prediction of ducted fan engine noise is presented. The key features of the BIEM are its versatility and the ability to compute rapidly any portion of the sound field without the need to compute the entire field. Governing equations for the BIEM are based on the assumptions that all acoustic processes are linear, generate spinning modes, and occur in a uniform flow field. An exterior boundary value problem (BVP) is defined that describes the scattering of incident sound by an engine duct with arbitrary profile. Boundary conditions on the duct walls are derived that allow for passive noise control treatment. The BVP is recast as a system of hypersingular boundary integral equations for the unknown duct surface quantities. BIEM solution methodology is demonstrated for the scattering of incident sound by a thin cylindrical duct with hard walls. Numerical studies are conducted for various engine parameters and continuous portions of the total pressure field are computed. Radiation and duct propagation results obtained are in agreement with the classical results of spinning mode theory for infinite ducts.
Klinman, Judith P
2014-01-01
The final arbiter of enzyme mechanism is the ability to establish and test a kinetic mechanism. Isotope effects play a major role in expanding the scope and insight derived from the Michaelis-Menten equation. The integration of isotope effects into the formalism of the Michaelis-Menten equation began in the 1970s and has continued until the present. This review discusses a family of eukaryotic copper proteins, including dopamine β-monooxygenase, tyramine β-monooxygenase and peptidylglycine α-amidating enzyme, which are responsible for the synthesis of neuroactive compounds, norepinephrine, octopamine and C-terminally carboxamidated peptides, respectively. The review highlights the results of studies showing how combining kinetic isotope effects with initial rate parameters permits the evaluation of: (a) the order of substrate binding to multisubstrate enzymes; (b) the magnitude of individual rate constants in complex, multistep reactions; (c) the identification of chemical intermediates; and (d) the role of nonclassical (tunnelling) behaviour in C-H activation.
Integrable pair-transition-coupled nonlinear Schrödinger equations.
Ling, Liming; Zhao, Li-Chen
2015-08-01
We study integrable coupled nonlinear Schrödinger equations with pair particle transition between components. Based on exact solutions of the coupled model with attractive or repulsive interaction, we predict that some new dynamics of nonlinear excitations can exist, such as the striking transition dynamics of breathers, new excitation patterns for rogue waves, topological kink excitations, and other new stable excitation structures. In particular, we find that nonlinear wave solutions of this coupled system can be written as a linear superposition of solutions for the simplest scalar nonlinear Schrödinger equation. Possibilities to observe them are discussed in a cigar-shaped Bose-Einstein condensate with two hyperfine states. The results would enrich our knowledge on nonlinear excitations in many coupled nonlinear systems with transition coupling effects, such as multimode nonlinear fibers, coupled waveguides, and a multicomponent Bose-Einstein condensate system.
Evaluating four-loop conformal Feynman integrals by D-dimensional differential equations
NASA Astrophysics Data System (ADS)
Eden, Burkhard; Smirnov, Vladimir A.
2016-10-01
We evaluate a four-loop conformal integral, i.e. an integral over four four-dimensional coordinates, by turning to its dimensionally regularized version and applying differential equations for the set of the corresponding 213 master integrals. To solve these linear differential equations we follow the strategy suggested by Henn and switch to a uniformly transcendental basis of master integrals. We find a solution to these equations up to weight eight in terms of multiple polylogarithms. Further, we present an analytical result for the given four-loop conformal integral considered in four-dimensional space-time in terms of single-valued harmonic polylogarithms. As a by-product, we obtain analytical results for all the other 212 master integrals within dimensional regularization, i.e. considered in D dimensions.
Mickens, R.E.
1997-12-12
The major thrust of this proposal was to continue our investigations of so-called non-standard finite-difference schemes as formulated by other authors. These schemes do not follow the standard rules used to model continuous differential equations by discrete difference equations. The two major aspects of this procedure consist of generalizing the definition of the discrete derivative and using a nonlocal model (on the computational grid or lattice) for nonlinear terms that may occur in the differential equations. Our aim was to investigate the construction of nonstandard finite-difference schemes for several classes of ordinary and partial differential equations. These equations are simple enough to be tractable, yet, have enough complexity to be both mathematically and scientifically interesting. It should be noted that all of these equations differential equations model some physical phenomena under an appropriate set of experimental conditions. The major goal of the project was to better understand the process of constructing finite-difference models for differential equations. In particular, it demonstrates the value of using nonstandard finite-difference procedures. A secondary goal was to construct and study a variety of analytical techniques that can be used to investigate the mathematical properties of the obtained difference equations. These mathematical procedures are of interest in their own right and should be a valuable contribution to the mathematics research literature in difference equations. All of the results obtained from the research done under this project have been published in the relevant research/technical journals or submitted for publication. Our expectation is that these results will lead to improved finite difference schemes for the numerical integration of both ordinary and partial differential equations. Section G of the Appendix gives a concise summary of the major results obtained under funding by the grant.
Hybrid function method for solving Fredholm and Volterra integral equations of the second kind
NASA Astrophysics Data System (ADS)
Hsiao, Chun-Hui
2009-08-01
Numerical solutions of Fredholm and Volterra integral equations of the second kind via hybrid functions, are proposed in this paper. Based upon some useful properties of hybrid functions, integration of the cross product, a special product matrix and a related coefficient matrix with optimal order, are applied to solve these integral equations. The main characteristic of this technique is to convert an integral equation into an algebraic; hence, the solution procedures are either reduced or simplified accordingly. The advantages of hybrid functions are that the values of n and m are adjustable as well as being able to yield more accurate numerical solutions than the piecewise constant orthogonal function, for the solutions of integral equations. We propose that the available optimal values of n and m can minimize the relative errors of the numerical solutions. The high accuracy and the wide applicability of the hybrid function approach will be demonstrated with numerical examples. The hybrid function method is superior to other piecewise constant orthogonal functions [W.F. Blyth, R.L. May, P. Widyaningsih, Volterra integral equations solved in Fredholm form using Walsh functions, Anziam J. 45 (E) (2004) C269-C282; M.H. Reihani, Z. Abadi, Rationalized Haar functions method for solving Fredholm and Volterra integral equations, J. Comp. Appl. Math. 200 (2007) 12-20] for these problems.
Neglected transport equations: extended Rankine-Hugoniot conditions and J -integrals for fracture
NASA Astrophysics Data System (ADS)
Davey, K.; Darvizeh, R.
2016-09-01
Transport equations in integral form are well established for analysis in continuum fluid dynamics but less so for solid mechanics. Four classical continuum mechanics transport equations exist, which describe the transport of mass, momentum, energy and entropy and thus describe the behaviour of density, velocity, temperature and disorder, respectively. However, one transport equation absent from the list is particularly pertinent to solid mechanics and that is a transport equation for movement, from which displacement is described. This paper introduces the fifth transport equation along with a transport equation for mechanical energy and explores some of the corollaries resulting from the existence of these equations. The general applicability of transport equations to discontinuous physics is discussed with particular focus on fracture mechanics. It is well established that bulk properties can be determined from transport equations by application of a control volume methodology. A control volume can be selected to be moving, stationary, mass tracking, part of, or enclosing the whole system domain. The flexibility of transport equations arises from their ability to tolerate discontinuities. It is insightful thus to explore the benefits derived from the displacement and mechanical energy transport equations, which are shown to be beneficial for capturing the physics of fracture arising from a displacement discontinuity. Extended forms of the Rankine-Hugoniot conditions for fracture are established along with extended forms of J -integrals.
Volume Integral Equations Applied to Circular and Square Cylinders
1992-12-01
Derivation of Exact Formula 2.1.1 TM Case inc ikx LetE - ze (1) E - z E 5’•< (2) 1 lz E -E + > f>a (3) 2 scat inc From Maxwell’s equation V X E...k H (k y)J’(k •) 0 n n o in o n 1 (14) 14 2.1.2 TE Case inc ^ ikx Let H - z e (1) H - z H • a (2) 1 lz H mH + H >a(3) 2 scat inc From Maxwell’s ... equation V X H -iweE (4) 1 ýH E - z i (5) i ?H E9 - z we (6) Using boundary conditions for t and ta at - a tan tan H - H (7) lz 2z E - E (8) 10 21 15 and
Variational Integration for Ideal MHD with Built-in Advection Equations
Zhou, Yao; Qin, Hong; Burby, J. W.; Bhattacharjee, A.
2014-08-05
Newcomb's Lagrangian for ideal MHD in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.
Variational integration for ideal magnetohydrodynamics with built-in advection equations
Zhou, Yao; Burby, J. W.; Bhattacharjee, A.; Qin, Hong
2014-10-15
Newcomb's Lagrangian for ideal magnetohydrodynamics (MHD) in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum-preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.
Random search algorithm for solving the nonlinear Fredholm integral equations of the second kind.
Hong, Zhimin; Yan, Zaizai; Yan, Jiao
2014-01-01
In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.
A comparison of the efficiency of numerical methods for integrating chemical kinetic rate equations
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
A comparison of the efficiency of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations is presented. The methods examined include two general-purpose codes EPISODE and LSODE and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an iterative solution of the algebraic energy conservation equation to compute the temperature can be more efficient than evaluating the temperature by integrating its time-derivative.
On a new semi-discrete integrable combination of Burgers and Sharma-Tasso-Olver equation
NASA Astrophysics Data System (ADS)
Zhao, Hai-qiong
2017-02-01
In this paper, a new semi-discrete integrable combination of Burgers and Sharma-Tasso-Olver equation is investigated. The underlying integrable structures like the Lax pair, the infinite number of conservation laws, the Darboux-Bäcklund transformation, and the solutions are presented in the explicit form. The theory of the semi-discrete equation including integrable properties yields the corresponding theory of the continuous counterpart in the continuous limit. Finally, numerical experiments are provided to demonstrate the effectiveness of the developed integrable semi-discretization algorithms.
On the stability of numerical integration routines for ordinary differential equations.
NASA Technical Reports Server (NTRS)
Glover, K.; Willems, J. C.
1973-01-01
Numerical integration methods for the solution of initial value problems for ordinary vector differential equations may be modelled as discrete time feedback systems. The stability criteria discovered in modern control theory are applied to these systems and criteria involving the routine, the step size and the differential equation are derived. Linear multistep, Runge-Kutta, and predictor-corrector methods are all investigated.
New solutions for two integrable cases of a generalized fifth-order nonlinear equation
NASA Astrophysics Data System (ADS)
Wazwaz, Abdul-Majid
2015-05-01
Multiple-complexiton solutions for a new generalized fifth-order nonlinear integrable equation are constructed with the help of the Hirota's method and the simplified Hirota's method. By extending the real parameters into complex parameters, nonsingular complexiton solutions are obtained for two specific coefficients of the new generalized equation.
Solution of coupled integral equations for quantum scattering in the presence of complex potentials
Franz, Jan
2015-01-15
In this paper, we present a method to compute solutions of coupled integral equations for quantum scattering problems in the presence of a complex potential. We show how the elastic and absorption cross sections can be obtained from the numerical solution of these equations in the asymptotic region at large radial distances.
Study of nonlinear waves described by the cubic Schroedinger equation
Walstead, A.E.
1980-03-12
The cubic Schroedinger equation (CSE) is ubiquitous as a model equation for the long-time evolution of finite-amplitude near-monochromatic dispersive waves. It incorporates the effects of the radiation field pressure on the constitutive properties of the supporting medium in a self-consistent manner. The properties of the uniformly transiating periodic wave solutions of the one-dimensional CSE are studied here. These (so-called cnoidal) waves are characterized by the values of four parameters. Whitham's averaged variational principle is used to derive a system of quasilinear evolution equations (the modulational equations) for the values of these parameters when they are slowly varying in space and time. Explicit expressions for the characteristic velocities of the modulational equations are obtained for the full set of cnoidal waves. Riemann invariants are obtained for several limits for the stable case, and growth rates are obtained for several limits, including the solitary wave chain, for the unstable case. The results for several nontrivial limiting cases agree with those obtained by independent methods by others. The dynamics of the CSE generalized to two spatial dimensions are studied for the unstable case. A large class of similarity solutions with cylindrical symmetry are obtained systematically using infinitesimal transformation group techniques. The methods are adapted to obtain the symmetries of the action functional of the CSE and to deduce nine integral invariants. A numerical study of the self-similar solutions reveals that they are modulationally unstable and that singularities dominate the dynamics of the CSE in two dimensions. The CSE is derived using perturbation theory for a specific problem in plasma physics: the evolution of the envelope of a near-monochromatic electromagnetic wave in a cold magnetized plasma. 13 figures, 2 tables.
NASA Astrophysics Data System (ADS)
M. H. M., Moussa; Rehab, M. El-Shiekh
2011-04-01
Based on the closed connections among the homogeneous balance (HB) method and Clarkson—Kruskal (CK) method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meantime it is shown that this leads to a direct reduction in the form of ordinary differential equation under some integrability conditions between the variable coefficients. Two different cases have been discussed, the search for solutions of those ordinary differential equations yielded many exact travelling and solitonic wave solutions in the form of hyperbolic and trigonometric functions under some constraints between the variable coefficients.
Applying integrals of motion to the numerical solution of differential equations
NASA Technical Reports Server (NTRS)
Vezewski, D. J.
1980-01-01
A method is developed for using the integrals of systems of nonlinear, ordinary, differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scalar or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.
Applying integrals of motion to the numerical solution of differential equations
NASA Technical Reports Server (NTRS)
Jezewski, D. J.
1979-01-01
A method is developed for using the integrals of systems of nonlinear, ordinary differential equations in a numerical integration process to control the local errors in these integrals and reduce the global errors of the solution. The method is general and can be applied to either scaler or vector integrals. A number of example problems, with accompanying numerical results, are used to verify the analysis and support the conjecture of global error reduction.
Monte Carlo Method for Solving the Fredholm Integral Equations of the Second Kind
NASA Astrophysics Data System (ADS)
ZhiMin, Hong; ZaiZai, Yan; JianRui, Chen
2012-12-01
This article is concerned with a numerical algorithm for solving approximate solutions of Fredholm integral equations of the second kind with random sampling. We use Simpson's rule for solving integral equations, which yields a linear system. The Monte Carlo method, based on the simulation of a finite discrete Markov chain, is employed to solve this linear system. To show the efficiency of the method, we use numerical examples. Results obtained by the present method indicate that the method is an effective alternate method.
1993-11-04
6. AUTHOR(S) P.P. Schmidt Indrani Bhattacharya- Kodali and Gregory Voth 7. PERFORMING ORGANIZATION NAME(S) AND AODRESS(ES) 8. PERIORMING ORGANIZATION...13. ABSTRACT (Maimum 200 words) The extended reference interaction site method (RISM) integral equation theory is applied to calculate the solvent...Integral Equation Calculation of Solvent Activation Free Energies for Electron and Proton Transfer Reactions Indrani Bhattacharya- Kodali and Gregory A. Voth
The ATOMFT integrator - Using Taylor series to solve ordinary differential equations
NASA Technical Reports Server (NTRS)
Berryman, Kenneth W.; Stanford, Richard H.; Breckheimer, Peter J.
1988-01-01
This paper discusses the application of ATOMFT, an integration package based on Taylor series solution with a sophisticated user interface. ATOMFT has the capabilities to allow the implementation of user defined functions and the solution of stiff and algebraic equations. Detailed examples, including the solutions to several astrodynamics problems, are presented. Comparisons with its predecessor ATOMCC and other modern integrators indicate that ATOMFT is a fast, accurate, and easy method to use to solve many differential equation problems.
NASA Astrophysics Data System (ADS)
Bednarcyk, Brett A.; Aboudi, Jacob; Arnold, Steven M.
2008-02-01
The radial return method is a well-known algorithm for integrating the classical plasticity equations. Mendelson presented an alternative method for integrating these equations in terms of the so-called plastic strain—total strain plasticity relations. In the present communication, it is shown that, although the two methods appear to be unrelated, they are actually equivalent. A table is provided demonstrating the step by step correspondence of the radial return and Mendelson algorithms in the case of isotropic hardening.
NASA Astrophysics Data System (ADS)
Tancredi, Lorenzo
2015-12-01
Integration by parts identities (IBPs) can be used to express large numbers of apparently different d-dimensional Feynman Integrals in terms of a small subset of so-called master integrals (MIs). Using the IBPs one can moreover show that the MIs fulfil linear systems of coupled differential equations in the external invariants. With the increase in number of loops and external legs, one is left in general with an increasing number of MIs and consequently also with an increasing number of coupled differential equations, which can turn out to be very difficult to solve. In this paper we show how studying the IBPs in fixed integer numbers of dimension d = n with n ∈ N one can extract the information useful to determine a new basis of MIs, whose differential equations decouple as d → n and can therefore be more easily solved as Laurent expansion in (d - n).
A study of dynamic energy equations for Stirling cycle analysis
NASA Technical Reports Server (NTRS)
Larson, V. H.
1983-01-01
An analytical and computer study of the dynamic energy equations that describe the physical phenomena that occurs in a Stirling cycle engine. The basic problem is set up in terms of a set o hyperbolic partial differential equations. The characteristic lines are determined. The equations are then transformed to ordinary differential equations that are valid along characteristic lines. Computer programs to solve the differential equations and to plot pertinent factors are described.
General linear response formula for non integrable systems obeying the Vlasov equation
NASA Astrophysics Data System (ADS)
Patelli, Aurelio; Ruffo, Stefano
2014-11-01
Long-range interacting N-particle systems get trapped into long-living out-of-equilibrium stationary states called quasi-stationary states (QSS). We study here the response to a small external perturbation when such systems are settled into a QSS. In the N → ∞ limit the system is described by the Vlasov equation and QSS are mapped into stable stationary solutions of such equation. We consider this problem in the context of a model that has recently attracted considerable attention, the Hamiltonian mean field (HMF) model. For such a model, stationary inhomogeneous and homogeneous states determine an integrable dynamics in the mean-field effective potential and an action-angle transformation allows one to derive an exact linear response formula. However, such a result would be of limited interest if restricted to the integrable case. In this paper, we show how to derive a general linear response formula which does not use integrability as a requirement. The presence of conservation laws (mass, energy, momentum, etc.) and of further Casimir invariants can be imposed a posteriori. We perform an analysis of the infinite time asymptotics of the response formula for a specific observable, the magnetization in the HMF model, as a result of the application of an external magnetic field, for two stationary stable distributions: the Boltzmann-Gibbs equilibrium distribution and the Fermi-Dirac one. When compared with numerical simulations the predictions of the theory are very good away from the transition energy from inhomogeneous to homogeneous states. Contribution to the Topical Issue "Theory and Applications of the Vlasov Equation", edited by Francesco Pegoraro, Francesco Califano, Giovanni Manfredi and Philip J. Morrison.
Integral Equation Space-Energy Flux Synthesis for Spherical Systems.
1979-09-01
flux distribution eigenfunction An eigenvalue associated with in(r) n(E) neutron number density in units...technique for obtaining the spatial and energy neutron flux distributions in multiplying systems. In IES, the integral form of the neutron transport... FLUX SYNTHESIS FOR SPHERICAL SYSTEMS I. Introduction The calculations of neutron flux distributions and neutron growth rate (a) for multiplying
Integrated optics technology study
NASA Technical Reports Server (NTRS)
Chen, B.; Findakly, T.; Innarella, R.
1982-01-01
The status and near term potential of materials and processes available for the fabrication of single mode integrated electro-optical components are discussed. Issues discussed are host material and orientation, waveguide formation, optical loss mechanisms, wavelength selection, polarization effects and control, laser to integrated optics coupling fiber optic waveguides to integrated optics coupling, sources, and detectors. Recommendations of the best materials, technology, and processes for fabrication of integrated optical components for communications and fiber gyro applications are given.
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.
A description of phase equilibria in nonideal systems with the use of integral equation theory
NASA Astrophysics Data System (ADS)
D'Yakonov, S. G.; Klinov, A. V.; D'Yakonov, G. S.
2009-06-01
Integral equation theory for partial distribution functions was used to consider a method for calculations of vapor-liquid phase equilibrium conditions in binary Lennard-Jones systems with substantial deviations from the Berthelot-Lorentz mixing rules. Possible sources of errors in pressure and chemical potential values and methods for refining the results were analyzed. The required accuracy of calculations can be reached using two parameters only, one in the closure to the Ornstein-Zernike equation and the other in the equation for the chemical potential. These parameters are determined independently from two thermodynamic equations.
Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions.
Chowdury, A; Kedziora, D J; Ankiewicz, A; Akhmediev, N
2015-02-01
We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schrödinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions.
Koga, James
2004-10-01
Usually the motion of an electron under the influence of electromagnetic fields is influenced to a small extent by radiation damping. With the advent of high power high irradiance lasers it has become possible to generate focused laser irradiances where electrons interacting with the laser become highly relativistic over very short time and spatial scales. By focusing petawatt class lasers to very small spot sizes the amount of radiation emitted by electrons can become very large. Resultingly, the damping of the electron motion by the emission of this radiation can become large. In order to study this problem a code is written to solve a set of equations describing the evolution of a strong electromagnetic wave interacting with a single electron. Usually the equation of motion of an electron including radiation damping under the influence of electromagnetic fields is derived from the Lorentz-Dirac equation treating the damping as a perturbation. We use this equation to integrate forward in time and use the Lorentz-Dirac equation to integrate backward in time. We show that for very short wavelength electromagnetic radiation deep in the quantum regime at high irradiances differences between the perturbation equation and Lorentz-Dirac can be seen. However, for electron motion in the classical regime the differences are negligible. For electron motion in the classical regime the first order damping equation is found to be very adequate.
On the maximal cut of Feynman integrals and the solution of their differential equations
NASA Astrophysics Data System (ADS)
Primo, Amedeo; Tancredi, Lorenzo
2017-03-01
The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in ɛ = (4 - d) / 2, where d are the space-time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exists no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.
On the solution of integral equations with a generalized cauchy kernel
NASA Technical Reports Server (NTRS)
Kaya, A. C.; Erdogan, F.
1986-01-01
In this paper a certain class of singular integral equations that may arise from the mixed boundary value problems in nonhomogeneous materials is considered. The distinguishing feature of these equations is that in addition to the Cauchy singularity, the kernels contain terms that are singular only at the end points. In the form of the singular integral equations adopted, the density function is a potential or a displacement and consequently the kernel has strong singularities of the form (t-x) sup-2, x sup n-2 (t+x) sup n, (n or = 2, 0x,tb). The complex function theory is used to determine the fundamental function of the problem for the general case and a simple numerical technique is described to solve the integral equation. Two examples from the theory of elasticity are then considered to show the application of the technique.
On the solution of integral equations with a generalized cauchy kernal
NASA Technical Reports Server (NTRS)
Kaya, A. C.; Erdogan, F.
1986-01-01
A certain class of singular integral equations that may arise from the mixed boundary value problems in nonhonogeneous materials is considered. The distinguishing feature of these equations is that in addition to the Cauchy singularity, the kernels contain terms that are singular only at the end points. In the form of the singular integral equations adopted, the density function is a potential or a displacement and consequently the kernal has strong singularities of the form (t-x)(-2), x(n-2) (t+x)(n), (n is = or 2, 0 x, t b). The complex function theory is used to determine the fundamental function of the problem for the general case and a simple numerical technique is described to solve the integral equation. Two examples from the theory of elasticity are then considered to show the application of the technique.
Thermodynamics and structure of a two-dimensional electrolyte by integral equation theory
Aupic, Jana; Urbic, Tomaz
2014-05-14
Monte Carlo simulations and integral equation theory were used to predict the thermodynamics and structure of a two-dimensional Coulomb fluid. We checked the possibility that integral equations reproduce Kosterlitz-Thouless and vapor-liquid phase transitions of the electrolyte and critical points. Integral equation theory results were compared to Monte Carlo data and the correctness of selected closure relations was assessed. Among selected closures hypernetted-chain approximation results matched computer simulation data best, but these equations unfortunately break down at temperatures well above the Kosterlitz-Thouless transition. The Kovalenko-Hirata closure produces results even at very low temperatures and densities, but no sign of phase transition was detected.
Eddy Current Analysis of Thin Metal Container in Induction Heating by Line Integral Equations
NASA Astrophysics Data System (ADS)
Fujita, Hagino; Ishibashi, Kazuhisa
In recent years, induction-heating cookers have been disseminated explosively. It is wished to commercialize flexible and disposable food containers that are available for induction heating. In order to develop a good quality food container that is heated moderately, it is necessary to analyze accurately eddy currents induced in a thin metal plate. The integral equation method is widely used for solving induction-heating problems. If the plate thickness approaches zero, the surface integral equations on the upper and lower plate surfaces tend to become the same and the equations become ill conditioned. In this paper, firstly, we derive line integral equations from the boundary integral equations on the assumption that the electromagnetic fields in metal are attenuated rapidly compared with those along the metal surface. Next, so as to test validity of the line integral equations, we solve the eddy current induced in a thin metal container in induction heating and obtain power density given to the container and impedance characteristics of the heating coil. We compare computed results with those by FEM.
NASA Astrophysics Data System (ADS)
Schulze-Halberg, Axel
2016-06-01
We construct supersymmetric partners of a quantum system featuring a class of trigonometric potentials that emerge from the spheroidal equation. Examples of both standard and confluent supersymmetric transformations are presented. Furthermore, we use integral formulas arising from the confluent supersymmetric formalism to derive new representations for single and multiple integrals of spheroidal functions.
ERIC Educational Resources Information Center
Santos-George, Arlene A.
2012-01-01
This dissertation empirically tested Tinto's student integration theory through structural equation modeling using a national sample of 2,847 first-time entering community college students. Tinto theorized that the more academically and socially integrated a student is to the college environment, the more likely the student will persist through…
Noncommutative Integrability of the Klein-Gordon and Dirac Equations in (2+1)-Dimensional Spacetime
NASA Astrophysics Data System (ADS)
Breev, A. I.; Shapovalov, A. V.
2017-03-01
Noncommutative integration of the Klein-Gordon and Dirac relativistic wave equations in (2+1)-dimensional Minkowski space is considered. It is shown that for all non-Abelian subalgebras of the (2+1)-dimensional Poincaré algebra the condition of noncommutative integrability is satisfied.
A novel approach to solve nonlinear Fredholm integral equations of the second kind.
Li, Hu; Huang, Jin
2016-01-01
In this paper, we present a novel approach to solve nonlinear Fredholm integral equations of the second kind. This algorithm is constructed by the integral mean value theorem and Newton iteration. Convergence and error analysis of the numerical solutions are given. Moreover, Numerical examples show the algorithm is very effective and simple.
Integrating chemical kinetic rate equations by selective use of stiff and nonstiff methods
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1985-01-01
The effect of switching between nonstiff and stiff methods on the efficiency of algorithms for integrating chemical kinetic rate equations was examined. Different integration methods were tested by application of the packaged code LSODE to four practical combustion kinetics problems. The problems describe adiabatic, and homogeneous gas phase combustion reactions. It is shown that selective use of nonstiff and stiff methods in different regimes of a typical batch combustion problem is faster than the use of either method for the entire problem. The implications which result in the development of fast integration techniques for combustion kinetic rate equations are discussed.
Integrating chemical kinetic rate equations by selective use of stiff and nonstiff methods
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1985-01-01
The effect of switching between nonstiff and stiff methods on the efficiency of algorithms for integrating chemical kinetic rate equations is presented. Different integration methods are tested by application of the packaged code LSODE to four practical combustion kinetics problems. The problems describe adiabatic, homogeneous gas-phase combustion reactions. It is shown that selective use of nonstiff and stiff methods in different regimes of a typical batch combustion problem is faster than the use of either method for the entire problem. The implications of this result to the development of fast integration techniques for combustion kinetic rate equations are discussed.
Cong, Wenxiang; Shen, Haiou; Cong, Alexander X; Wang, Ge
2008-01-01
We present a generalized Delta-Eddington phase function to simplify the radiative transfer equation to integral equations with respect to both photon fluence rate and flux vector. The photon fluence rate and flux can be solved from the system of integral equations. By comparing to the Monte Carlo simulation results, the solutions of the system of integral equations accurately model the photon propagation in biological tissue over a wide range of optical parameters.
The Reduction of Ducted Fan Engine Noise Via A Boundary Integral Equation Method
NASA Technical Reports Server (NTRS)
Tweed, J.; Dunn, M.
1997-01-01
The development of a Boundary Integral Equation Method (BIEM) for the prediction of ducted fan engine noise is discussed. The method is motivated by the need for an efficient and versatile computational tool to assist in parametric noise reduction studies. In this research, the work in reference 1 was extended to include passive noise control treatment on the duct interior. The BEM considers the scattering of incident sound generated by spinning point thrust dipoles in a uniform flow field by a thin cylindrical duct. The acoustic field is written as a superposition of spinning modes. Modal coefficients of acoustic pressure are calculated term by term. The BEM theoretical framework is based on Helmholtz potential theory. A boundary value problem is converted to a boundary integral equation formulation with unknown single and double layer densities on the duct wall. After solving for the unknown densities, the acoustic field is easily calculated. The main feature of the BIEM is the ability to compute any portion of the sound field without the need to compute the entire field. Other noise prediction methods such as CFD and Finite Element methods lack this property. Additional BIEM attributes include versatility, ease of use, rapid noise predictions, coupling of propagation and radiation both forward and aft, implementable on midrange personal computers, and valid over a wide range of frequencies.
Iwaki, Takafumi; Shew, Chwen-Yang; Gumbs, Godfrey
2005-09-22
The structure of two-dimensional (2D) hard-sphere fluids on a cylindrical surface is investigated by means of the Ornstein-Zernike integral equation with the Percus-Yevick and the hypernetted-chain approximation. The 2D cylindrical coordinate breaks the spherical symmetry. Hence, the pair-correlation function is reformulated as a two-variable function to account for the packing along and around the cylinder. Detailed pair-correlation function calculations based on the two integral equation theories are compared with Monte Carlo simulations. In general, the Percus-Yevick theory is more accurate than the hypernetted-chain theory, but exceptions are observed for smaller cylinders. Moreover, analysis of the angular-dependent contact values shows that particles are preferentially packed anisotropically. The origin of such an anisotropic packing is driven by the entropic effect because the energy of all the possible system configurations of a dense hard-sphere fluid is the same. In addition, the anisotropic packing observed in our model studies serves as a basis for linking the close packing with the morphology of an ordered structure for particles adsorbed onto a cylindrical nanotube.
NASA Technical Reports Server (NTRS)
Sidi, A.; Israeli, M.
1986-01-01
High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.
A Tensor-Train accelerated solver for integral equations in complex geometries
NASA Astrophysics Data System (ADS)
Corona, Eduardo; Rahimian, Abtin; Zorin, Denis
2017-04-01
We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest O (log N) and once the inverse is computed, it can be applied in O (Nlog N) . We analyze the QTT ranks for hierarchically low rank matrices and discuss its relationship to commonly used hierarchical compression techniques such as FMM and HSS. We prove that the QTT ranks are bounded for translation-invariant systems and argue that this behavior extends to non-translation invariant volume and boundary integrals. For volume integrals, the QTT decomposition provides an efficient direct solver requiring significantly less memory compared to other fast direct solvers. We present results demonstrating the remarkable performance of the QTT-based solver when applied to both translation and non-translation invariant volume integrals in 3D. For boundary integral equations, we demonstrate that using a QTT decomposition to construct preconditioners for a Krylov subspace method leads to an efficient and robust solver with a small memory footprint. We test the QTT preconditioners in the iterative solution of an exterior elliptic boundary value problem (Laplace) formulated as a boundary integral equation in complex, multiply connected geometries.
NASA Astrophysics Data System (ADS)
Myrzakulov, R.; Mamyrbekova, G. K.; Nugmanova, G. N.; Yesmakhanova, K. R.; Lakshmanan, M.
2014-06-01
Motion of curves and surfaces in R3 lead to nonlinear evolution equations which are often integrable. They are also intimately connected to the dynamics of spin chains in the continuum limit and integrable soliton systems through geometric and gauge symmetric connections/equivalence. Here we point out the fact that a more general situation in which the curves evolve in the presence of additional self-consistent vector potentials can lead to interesting generalized spin systems with self-consistent potentials or soliton equations with self-consistent potentials. We obtain the general form of the evolution equations of underlying curves and report specific examples of generalized spin chains and soliton equations. These include principal chiral model and various Myrzakulov spin equations in (1+1) dimensions and their geometrically equivalent generalized nonlinear Schrödinger (NLS) family of equations, including Hirota-Maxwell-Bloch equations, all in the presence of self-consistent potential fields. The associated gauge equivalent Lax pairs are also presented to confirm their integrability.
Wiese, Kay Jörg
2016-04-01
We derive and study two different formalisms used for nonequilibrium processes: the coherent-state path integral, and an effective, coarse-grained stochastic equation of motion. We first study the coherent-state path integral and the corresponding field theory, using the annihilation process A+A→A as an example. The field theory contains counterintuitive quartic vertices. We show how they can be interpreted in terms of a first-passage problem. Reformulating the coherent-state path integral as a stochastic equation of motion, the noise generically becomes imaginary. This renders it not only difficult to interpret, but leads to convergence problems at finite times. We then show how alternatively an effective coarse-grained stochastic equation of motion with real noise can be constructed. The procedure is similar in spirit to the derivation of the mean-field approximation for the Ising model, and the ensuing construction of its effective field theory. We finally apply our findings to stochastic Manna sandpiles. We show that the coherent-state path integral is inappropriate, or at least inconvenient. As an alternative, we derive and solve its mean-field approximation, which we then use to construct a coarse-grained stochastic equation of motion with real noise.
Yang, Yunqing; Yan, Zhenya; Malomed, Boris A
2015-10-01
We analytically study rogue-wave (RW) solutions and rational solitons of an integrable fifth-order nonlinear Schrödinger (FONLS) equation with three free parameters. It includes, as particular cases, the usual NLS, Hirota, and Lakshmanan-Porsezian-Daniel equations. We present continuous-wave (CW) solutions and conditions for their modulation instability in the framework of this model. Applying the Darboux transformation to the CW input, novel first- and second-order RW solutions of the FONLS equation are analytically found. In particular, trajectories of motion of peaks and depressions of profiles of the first- and second-order RWs are produced by means of analytical and numerical methods. The solutions also include newly found rational and W-shaped one- and two-soliton modes. The results predict the corresponding dynamical phenomena in extended models of nonlinear fiber optics and other physically relevant integrable systems.
Subprograms for integrating the equations of motion of satellites. FORTRAN 4
NASA Technical Reports Server (NTRS)
Prokhorenko, V. I.
1980-01-01
The subprograms for the formation of the right members of the equations of motion of artificial Earth satellites (AES), integration of systems of differential equations by Adams' method, and the calculation of the values of various functions from the AES parameters of motion are described. These subprograms are written in the FORTRAN 4 language and constitute an essential part of the package of applied programs for the calculation of navigational parameters AES.
Computation of transonic vortex flows past delta wings Integral equation approach
NASA Technical Reports Server (NTRS)
Kandil, O. A.; Yates, E. C., Jr.
1985-01-01
The steady full-potential equation is written in the form of Poisson's equation, and the solution of the velocity field is expressed in terms of an integral equation. The solution consists of a surface integral of vorticity distribution on the wing and its free-vortex sheets and a volume integral of source distribution within a volume around the wing and its free-vortex sheets. The solution is obtained through successive iteration cycles. The source distribution is computed by using a mixed finite-difference scheme of the Murman-Cole type. The method is applied to delta wings. Numerical examples show that a conical shock is captured on the suction side of the wing. It is attached to the lower surface of the leading-edge vortex but does not necessarily reach to the wing surface.
Analyzing diffraction gratings by a boundary integral equation Neumann-to-Dirichlet map method.
Wu, Yumao; Lu, Ya Yan
2009-11-01
For analyzing diffraction gratings, a new method is developed based on dividing one period of the grating into homogeneous subdomains and computing the Neumann-to-Dirichlet (NtD) maps for these subdomains by boundary integral equations. For a subdomain, the NtD operator maps the normal derivative of the wave field to the wave field on its boundary. The integral operators used in this method are simple to approximate, since they involve only the standard Green's function of the Helmholtz equation in homogeneous media. The method retains the advantages of existing boundary integral equation methods for diffraction gratings but avoids the quasi-periodic Green's functions that are expensive to evaluate.
Zhang, D.S.; Wei, G.W.; Kouri, D.J. ); Hoffman, D.K. ); Gorman, M.; Palacios, A. ); Gunaratne, G.H. The Institute of Fundamental Studies, Kandy )
1999-09-01
An algorithm is presented to integrate nonlinear partial differential equations, which is particularly useful when accurate estimation of spatial derivatives is required. It is based on an analytic approximation method, referred to as distributed approximating functionals (DAF[close quote]s), which can be used to estimate a function and a finite number of derivatives with a specified accuracy. As an application, the Kuramoto-Sivashinsky (KS) equation is integrated in polar coordinates. Its integration requires accurate estimation of spatial derivatives, particularly close to the origin. Several stationary and nonstationary solutions of the KS equation are presented, and compared with analogous states observed in the combustion front of a circular burner. A two-ring, nonuniform counter-rotating state has been obtained in a KS model simulation of such a burner. [copyright] [ital 1999] [ital The American Physical Society
Lopez, L F; Coutinho, F A
2000-03-01
In this paper, we show that the positive solution of a nonlinear integral equation which appears in classical SIR epidemiological models is unique. The demonstration of this fact is necessary to justify the correctness of any approximate or numerical solution. The SIR epidemiological model is used only for simplicity. In fact, the methods used can be easily extended to prove the existence and uniqueness of the more involved integral equations that appear when more biological realities are considered. Thus the inclusion of a latent class (SLIR models) and models incorporating variability in the infectiousness with duration of the infection and spatial distribution lead to integral equations to which the results derived in this paper apply immediately.
NASA Astrophysics Data System (ADS)
Kiesewetter, Simon; Drummond, Peter D.
2017-03-01
A variance reduction method for stochastic integration of Fokker-Planck equations is derived. This unifies the cumulant hierarchy and stochastic equation approaches to obtaining moments, giving a performance superior to either. We show that the brute force method of reducing sampling error by just using more trajectories in a sampled stochastic equation is not the best approach. The alternative of using a hierarchy of moment equations is also not optimal, as it may converge to erroneous answers. Instead, through Bayesian conditioning of the stochastic noise on the requirement that moment equations are satisfied, we obtain improved results with reduced sampling errors for a given number of stochastic trajectories. The method used here converges faster in time-step than Ito-Euler algorithms. This parallel optimized sampling (POS) algorithm is illustrated by several examples, including a bistable nonlinear oscillator case where moment hierarchies fail to converge.
Symmetries, Integrability and Exact Solutions to the (2+1)-Dimensional Benney Types of Equations
NASA Astrophysics Data System (ADS)
Liu, Han-Ze; Xin, Xiang-Peng
2016-08-01
This paper is concerned with the (2+1)-dimensional Benney types of equations. By the complete Lie group classification method, all of the point symmetries of the Benney types of equations are obtained, and the integrable condition of the equation is given. Then, the symmetry reductions and exact solutions to the (2+1)-dimensional nonlinear wave equations are presented. Especially, the shock wave solutions of the Benney equations are investigated by the symmetry reduction and trial function method. Supported by the National Natural Science Foundation of China under Grant Nos. 11171041 and 11505090, Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No. BS2015SF009, and the doctorial foundation of Liaocheng University under Grant No. 31805
NASA Astrophysics Data System (ADS)
Rong, Loh Jian; Chang, Phang
2016-02-01
In this paper, we first define generalized shifted Jacobi polynomial on interval and then use it to define Jacobi wavelet. Then, the operational matrix of fractional integration for Jacobi wavelet is being derived to solve fractional differential equation and fractional integro-differential equation. This method can be seen as a generalization of other orthogonal wavelet operational methods, e.g. Legendre wavelets, Chebyshev wavelets of 1st kind, Chebyshev wavelets of 2nd kind, etc. which are special cases of the Jacobi wavelets. We apply our method to a special type of fractional integro-differential equation of Fredholm type.
Equation free projective integration: A novel scheme for modeling multiscale processes in plasmas
NASA Astrophysics Data System (ADS)
Shay, M. A.; Dorland, B.; Drake, J.; Jemella, B.; Stantchev, G.; Cowley, S.
2005-05-01
We examine a novel simulation scheme called equation free projective integration1 which has the potential to allow global simulations of plasmas while still including the global effects of microscale physics. These simulation codes would be ideal for such multiscale problems as the Earth's magnetosphere, tokamaks, and the solar corona. In this method, the global plasma variables stepped forward in time are not time-integrated directly using dynamical differential equations, hence the name "equation free." Instead, these variables are represented on a microgrid using a kinetic simulation. This microsimulation is integrated forward long enough to determine the time derivatives of the global plasma variables, which are then used to integrate forward the global variables with much larger time steps. Results will be presented of the successful application of equation free to 1-D ion acoustic wave steepening. In addition, initial results of this technique applied to reconnection will also be discussed. 1 I. G. Kevrekidis et. al., "Equation-free multiscale computation: Enabling microscopic simulators to perform system-level tasks," arXiv:physics/0209043.
Fan, Zongwei; Mei, Deqing; Yang, Keji; Chen, Zichen
2014-12-01
To eliminate the limitations of the conventional sound field separation methods which are only applicable to regular surfaces, a sound field separation method based on combined integral equations is proposed to separate sound fields directly in the spatial domain. In virtue of the Helmholtz integral equations for the incident and scattering fields outside a sound scatterer, combined integral equations are derived for sound field separation, which build the quantitative relationship between the sound fields on two arbitrary separation surfaces enclosing the sound scatterer. Through boundary element discretization of the two surfaces, corresponding systems of linear equations are obtained for practical application. Numerical simulations are performed for sound field separation on different shaped surfaces. The influences induced by the aspect ratio of the separation surfaces and the signal noise in the measurement data are also investigated. The separated incident and scattering sound fields agree well with the original corresponding fields described by analytical expressions, which validates the effectiveness and accuracy of the combined integral equations based separation method.
Volume integrals associated with the inhomogeneous Helmholtz equation. Part 1: Ellipsoidal region
NASA Technical Reports Server (NTRS)
Fu, L. S.; Mura, T.
1983-01-01
Problems of wave phenomena in fields of acoustics, electromagnetics and elasticity are often reduced to an integration of the inhomogeneous Helmholtz equation. Results are presented for volume integrals associated with the Helmholtz operator, nabla(2) to alpha(2), for the case of an ellipsoidal region. By using appropriate Taylor series expansions and multinomial theorem, these volume integrals are obtained in series form for regions r 4' and r r', where r and r' are distances from the origin to the point of observation and source, respectively. Derivatives of these integrals are easily evaluated. When the wave number approaches zero, the results reduce directly to the potentials of variable densities.
NASA Astrophysics Data System (ADS)
Zhao, Huaqing
There are two major objectives of this thesis work. One is to study theoretically the fracture and fatigue behavior of both homogeneous and functionally graded materials, with or without crack bridging. The other is to further develop the singular integral equation approach in solving mixed boundary value problems. The newly developed functionally graded materials (FGMs) have attracted considerable research interests as candidate materials for structural applications ranging from aerospace to automobile to manufacturing. From the mechanics viewpoint, the unique feature of FGMs is that their resistance to deformation, fracture and damage varies spatially. In order to guide the microstructure selection and the design and performance assessment of components made of functionally graded materials, in this thesis work, a series of theoretical studies has been carried out on the mode I stress intensity factors and crack opening displacements for FGMs with different combinations of geometry and material under various loading conditions, including: (1) a functionally graded layer under uniform strain, far field pure bending and far field axial loading, (2) a functionally graded coating on an infinite substrate under uniform strain, and (3) a functionally graded coating on a finite substrate under uniform strain, far field pure bending and far field axial loading. In solving crack problems in homogeneous and non-homogeneous materials, a very powerful singular integral equation (SEE) method has been developed since 1960s by Erdogan and associates to solve mixed boundary value problems. However, some of the kernel functions developed earlier are incomplete and possibly erroneous. In this thesis work, mode I fracture problems in a homogeneous strip are reformulated and accurate singular Cauchy type kernels are derived. Very good convergence rates and consistency with standard data are achieved. Other kernel functions are subsequently developed for mode I fracture in
Integrable discretisations for a class of nonlinear Schrödinger equations on Grassmann algebras
NASA Astrophysics Data System (ADS)
Grahovski, Georgi G.; Mikhailov, Alexander V.
2013-12-01
Integrable discretisations for a class of coupled (super) nonlinear Schrödinger (NLS) type of equations are presented. The class corresponds to a Lax operator with entries in a Grassmann algebra. Elementary Darboux transformations are constructed. As a result, Grassmann generalisations of the Toda lattice and the NLS dressing chain are obtained. The compatibility (Bianchi commutativity) of these Darboux transformations leads to integrable Grassmann generalisations of the difference Toda and NLS equations. The resulting systems will have discrete Lax representations provided by the set of two consistent elementary Darboux transformations. For the two discrete systems obtained, initial value and initial-boundary problems are formulated.
Numerical modelling of qualitative behaviour of solutions to convolution integral equations
NASA Astrophysics Data System (ADS)
Ford, Neville J.; Diogo, Teresa; Ford, Judith M.; Lima, Pedro
2007-08-01
We consider the qualitative behaviour of solutions to linear integral equations of the formwhere the kernel k is assumed to be either integrable or of exponential type. After a brief review of the well-known Paley-Wiener theory we give conditions that guarantee that exact and approximate solutions of (1) are of a specific exponential type. As an example, we provide an analysis of the qualitative behaviour of both exact and approximate solutions of a singular Volterra equation with infinitely many solutions. We show that the approximations of neighbouring solutions exhibit the correct qualitative behaviour.
Dynamical Behavior of Solution in Integrable Nonlocal Lakshmanan—Porsezian—Daniel Equation
NASA Astrophysics Data System (ADS)
Liu, Wei; Qiu, De-Qin; Wu, Zhi-Wei; He, Jing-Song
2016-06-01
The integrable nonlocal Lakshmanan—Porsezian—Daniel (LPD) equation which has the higher-order terms (dispersions and nonlinear effects) is first introduced. We demonstrate the integrability of the nonlocal LPD equation, provide its Lax pair, and present its rational soliton solutions and self-potential function by using the degenerate Darboux transformation. From the numerical plots of solutions, the compression effects of the real refractive index profile and the gain-or-loss distribution produced by δ are discussed. Supported by the National Natural Science Foundation of China under Grant No. 11271210 and the K.C. Wong Magna Fund in Ningbo University
The statistical theory of the fracture of fragile bodies. Part 2: The integral equation method
NASA Technical Reports Server (NTRS)
Kittl, P.
1984-01-01
It is demonstrated how with the aid of a bending test, the Weibull fracture risk function can be determined - without postulating its analytical form - by resolving an integral equation. The respective solutions for rectangular and circular section beams are given. In the first case the function is expressed as an algorithm and in the second, in the form of series. Taking into account that the cumulative fracture probability appearing in the solution to the integral equation must be continuous and monotonically increasing, any case of fabrication or selection of samples can be treated.
A wavelet-based computational method for solving stochastic Itô–Volterra integral equations
Mohammadi, Fakhrodin
2015-10-01
This paper presents a computational method based on the Chebyshev wavelets for solving stochastic Itô–Volterra integral equations. First, a stochastic operational matrix for the Chebyshev wavelets is presented and a general procedure for forming this matrix is given. Then, the Chebyshev wavelets basis along with this stochastic operational matrix are applied for solving stochastic Itô–Volterra integral equations. Convergence and error analysis of the Chebyshev wavelets basis are investigated. To reveal the accuracy and efficiency of the proposed method some numerical examples are included.
NASA Astrophysics Data System (ADS)
Davidenko, V. D.; Zinchenko, A. S.; Harchenko, I. K.
2016-12-01
Integral equations for the shape functions in the adiabatic, quasi-static, and improved quasi-static approximations are presented. The approach to solving these equations by the Monte Carlo method is described.
NASA Astrophysics Data System (ADS)
Gerini, G.; Visser, H. J.
In this paper we present an efficient theoretical formulation for a full-wave analysis of phased arrays conformal to cylindrical structures. The theory is based on an integral equation formulation and the Unit Cell Approach. Thanks to its generality and efficiency, this method represents a good starting point for the development of accurate CAD tools for the analysis of integrated cylindrical structures including radome, coaxial excitations and tuning elements in waveguide.
Altürk, Ahmet
2016-01-01
Mean value theorems for both derivatives and integrals are very useful tools in mathematics. They can be used to obtain very important inequalities and to prove basic theorems of mathematical analysis. In this article, a semi-analytical method that is based on weighted mean-value theorem for obtaining solutions for a wide class of Fredholm integral equations of the second kind is introduced. Illustrative examples are provided to show the significant advantage of the proposed method over some existing techniques.
The quench map in an integrable classical field theory: nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Caudrelier, Vincent; Doyon, Benjamin
2016-11-01
We study the non-equilibrium dynamics obtained by an abrupt change (a quench) in the parameters of an integrable classical field theory, the nonlinear Schrödinger equation. We first consider explicit one-soliton examples, which we fully describe by solving the direct part of the inverse scattering problem. We then develop some aspects of the general theory using elements of the inverse scattering method. For this purpose, we introduce the quench map which acts on the space of scattering data and represents the change of parameter with fixed field configuration (initial condition). We describe some of its analytic properties by implementing a higher level version of the inverse scattering method, and we discuss the applications of Darboux-Bäcklund transformations, Gelfand-Levitan-Marchenko equations and the Rosales series solution to a related, dual quench problem. Finally, we comment on the interplay between quantum and classical tools around the theme of quenches and on the usefulness of the quantization of our classical approach to the quantum quench problem.
Solving the hypersingular boundary integral equation for the Burton and Miller formulation.
Langrenne, Christophe; Garcia, Alexandre; Bonnet, Marc
2015-11-01
This paper presents an easy numerical implementation of the Burton and Miller (BM) formulation, where the hypersingular Helmholtz integral is regularized by identities from the associated Laplace equation and thus needing only the evaluation of weakly singular integrals. The Helmholtz equation and its normal derivative are combined directly with combinations at edge or corner collocation nodes not used when the surface is not smooth. The hypersingular operators arising in this process are regularized and then evaluated by an indirect procedure based on discretized versions of the Calderón identities linking the integral operators for associated Laplace problems. The method is valid for acoustic radiation and scattering problems involving arbitrarily shaped three-dimensional bodies. Unlike other approaches using direct evaluation of hypersingular integrals, collocation points still coincide with mesh nodes, as is usual when using conforming elements. Using higher-order shape functions (with the boundary element method model size kept fixed) reduces the overall numerical integration effort while increasing the solution accuracy. To reduce the condition number of the resulting BM formulation at low frequencies, a regularized version α = ik/(k(2 )+ λ) of the classical BM coupling factor α = i/k is proposed. Comparisons with the combined Helmholtz integral equation Formulation method of Schenck are made for four example configurations, two of them featuring non-smooth surfaces.
Functional integral derivation of the kinetic equation of two-dimensional point vortices
NASA Astrophysics Data System (ADS)
Fouvry, Jean-Baptiste; Chavanis, Pierre-Henri; Pichon, Christophe
2016-08-01
We present a brief derivation of the kinetic equation describing the secular evolution of point vortices in two-dimensional hydrodynamics, by relying on a functional integral formalism. We start from Liouville's equation which describes the exact dynamics of a two-dimensional system of point vortices. At the order 1 / N, the evolution of the system is characterised by the first two equations of the BBGKY hierarchy involving the system's 1-body distribution function and its 2-body correlation function. Thanks to the introduction of auxiliary fields, these two evolution constraints may be rewritten as a functional integral. When functionally integrated over the 2-body correlation function, this rewriting leads to a new constraint coupling the 1-body distribution function and the two auxiliary fields. Once inverted, this constraint provides, through a new route, the closed non-linear kinetic equation satisfied by the 1-body distribution function. Such a method sheds new lights on the origin of these kinetic equations complementing the traditional derivation methods.
Wilson, D Scott; Lee, Lloyd L
2005-07-22
We explore the vapor-liquid phase behavior of binary mixtures of Lennard-Jones-type molecules where one component is supercritical, given the system temperature. We apply the self-consistency approach to the Ornstein-Zernike integral equations to obtain the correlation functions. The consistency checks include not only thermodynamic consistencies (pressure consistency and Gibbs-Duhem consistency), but also pointwise consistencies, such as the zero-separation theorems on the cavity functions. The consistencies are enforced via the bridge functions in the closure which contain adjustable parameters. The full solution requires the values of not only the monomer chemical potentials, but also the dimer chemical potentials present in the zero-separation theorems. These are evaluated by the direct chemical-potential formula [L. L. Lee, J. Chem. Phys. 97, 8606 (1992)] that does not require temperature nor density integration. In order to assess the integral equation accuracy, molecular-dynamics simulations are carried out alongside the states studied. The integral equation results compare well with simulation data. In phase calculations, it is important to have pressure consistency and valid chemical potentials, since the matching of phase boundaries requires the equality of the pressures and chemical potentials of both the liquid and vapor phases. The mixtures studied are methane-type and pentane-type molecules, both characterized by effective Lennard-Jones potentials. Calculations on one isotherm show that the integral equation approach yields valid answers as compared with the experimental data of Sage and Lacey. To study vapor-liquid phase behavior, it is necessary to use consistent theories; any inconsistencies, especially in pressure, will vitiate the phase boundary calculations.
NASA Astrophysics Data System (ADS)
Liu, Hanze; Xin, Xiangpeng; Wang, Zenggui; Liu, Xiqiang
2017-03-01
This paper is concerned with the Bäcklund transformations (BTs) of the nonlinear evolution equations (NLEEs). Based on the homogeneous balance principle (HBP), the existence of the BT of the generalized Burgers'-KdV (B-KdV) equation is classified, then the BTs of the nonlinear equations are given. In general, the method can be used to construct BTs of the nonlinear evolution equations in polynomial form. Furthermore, the integrability and exact explicit solutions to the nonlinear equations are investigated.
Chen, I L; Chen, J T; Kuo, S R; Liang, M T
2001-03-01
Integral equation methods have been widely used to solve interior eigenproblems and exterior acoustic problems (radiation and scattering). It was recently found that the real-part boundary element method (BEM) for the interior problem results in spurious eigensolutions if the singular (UT) or the hypersingular (LM) equation is used alone. The real-part BEM results in spurious solutions for interior problems in a similar way that the singular integral equation (UT method) results in fictitious solutions for the exterior problem. To solve this problem, a Combined Helmholtz Exterior integral Equation Formulation method (CHEEF) is proposed. Based on the CHEEF method, the spurious solutions can be filtered out if additional constraints from the exterior points are chosen carefully. Finally, two examples for the eigensolutions of circular and rectangular cavities are considered. The optimum numbers and proper positions for selecting the points in the exterior domain are analytically studied. Also, numerical experiments were designed to verify the analytical results. It is worth pointing out that the nodal line of radiation mode of a circle can be rotated due to symmetry, while the nodal line of the rectangular is on a fixed position.
An efficient and fast parallel method for Volterra integral equations of Abel type
NASA Astrophysics Data System (ADS)
Capobianco, Giovanni; Conte, Dajana
2006-05-01
In this paper we present an efficient and fast parallel waveform relaxation method for Volterra integral equations of Abel type, obtained by reformulating a nonstationary waveform relaxation method for systems of equations with linear coefficient constant kernel. To this aim we consider the Laplace transform of the equation and here we apply the recurrence relation given by the Chebyshev polynomial acceleration for algebraic linear systems. Back in the time domain, we obtain a three term recursion which requires, at each iteration, the evaluation of convolution integrals, where only the Laplace transform of the kernel is known. For this calculation we can use a fast convolution algorithm. Numerical experiments have been done also on problems where it is not possible to use the original nonstationary method, obtaining good results in terms of improvement of the rate of convergence with respect the stationary method.
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)
Tong, Mei Song; Yang, Kuo; Sheng, Wei Tian; Zhu, Zhen Ying
2013-12-01
Reconstruction of unknown objects by microwave illumination requires efficient inversion for measured electromagnetic scattering data. In the integral equation approach for reconstructing dielectric objects based on the Born iterative method or its variations, the volume integral equations are involved because the imaging domain is fully inhomogeneous. When solving the forward scattering integral equation, the Nyström method is used because the traditional method of moments may be inconvenient due to the inhomogeneity of the imaging domain. The benefits of the Nyström method include the simple implementation without using any basis and testing functions and low requirement on geometrical discretization. When solving the inverse scattering integral equation, the Gauss-Newton minimization approach with a line search method (LSM) and multiplicative regularization method (MRM) is employed. The LSM can optimize the search of step size in each iteration, whereas the MRM may reduce the number of numerical experiments for choosing the regularization parameter. Numerical examples for reconstructing typical dielectric objects under limited observation angles are presented to illustrate the inversion approach.
The Transmission Line as a Simple Example for Introducing Integral Equations to Undergraduates
ERIC Educational Resources Information Center
Rothwell, E. J.
2009-01-01
Integral equations are becoming a common means for describing problems in electromagnetics, and so it is important to expose students to methods for their solution. Typically this is done using examples in antennas, scattering, or electrostatics. Unfortunately, many difficult issues arise in the formulation and solution of the associated…
Integration processes of ordinary differential equations based on Laguerre-Radau interpolations
NASA Astrophysics Data System (ADS)
Guo, Ben-Yu; Wang, Zhong-Qing; Tian, Hong-Jiong; Wang, Li-Lian
2008-03-01
In this paper, we propose two integration processes for ordinary differential equations based on modified Laguerre-Radau interpolations, which are very efficient for long-time numerical simulations of dynamical systems. The global convergence of proposed algorithms are proved. Numerical results demonstrate the spectral accuracy of these new approaches and coincide well with theoretical analysis.
A Nyström interpolant for some weakly singular linear Volterra integral equations
NASA Astrophysics Data System (ADS)
Baratella, Paola
2009-09-01
We consider a second kind weakly singular Volterra integral equation defined by a non-compact operator and derive a Nyström type interpolant of the solution based on Gauss-Radau nodes. Assuming the stability of the interpolant, which is confirmed by the numerical tests, we derive convergence estimates.
NASA Astrophysics Data System (ADS)
Muthuvalu, Mohana Sundaram
2016-06-01
In this paper, performance analysis of the preconditioned Gauss-Seidel iterative methods for solving dense linear system arise from Fredholm integral equations of the second kind is investigated. The formulation and implementation of the preconditioned Gauss-Seidel methods are presented. Numerical results are included in order to verify the performance of the methods.
AN INTEGRAL EQUATION REPRESENTATION OF WIDE-BAND ELECTROMAGNETIC SCATTERING BY THIN SHEETS
An efficient, accurate numerical modeling scheme has been developed, based on the integral equation solution to compute electromagnetic (EM) responses of thin sheets over a wide frequency band. The thin-sheet approach is useful for simulating the EM response of a fracture system ...
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1984-01-01
The efficiency and accuracy of several algorithms recently developed for the efficient numerical integration of stiff ordinary differential equations are compared. The methods examined include two general-purpose codes, EPISODE and LSODE, and three codes (CHEMEQ, CREK1D, and GCKP84) developed specifically to integrate chemical kinetic rate equations. The codes are applied to two test problems drawn from combustion kinetics. The comparisons show that LSODE is the fastest code currently available for the integration of combustion kinetic rate equations. An important finding is that an interactive solution of the algebraic energy conservation equation to compute the temperature does not result in significant errors. In addition, this method is more efficient than evaluating the temperature by integrating its time derivative. Significant reductions in computational work are realized by updating the rate constants (k = at(supra N) N exp(-E/RT) only when the temperature change exceeds an amount delta T that is problem dependent. An approximate expression for the automatic evaluation of delta T is derived and is shown to result in increased efficiency.
NASA Technical Reports Server (NTRS)
Jamnejad, V.; Cwik, T.; Zuffada, C.
1994-01-01
A coupled finite element-combined field integral equation technique was originally developed for solving scattering problems involving inhomogeneous objects of arbitrary shape and large dimensions in wavelength.
NASA Technical Reports Server (NTRS)
Barker, L. E., Jr.; Bowles, R. L.; Williams, L. H.
1973-01-01
High angular rates encountered in real-time flight simulation problems may require a more stable and accurate integration method than the classical methods normally used. A study was made to develop a general local linearization procedure of integrating dynamic system equations when using a digital computer in real-time. The procedure is specifically applied to the integration of the quaternion rate equations. For this application, results are compared to a classical second-order method. The local linearization approach is shown to have desirable stability characteristics and gives significant improvement in accuracy over the classical second-order integration methods.
Connectivity as an alternative to boundary integral equations: Construction of bases
Herrera, Ismael; Sabina, Federico J.
1978-01-01
In previous papers Herrera developed a theory of connectivity that is applicable to the problem of connecting solutions defined in different regions, which occurs when solving partial differential equations and many problems of mechanics. In this paper we explain how complete connectivity conditions can be used to replace boundary integral equations in many situations. We show that completeness is satisfied not only in steady-state problems such as potential, reduced wave equation and static and quasi-static elasticity, but also in time-dependent problems such as heat and wave equations and dynamical elasticity. A method to obtain bases of connectivity conditions, which are independent of the regions considered, is also presented. PMID:16592522
New integration techniques for chemical kinetic rate equations. 2: Accuracy comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1985-01-01
A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general purpose codes EEPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREKID, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas phase chemical reactions at constant pressure, and include all three combustion regimes: induction, heat release, and equilibration. The comparisons show that LSODE is the most efficient code - in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.
New integration techniques for chemical kinetic rate equations. II - Accuracy comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1986-01-01
A comparison of the accuracy of several techniques recently developed for solving stiff differential equations is presented. The techniques examined include two general purpose codes EEPISODE and LSODE developed for an arbitrary system of ordinary differential equations, and three specialized codes CHEMEQ, CREKID, and GCKP84 developed specifically to solve chemical kinetic rate equations. The accuracy comparisons are made by applying these solution procedures to two practical combustion kinetics problems. Both problems describe adiabatic, homogeneous, gas phase chemical reactions at constant pressure, and include all three combustion regimes: induction heat release, and equilibration. The comparisons show that LSODE is the most efficient code - in the sense that it requires the least computational work to attain a specified accuracy level. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for the temperature can be more accurate and efficient than computing the temperature by integrating its time derivative.
Structural equation modeling for observational studies
Grace, J.B.
2008-01-01
Structural equation modeling (SEM) represents a framework for developing and evaluating complex hypotheses about systems. This method of data analysis differs from conventional univariate and multivariate approaches familiar to most biologists in several ways. First, SEMs are multiequational and capable of representing a wide array of complex hypotheses about how system components interrelate. Second, models are typically developed based on theoretical knowledge and designed to represent competing hypotheses about the processes responsible for data structure. Third, SEM is conceptually based on the analysis of covariance relations. Most commonly, solutions are obtained using maximum-likelihood solution procedures, although a variety of solution procedures are used, including Bayesian estimation. Numerous extensions give SEM a very high degree of flexibility in dealing with nonnormal data, categorical responses, latent variables, hierarchical structure, multigroup comparisons, nonlinearities, and other complicating factors. Structural equation modeling allows researchers to address a variety of questions about systems, such as how different processes work in concert, how the influences of perturbations cascade through systems, and about the relative importance of different influences. I present 2 example applications of SEM, one involving interactions among lynx (Lynx pardinus), mongooses (Herpestes ichneumon), and rabbits (Oryctolagus cuniculus), and the second involving anuran species richness. Many wildlife ecologists may find SEM useful for understanding how populations function within their environments. Along with the capability of the methodology comes a need for care in the proper application of SEM.
Multiple Integration of the Heat-Conduction Equation for a Space Bounded From the Inside
NASA Astrophysics Data System (ADS)
Kot, V. A.
2016-03-01
An N-fold integration of the heat-conduction equation for a space bounded from the inside has been performed using a system of identical equalities with definition of the temperature function by a power polynomial with an exponential factor. It is shown that, in a number of cases, the approximate solutions obtained can be considered as exact because their errors comprise hundredths and thousandths of a percent. The method proposed for N-fold integration represents an alternative to classical integral transformations.
Giurgiutiu, V.; Ionita, A.; Dillard, D.A.; Graffeo, J.K.
1996-12-31
Fracture mechanics analysis of adhesively bonded joints has attracted considerable attention in recent years. A possible approach to the analysis of adhesive layer cracks is to study a brittle adhesive between 2 elastic half-planes representing the substrates. A 2-material 3-region elasticity problem is set up and has to be solved. A modeling technique based on the work of Fleck, Hutchinson, and Suo is used. Two complex potential problems using Muskelishvili`s formulation are set up for the 3-region, 2-material model: (a) a distribution of edge dislocations is employed to simulate the crack and its near field; and (b) a crack-free problem is used to simulate the effect of the external loading applied in the far field. Superposition of the two problems is followed by matching tractions and displacements at the bimaterial boundaries. The Cauchy principal value integral is used to treat the singularities. Imposing the traction-free boundary conditions over the entire crack length yielded a linear system of two integral equations. The parameters of the problem are Dundurs` elastic mismatch coefficients, {alpha} and {beta}, and the ratio c/H representing the geometric position of the crack in the adhesive layer.
On the symplectic integration of the discrete nonlinear Schrödinger equation with disorder
NASA Astrophysics Data System (ADS)
Gerlach, E.; Meichsner, J.; Skokos, C.
2016-09-01
We present several methods, which utilize symplectic integration techniques based on two and three part operator splitting, for numerically solving the equations of motion of the disordered, discrete nonlinear Schrödinger (DDNLS) equation, and compare their efficiency. Our results suggest that the most suitable methods for the very long time integration of this one-dimensional Hamiltonian lattice model with many degrees of freedom (of the order of a few hundreds) are the ones based on three part splits of the system's Hamiltonian. Two part split techniques can be preferred for relatively small lattices having up to N ≈ 70 sites. An advantage of the latter methods is the better conservation of the system's second integral, i.e. the wave packet's norm.
Multi-off-grid methods in multi-step integration of ordinary differential equations
NASA Technical Reports Server (NTRS)
Beaudet, P. R.
1974-01-01
Description of methods of solving first- and second-order systems of differential equations in which all derivatives are evaluated at off-grid locations in order to circumvent the Dahlquist stability limitation on the order of on-grid methods. The proposed multi-off-grid methods require off-grid state predictors for the evaluation of the n derivatives at each step. Progressing forward in time, the off-grid states are predicted using a linear combination of back on-grid state values and off-grid derivative evaluations. A comparison is made between the proposed multi-off-grid methods and the corresponding Adams and Cowell on-grid integration techniques in integrating systems of ordinary differential equations, showing a significant reduction in the error at larger step sizes in the case of the multi-off-grid integrator.
Mardanov, M J; Mahmudov, N I; Sharifov, Y A
2014-01-01
We study a boundary value problem for the system of nonlinear impulsive fractional differential equations of order α (0 < α ≤ 1) involving the two-point and integral boundary conditions. Some new results on existence and uniqueness of a solution are established by using fixed point theorems. Some illustrative examples are also presented. We extend previous results even in the integer case α = 1.
Integrability and a new breed of solitons of an NLS type equation in 2+1 dimensions
NASA Astrophysics Data System (ADS)
Jiang, Zhuhan; Bullough, R. K.
1994-07-01
The integrability via Painlevé analysis as well as conservation laws is established for a new type of generalized NLS equation in 2+1 dimensions. By adapting ZS-AKNS' spectral problem to the system, we study a new breed of solitons, corresponding to the discrete spectral parameters that depend on both time and space. Explicit single and double solitons are also given.
NASA Astrophysics Data System (ADS)
Agachev, J. R.; Galimyanov, A. F.
2016-11-01
In this paper the method of mechanical quadrature solutions fractional integral equation. Computational scheme quadrature method is based on the quadrature formula of rectangles with equidistant nodes, which is the formula of the highest trigonometric degree of accuracy, using a regularizing parameter. This decision is taken for the approximate trigonometric interpolation polynomial constructed from the values that make up the solution of the quadrature method. The substantiation of the method in Holder spaces.
Ciraolo, Giulio Gargano, Francesco Sciacca, Vincenzo
2013-08-01
We study a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. Our approach is based on the minimization of an integral functional arising from a volume integral formulation of the radiation condition. The index of refraction does not need to be constant at infinity and may have some angular dependency as well as perturbations. We prove analytical results on the convergence of the approximate solution. Numerical examples for different shapes of the artificial boundary and for non-constant indexes of refraction will be presented.
NASA Astrophysics Data System (ADS)
Absi, Noureddine; Hoggan, Philip
The integral bottleneck in evaluating molecular energies arises from the two-electron contributions. These are difficult and time-consuming to evaluate, especially over exponential type orbitals, used here to ensure the correct behavior of atomic orbitals. The two-center two-electron integrals are essential to describe atom pairs in molecules and distinguish those that are bound. In this work on analytical integration, it is shown that the two-center Coulomb integrals involved can be expressed as one-electron kinetic energy-like integrals. This is accomplished using the fact that the Coulomb operator is a Green's function of the Laplacian. The ensuing integrals may be further simplified by defining spectral forms for the one-electron potential satisfying Poisson's equation therein. A sum of overlap integrals with the atomic orbital energy eigenvalue as a factor is then obtained to give the Coulomb energy. This is most easily evaluated by direct integration. The orbitals involved in three and four center integrals are translated to two centers. This is discussed very briefly. The evaluation of exchange energy is a straightforward extension of this work. The summation coefficients in spectral forms are evaluated analytically from Gaunt coefficients. The Poisson method may be used to calculate Coulomb energy integrals efficiently. For a single processor, gains of CPU time for a given chemical accuracy exceed a factor of 4. This method lends itself to efficient evaluation on a parallel computer.
Numerical solution of random singular integral equation appearing in crack problems
NASA Technical Reports Server (NTRS)
Sambandham, M.; Srivatsan, T. S.; Bharucha-Reid, A. T.
1986-01-01
The solution of several elasticity problems, and particularly crack problems, can be reduced to the solution of one-dimensional singular integral equations with a Cauchy-type kernel or to a system of uncoupled singular integral equations. Here a method for the numerical solution of random singular integral equations of Cauchy type is presented. The solution technique involves a Chebyshev series approximation, the coefficients of which are the solutions of a system of random linear equations. This method is applied to the problem of periodic array of straight cracks inside an infinite isotropic elastic medium and subjected to a nonuniform pressure distribution along the crack edges. The statistical properties of the random solution are evaluated numerically, and the random solution is used to determine the values of the stress-intensity factors at the crack tips. The error, expressed as the difference between the mean of the random solution and the deterministic solution, is established. Values of stress-intensity factors at the crack tip for different random input functions are presented.
Accurate integral equation theory for the central force model of liquid water and ionic solutions
NASA Astrophysics Data System (ADS)
Ichiye, Toshiko; Haymet, A. D. J.
1988-10-01
The atom-atom pair correlation functions and thermodynamics of the central force model of water, introduced by Lemberg, Stillinger, and Rahman, have been calculated accurately by an integral equation method which incorporates two new developments. First, a rapid new scheme has been used to solve the Ornstein-Zernike equation. This scheme combines the renormalization methods of Allnatt, and Rossky and Friedman with an extension of the trigonometric basis-set solution of Labik and co-workers. Second, by adding approximate ``bridge'' functions to the hypernetted-chain (HNC) integral equation, we have obtained predictions for liquid water in which the hydrogen bond length and number are in good agreement with ``exact'' computer simulations of the same model force laws. In addition, for dilute ionic solutions, the ion-oxygen and ion-hydrogen coordination numbers display both the physically correct stoichiometry and good agreement with earlier simulations. These results represent a measurable improvement over both a previous HNC solution of the central force model and the ex-RISM integral equation solutions for the TIPS and other rigid molecule models of water.
ERIC Educational Resources Information Center
Yu, Baohua
2013-01-01
This study examined the interrelationships of integrative motivation, competence in second language (L2) communication, sociocultural adaptation, academic adaptation and persistence of international students at an Australian university. Structural equation modelling demonstrated that the integrative motivation of international students has a…
On the solution of integral equations with a generalized Cauchy kernel
NASA Technical Reports Server (NTRS)
Kaya, A. C.; Erdogan, F.
1987-01-01
A numerical technique is developed analytically to solve a class of singular integral equations occurring in mixed boundary-value problems for nonhomogeneous elastic media with discontinuities. The approach of Kaya and Erdogan (1987) is extended to treat equations with generalized Cauchy kernels, reformulating the boundary-value problems in terms of potentials as the unknown functions. The numerical implementation of the solution is discussed, and results for an epoxy-Al plate with a crack terminating at the interface and loading normal to the crack are presented in tables.
NASA Technical Reports Server (NTRS)
Rosenbaum, J. S.
1976-01-01
If a system of ordinary differential equations represents a property conserving system that can be expressed linearly (e.g., conservation of mass), it is then desirable that the numerical integration method used conserve the same quantity. It is shown that both linear multistep methods and Runge-Kutta methods are 'conservative' and that Newton-type methods used to solve the implicit equations preserve the inherent conservation of the numerical method. It is further shown that a method used by several authors is not conservative.
Temperature-dependent isovector pairing gap equations using a path integral approach
Fellah, M.; Allal, N. H.; Belabbas, M.; Oudih, M. R.; Benhamouda, N.
2007-10-15
Temperature-dependent isovector neutron-proton (np) pairing gap equations have been established by means of the path integral approach. These equations generalize the BCS ones for the pairing between like particles at finite temperature. The method has been numerically tested using the one-level model. It has been shown that the gap parameter {delta}{sub np} has a behavior analogous to that of {delta}{sub nn} and {delta}{sub pp} as a function of the temperature: one notes the presence of a critical temperature. Moreover, it has been shown that the isovector pairing effects remain beyond the critical temperature that corresponds to the pairing between like particles.
NASA Technical Reports Server (NTRS)
Nixon, D.
1978-01-01
The linear transonic perturbation integral equation previously derived for nonlifting airfoils is formulated for lifting cases. In order to treat shock wave motions, a strained coordinate system is used in which the shock location is invariant. The tangency boundary conditions are either formulated using the thin airfoil approximation or by using the analytic continuation concept. A direct numerical solution to this equation is derived in contrast to the iterative scheme initially used, and results of both lifting and nonlifting examples indicate that the method is satisfactory.
Estimates for a class of oscillatory integrals and decay rates for wave-type equations.
Arnold, Anton; Kim, Jinmyong; Yao, Xiaohua
2012-10-01
This paper investigates higher order wave-type equations of the form [Formula: see text], where the symbol [Formula: see text] is a real, non-degenerate elliptic polynomial of the order [Formula: see text] on [Formula: see text]. Using methods from harmonic analysis, we first establish global pointwise time-space estimates for a class of oscillatory integrals that appear as the fundamental solutions to the Cauchy problem of such wave equations. These estimates are then used to establish (pointwise-in-time) [Formula: see text] estimates on the wave solution in terms of the initial conditions.
On a method for constructing the Lax pairs for nonlinear integrable equations
NASA Astrophysics Data System (ADS)
Habibullin, I. T.; Khakimova, A. R.; Poptsova, M. N.
2016-01-01
We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.
On integration of a multidimensional version of n-wave type equation
NASA Astrophysics Data System (ADS)
Zenchuk, A. I.
2014-12-01
We represent a version of multidimensional quasilinear partial differential equation (PDE) together with large manifold of particular solutions given in an integral form. The dimensionality of constructed PDE can be arbitrary. We call it the n-wave type PDE, although the structure of its nonlinearity differs from that of the classical completely integrable (2+1)-dimensional n-wave equation. The richness of solution space to such a PDE is characterized by a set of arbitrary functions of several variables. However, this richness is not enough to provide the complete integrability, which is shown explicitly. We describe a class of multi-solitary wave solutions in details. Among examples of explicit particular solutions, we represent a lump-lattice solution depending on five independent variables. In Appendix, as an important supplemental material, we show that our nonlinear PDE is reducible from the more general multidimensional PDE which can be derived using the dressing method based on the linear integral equation with the kernel of a special type (a modification of the ∂ ¯ -problem). The dressing algorithm gives us a key for construction of higher order PDEs, although they are not discussed in this paper.
NASA Astrophysics Data System (ADS)
Mukherjee, Abhik; Janaki, M. S.; Kundu, Anjan
2015-07-01
A new, completely integrable, two dimensional evolution equation is derived for an ion acoustic wave propagating in a magnetized, collisionless plasma. The equation is a multidimensional generalization of a modulated wavepacket with weak transverse propagation, which has resemblance to nonlinear Schrödinger (NLS) equation and has a connection to Kadomtsev-Petviashvili equation through a constraint relation. Higher soliton solutions of the equation are derived through Hirota bilinearization procedure, and an exact lump solution is calculated exhibiting 2D structure. Some mathematical properties demonstrating the completely integrable nature of this equation are described. Modulational instability using nonlinear frequency correction is derived, and the corresponding growth rate is calculated, which shows the directional asymmetry of the system. The discovery of this novel (2+1) dimensional integrable NLS type equation for a magnetized plasma should pave a new direction of research in the field.
Mukherjee, Abhik Janaki, M. S. Kundu, Anjan
2015-07-15
A new, completely integrable, two dimensional evolution equation is derived for an ion acoustic wave propagating in a magnetized, collisionless plasma. The equation is a multidimensional generalization of a modulated wavepacket with weak transverse propagation, which has resemblance to nonlinear Schrödinger (NLS) equation and has a connection to Kadomtsev-Petviashvili equation through a constraint relation. Higher soliton solutions of the equation are derived through Hirota bilinearization procedure, and an exact lump solution is calculated exhibiting 2D structure. Some mathematical properties demonstrating the completely integrable nature of this equation are described. Modulational instability using nonlinear frequency correction is derived, and the corresponding growth rate is calculated, which shows the directional asymmetry of the system. The discovery of this novel (2+1) dimensional integrable NLS type equation for a magnetized plasma should pave a new direction of research in the field.
Schmidt, Rita; Webb, Andrew
2016-01-01
Electrical Properties Tomography (EPT) using MRI is a technique that has been developed to provide a new contrast mechanism for in vivo imaging. Currently the most common method relies on the solution of the homogeneous Helmholtz equation, which has limitations in accurate estimation at tissue interfaces. A new method proposed in this work combines a Maxwell's integral equation representation of the problem, and the use of high permittivity materials (HPM) to control the RF field, in order to reconstruct the electrical properties image. The magnetic field is represented by an integral equation considering each point as a contrast source. This equation can be solved in an inverse method. In this study we use a reference simulation or scout scan of a uniform phantom to provide an initial estimate for the inverse solution, which allows the estimation of the complex permittivity within a single iteration. Incorporating two setups with and without the HPM improves the reconstructed result, especially with respect to the very low electric field in the center of the sample. Electromagnetic simulations of the brain were performed at 3T to generate the B1(+) field maps and reconstruct the electric properties images. The standard deviations of the relative permittivity and conductivity were within 14% and 18%, respectively for a volume consisting of white matter, gray matter and cerebellum.
NASA Astrophysics Data System (ADS)
Schmidt, Rita; Webb, Andrew
2016-01-01
Electrical Properties Tomography (EPT) using MRI is a technique that has been developed to provide a new contrast mechanism for in vivo imaging. Currently the most common method relies on the solution of the homogeneous Helmholtz equation, which has limitations in accurate estimation at tissue interfaces. A new method proposed in this work combines a Maxwell's integral equation representation of the problem, and the use of high permittivity materials (HPM) to control the RF field, in order to reconstruct the electrical properties image. The magnetic field is represented by an integral equation considering each point as a contrast source. This equation can be solved in an inverse method. In this study we use a reference simulation or scout scan of a uniform phantom to provide an initial estimate for the inverse solution, which allows the estimation of the complex permittivity within a single iteration. Incorporating two setups with and without the HPM improves the reconstructed result, especially with respect to the very low electric field in the center of the sample. Electromagnetic simulations of the brain were performed at 3 T to generate the B1+ field maps and reconstruct the electric properties images. The standard deviations of the relative permittivity and conductivity were within 14% and 18%, respectively for a volume consisting of white matter, gray matter and cerebellum.
Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations
NASA Astrophysics Data System (ADS)
Chvartatskyi, Oleksandr; Dimakis, Aristophanes; Müller-Hoissen, Folkert
2016-08-01
We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.
Integral equations with modified fundamental solution in time-harmonic electromagnetic scattering
NASA Astrophysics Data System (ADS)
Jost, Gabriele
The exterior boundary-value problem for time-harmonic EM reflection from a perfect conductor is investigated analytically. The technique developed by Ursell (1973 and 1974) and Jones (1974) for the scalar Helmholtz equation, based on the addition of outgoing waves to the fundamental free-space solution, is extended to overcome the problem of nonuniqueness in the case of the reduced Maxwell equations. The reduction of the boundary-value problem to a Fredholm integral equation of the second kind and the coefficient selection criteria are explained in detail, and numerical results showing the dependence of the condition number on the frequency are presented in extensive graphs. The present approach is found to be well suited to small wave numbers.
Integrable multi-component generalization of a modified short pulse equation
NASA Astrophysics Data System (ADS)
Matsuno, Yoshimasa
2016-11-01
We propose a multi-component generalization of the modified short pulse (SP) equation which was derived recently as a reduction of Feng's two-component SP equation. Above all, we address the two-component system in depth. We obtain the Lax pair, an infinite number of conservation laws and multisoliton solutions for the system, demonstrating its integrability. Subsequently, we show that the two-component system exhibits cusp solitons and breathers for which the detailed analysis is performed. Specifically, we explore the interaction process of two cusp solitons and derive the formula for the phase shift. While cusp solitons are singular solutions, smooth breather solutions are shown to exist, provided that the parameters characterizing the solutions satisfy certain conditions. Last, we discuss the relation between the proposed system and existing two-component SP equations.
Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms.
Chowdury, A; Kedziora, D J; Ankiewicz, A; Akhmediev, N
2014-09-01
We present the fifth-order equation of the nonlinear Schrödinger hierarchy. This integrable partial differential equation contains fifth-order dispersion and nonlinear terms related to it. We present the Lax pair and use Darboux transformations to derive exact expressions for the most representative soliton solutions. This set includes two-soliton collisions and the degenerate case of the two-soliton solution, as well as beating structures composed of two or three solitons. Ultimately, the new quintic operator and the terms it adds to the standard nonlinear Schrödinger equation (NLSE) are found to primarily affect the velocity of solutions, with complicated flow-on effects. Furthermore, we present a new structure, composed of coincident equal-amplitude solitons, which cannot exist for the standard NLSE.
Error analysis of exponential integrators for oscillatory second-order differential equations
NASA Astrophysics Data System (ADS)
Grimm, Volker; Hochbruck, Marlis
2006-05-01
In this paper, we analyse a family of exponential integrators for second-order differential equations in which high-frequency oscillations in the solution are generated by a linear part. Conditions are given which guarantee that the integrators allow second-order error bounds independent of the product of the step size with the frequencies. Our convergence analysis generalizes known results on the mollified impulse method by García-Archilla, Sanz-Serna and Skeel (1998, SIAM J. Sci. Comput. 30 930-63) and on Gautschi-type exponential integrators (Hairer E, Lubich Ch and Wanner G 2002 Geometric Numerical Integration (Berlin: Springer), Hochbruck M and Lubich Ch 1999 Numer. Math. 83 403-26).
Integral-equation approach to the weak-field asymptotic theory of tunneling ionization
NASA Astrophysics Data System (ADS)
Dnestryan, Andrey I.; Tolstikhin, Oleg I.
2016-03-01
An integral equation approach to the weak-field asymptotic theory (WFAT) of tunneling ionization is developed. An integral representation for the exact partial amplitudes of ionization into parabolic channels is derived. The WFAT expansion for the ionization rate follows immediately from this relation. Integral representations for the coefficients in the expansion are obtained. The integrals accumulate where the ionizing orbital has large amplitude and are not sensitive to its behavior in the asymptotic region. Hence, these formulas enable one to reliably calculate the WFAT coefficients even if the orbital is represented by an expansion in Gaussian basis, as is usually the case in standard software packages for electronic structure calculations. This development is expected to greatly simplify the implementation of the WFAT for polyatomic molecules, and thus facilitate its growing applications in strong-field physics.
An integral equation formulation for rigid bodies in Stokes flow in three dimensions
NASA Astrophysics Data System (ADS)
Corona, Eduardo; Greengard, Leslie; Rachh, Manas; Veerapaneni, Shravan
2017-03-01
We present a new derivation of a boundary integral equation (BIE) for simulating the three-dimensional dynamics of arbitrarily-shaped rigid particles of genus zero immersed in a Stokes fluid, on which are prescribed forces and torques. Our method is based on a single-layer representation and leads to a simple second-kind integral equation. It avoids the use of auxiliary sources within each particle that play a role in some classical formulations. We use a spectrally accurate quadrature scheme to evaluate the corresponding layer potentials, so that only a small number of spatial discretization points per particle are required. The resulting discrete sums are computed in O (n) time, where n denotes the number of particles, using the fast multipole method (FMM). The particle positions and orientations are updated by a high-order time-stepping scheme. We illustrate the accuracy, conditioning and scaling of our solvers with several numerical examples.
{{{Z}}_{N}} graded discrete Lax pairs and integrable difference equations
NASA Astrophysics Data System (ADS)
Fordy, Allan P.; Xenitidis, Pavlos
2017-04-01
We introduce a class of {{{Z}}N} graded discrete Lax pairs, with N× N matrices, linear in the spectral parameter. We give a classification scheme for such Lax pairs and the associated integrable lattice systems. We present two potential forms and completely classify the generic case. Many well known examples belong to our scheme for N = 2, so many of our systems may be regarded as generalisations of these. Even at N = 3, several new integrable systems arise. A decomposable case gives rise to interesting coupled systems of lower dimensional equations. Many of our equations are mutually compatible, so can be used together to form ‘coloured’ lattices.
Global integral gradient bounds for quasilinear equations below or near the natural exponent
NASA Astrophysics Data System (ADS)
Phuc, Nguyen Cong
2014-10-01
We obtain sharp integral potential bounds for gradients of solutions to a wide class of quasilinear elliptic equations with measure data. Our estimates are global over bounded domains that satisfy a mild exterior capacitary density condition. They are obtained in Lorentz spaces whose degrees of integrability lie below or near the natural exponent of the operator involved. As a consequence, nonlinear Calderón-Zygmund type estimates below the natural exponent are also obtained for -superharmonic functions in the whole space ℝ n . This answers a question raised in our earlier work (On Calderón-Zygmund theory for p- and -superharmonic functions, to appear in Calc. Var. Partial Differential Equations, DOI 10.1007/s00526-011-0478-8) and thus greatly improves the result there.
Romá, Federico; Cugliandolo, Leticia F; Lozano, Gustavo S
2014-08-01
We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.
A fast and well-conditioned spectral method for singular integral equations
NASA Astrophysics Data System (ADS)
Slevinsky, Richard Mikael; Olver, Sheehan
2017-03-01
We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This is accomplished by utilizing low rank approximations for sparse representations of the bivariate kernels. The resulting system can be solved in O (m2 n) operations using an adaptive QR factorization, where m is the bandwidth and n is the optimal number of unknowns needed to resolve the true solution. The complexity is reduced to O (mn) operations by pre-caching the QR factorization when the same operator is used for multiple right-hand sides. Stability is proved by showing that the resulting linear operator can be diagonally preconditioned to be a compact perturbation of the identity. Applications considered include the Faraday cage, and acoustic scattering for the Helmholtz and gravity Helmholtz equations, including spectrally accurate numerical evaluation of the far- and near-field solution. The JULIA software package SingularIntegralEquations.jl implements our method with a convenient, user-friendly interface.
NASA Astrophysics Data System (ADS)
Jiang, Tian; Zhang, Yong-Tao
2016-04-01
Implicit integration factor (IIF) methods were developed in the literature for solving time-dependent stiff partial differential equations (PDEs). Recently, IIF methods were combined with weighted essentially non-oscillatory (WENO) schemes in Jiang and Zhang (2013) [19] to efficiently solve stiff nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the non-stiff hyperbolic part of the system. To efficiently calculate large matrix exponentials, Krylov subspace approximation is directly applied to the implicit integration factor (IIF) methods. So far, the IIF methods developed in the literature are multistep methods. In this paper, we develop Krylov single-step IIF-WENO methods for solving stiff advection-diffusion-reaction equations. The methods are designed carefully to avoid generating positive exponentials in the matrix exponentials, which is necessary for the stability of the schemes. We analyze the stability and truncation errors of the single-step IIF schemes. Numerical examples of both scalar equations and systems are shown to demonstrate the accuracy, efficiency and robustness of the new methods.
On Generating Discrete Integrable Systems via Lie Algebras and Commutator Equations
NASA Astrophysics Data System (ADS)
Zhang, Yu-Feng; Tam, Honwah
2016-03-01
In the paper, we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly. By the approach the various loop algebras of the Lie algebra A1 are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained, respectively. A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy. Finally, via two different enlarging Lie algebras of the Lie algebra A1, we derive two resulting differential-difference integrable couplings of the Toda hierarchy, of course, they are all various discrete expanding integrable models of the Toda hierarchy. When the introduced spectral matrices are higher degrees, the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple. Supported by the National Natural Science Foundation of China under Grant No. 11371361, the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and Hong Kong Research Grant Council under Grant No. HKBU202512, as well as the Natural Science Foundation of Shandong Province under Grant No. ZR2013AL016
NASA Astrophysics Data System (ADS)
Bomont, Jean-Marc; Pastore, Giorgio
2015-09-01
We propose and discuss a straightforward search protocol for the glass-like solutions of the integral equations of the two-replica approach to the random first-order transition theory of the liquid-glass transition. The new numerical strategy supplements those recently introduced by Jean-Pierre Hansen and ourselves. A few results for inverse power (1/r12) fluid are discussed and critically compared with results from other approaches.
NASA Technical Reports Server (NTRS)
Jones, J. E.; Richmond, J. H.
1974-01-01
An integral equation formulation is applied to predict pitch- and roll-plane radiation patterns of a thin VHF/UHF (very high frequency/ultra high frequency) annular slot communications antenna operating at several locations in the nose region of the space shuttle orbiter. Digital computer programs used to compute radiation patterns are given and the use of the programs is illustrated. Experimental verification of computed patterns is given from measurements made on 1/35-scale models of the orbiter.
On the Numerical Solution of the Integral Equation Formulation for Transient Structural Synthesis
2014-09-01
history of integral equations dates back to the early nineteenth century when the profound mathematical insights of Newton and Leibniz were being...matrix. As shown in [10], the element stiffness matrix is as follows: 2 2 3 2 2 12 6 12 6 6 4 6 2 12 6 12 6 6 2 6 4 e l l l l l lEI K l ll l l l l
NASA Technical Reports Server (NTRS)
Hu, Fang Q.; Pizzo, Michelle E.; Nark, Douglas M.
2016-01-01
Based on the time domain boundary integral equation formulation of the linear convective wave equation, a computational tool dubbed Time Domain Fast Acoustic Scattering Toolkit (TD-FAST) has recently been under development. The time domain approach has a distinct advantage that the solutions at all frequencies are obtained in a single computation. In this paper, the formulation of the integral equation, as well as its stabilization by the Burton-Miller type reformulation, is extended to cases of a constant mean flow in an arbitrary direction. In addition, a "Source Surface" is also introduced in the formulation that can be employed to encapsulate regions of noise sources and to facilitate coupling with CFD simulations. This is particularly useful for applications where the noise sources are not easily described by analytical source terms. Numerical examples are presented to assess the accuracy of the formulation, including a computation of noise shielding by a thin barrier motivated by recent Historical Baseline F31A31 open rotor noise shielding experiments. Furthermore, spatial resolution requirements of the time domain boundary element method are also assessed using point per wavelength metrics. It is found that, using only constant basis functions and high-order quadrature for surface integration, relative errors of less than 2% may be obtained when the surface spatial resolution is 5 points-per-wavelength (PPW) or 25 points-per-wavelength squared (PPW2).
Villena, Jorge Fernandez; Polimeridis, Athanasios G; Eryaman, Yigitcan; Adalsteinsson, Elfar; Wald, Lawrence L; White, Jacob K; Daniel, Luca
2016-11-01
A fast frequency domain full-wave electromagnetic simulation method is introduced for the analysis of MRI coils loaded with the realistic human body models. The approach is based on integral equation methods decomposed into two domains: 1) the RF coil array and shield, and 2) the human body region where the load is placed. The analysis of multiple coil designs is accelerated by introducing the precomputed magnetic resonance Green functions (MRGFs), which describe how the particular body model used responds to the incident fields from external sources. These MRGFs, which are precomputed once for a given body model, can be combined with any integral equation solver and reused for the analysis of many coil designs. This approach provides a fast, yet comprehensive, analysis of coil designs, including the port S-parameters and the electromagnetic field distribution within the inhomogeneous body. The method solves the full-wave electromagnetic problem for a head array in few minutes, achieving a speed up of over 150 folds with root mean square errors in the electromagnetic field maps smaller than 0.4% when compared to the unaccelerated integral equation-based solver. This enables the characterization of a large number of RF coil designs in a reasonable time, which is a first step toward an automatic optimization of multiple parameters in the design of transmit arrays, as illustrated in this paper, but also receive arrays.
Numerical analysis of composite STEEL-CONCRETE SECTIONS using integral equation of Volterra
NASA Astrophysics Data System (ADS)
Partov, Doncho; Kantchev, Vesselin
2011-09-01
The paper presents analysis of the stress and deflections changes due to creep in statically determinate composite steel-concrete beam. The mathematical model involves the equation of equilibrium, compatibility and constitutive relationship, i.e. an elastic law for the steel part and an integral-type creep law of Boltzmann — Volterra for the concrete part. On the basis of the theory of the viscoelastic body of Arutyunian-Trost-Bažant for determining the redistribution of stresses in beam section between concrete plate and steel beam with respect to time "t", two independent Volterra integral equations of the second kind have been derived. Numerical method based on linear approximation of the singular kernal function in the integral equation is presented. Example with the model proposed is investigated. The creep functions is suggested by the model CEB MC90-99 and the "ACI 209R-92 model. The elastic modulus of concrete E c (t) is assumed to be constant in time `t'. The obtained results from the both models are compared.
Numerical analysis of composite STEEL-CONCRETE SECTIONS using integral equation of Volterra
NASA Astrophysics Data System (ADS)
Partov, Doncho; Kantchev, Vesselin
2011-09-01
The paper presents analysis of the stress and deflections changes due to creep in statically determinate composite steel-concrete beam. The mathematical model involves the equation of equilibrium, compatibility and constitutive relationship, i.e. an elastic law for the steel part and an integral-type creep law of Boltzmann — Volterra for the concrete part. On the basis of the theory of the viscoelastic body of Arutyunian-Trost-Bažant for determining the redistribution of stresses in beam section between concrete plate and steel beam with respect to time "t", two independent Volterra integral equations of the second kind have been derived. Numerical method based on linear approximation of the singular kernal function in the integral equation is presented. Example with the model proposed is investigated. The creep functions is suggested by the model CEB MC90-99 and the "ACI 209R-92 model. The elastic modulus of concrete Ec(t) is assumed to be constant in time `t'. The obtained results from the both models are compared.
NASA Technical Reports Server (NTRS)
Hu, Fang Q.
1994-01-01
It is known that the exact analytic solutions of wave scattering by a circular cylinder, when they exist, are not in a closed form but in infinite series which converges slowly for high frequency waves. In this paper, we present a fast number solution for the scattering problem in which the boundary integral equations, reformulated from the Helmholtz equation, are solved using a Fourier spectral method. It is shown that the special geometry considered here allows the implementation of the spectral method to be simple and very efficient. The present method differs from previous approaches in that the singularities of the integral kernels are removed and dealt with accurately. The proposed method preserves the spectral accuracy and is shown to have an exponential rate of convergence. Aspects of efficient implementation using FFT are discussed. Moreover, the boundary integral equations of combined single and double-layer representation are used in the present paper. This ensures the uniqueness of the numerical solution for the scattering problem at all frequencies. Although a strongly singular kernel is encountered for the Neumann boundary conditions, we show that the hypersingularity can be handled easily in the spectral method. Numerical examples that demonstrate the validity of the method are also presented.
Allometric equations for integrating remote sensing imagery into forest monitoring programmes.
Jucker, Tommaso; Caspersen, John; Chave, Jérôme; Antin, Cécile; Barbier, Nicolas; Bongers, Frans; Dalponte, Michele; van Ewijk, Karin Y; Forrester, David I; Haeni, Matthias; Higgins, Steven I; Holdaway, Robert J; Iida, Yoshiko; Lorimer, Craig; Marshall, Peter L; Momo, Stéphane; Moncrieff, Glenn R; Ploton, Pierre; Poorter, Lourens; Rahman, Kassim Abd; Schlund, Michael; Sonké, Bonaventure; Sterck, Frank J; Trugman, Anna T; Usoltsev, Vladimir A; Vanderwel, Mark C; Waldner, Peter; Wedeux, Beatrice M M; Wirth, Christian; Wöll, Hannsjörg; Woods, Murray; Xiang, Wenhua; Zimmermann, Niklaus E; Coomes, David A
2017-01-01
Remote sensing is revolutionizing the way we study forests, and recent technological advances mean we are now able - for the first time - to identify and measure the crown dimensions of individual trees from airborne imagery. Yet to make full use of these data for quantifying forest carbon stocks and dynamics, a new generation of allometric tools which have tree height and crown size at their centre are needed. Here, we compile a global database of 108753 trees for which stem diameter, height and crown diameter have all been measured, including 2395 trees harvested to measure aboveground biomass. Using this database, we develop general allometric models for estimating both the diameter and aboveground biomass of trees from attributes which can be remotely sensed - specifically height and crown diameter. We show that tree height and crown diameter jointly quantify the aboveground biomass of individual trees and find that a single equation predicts stem diameter from these two variables across the world's forests. These new allometric models provide an intuitive way of integrating remote sensing imagery into large-scale forest monitoring programmes and will be of key importance for parameterizing the next generation of dynamic vegetation models.
Solvation effects on chemical shifts by embedded cluster integral equation theory.
Frach, Roland; Kast, Stefan M
2014-12-11
The accurate computational prediction of nuclear magnetic resonance (NMR) parameters like chemical shifts represents a challenge if the species studied is immersed in strongly polarizing environments such as water. Common approaches to treating a solvent in the form of, e.g., the polarizable continuum model (PCM) ignore strong directional interactions such as H-bonds to the solvent which can have substantial impact on magnetic shieldings. We here present a computational methodology that accounts for atomic-level solvent effects on NMR parameters by extending the embedded cluster reference interaction site model (EC-RISM) integral equation theory to the prediction of chemical shifts of N-methylacetamide (NMA) in aqueous solution. We examine the influence of various so-called closure approximations of the underlying three-dimensional RISM theory as well as the impact of basis set size and different treatment of electrostatic solute-solvent interactions. We find considerable and systematic improvement over reference PCM and gas phase calculations. A smaller basis set in combination with a simple point charge model already yields good performance which can be further improved by employing exact electrostatic quantum-mechanical solute-solvent interaction energies. A larger basis set benefits more significantly from exact over point charge electrostatics, which can be related to differences of the solvent's charge distribution.
Schmidt, Matthew; Constable, Steve; Ing, Christopher; Roy, Pierre-Nicholas
2014-06-21
We developed and studied the implementation of trial wavefunctions in the newly proposed Langevin equation Path Integral Ground State (LePIGS) method [S. Constable, M. Schmidt, C. Ing, T. Zeng, and P.-N. Roy, J. Phys. Chem. A 117, 7461 (2013)]. The LePIGS method is based on the Path Integral Ground State (PIGS) formalism combined with Path Integral Molecular Dynamics sampling using a Langevin equation based sampling of the canonical distribution. This LePIGS method originally incorporated a trivial trial wavefunction, ψ{sub T}, equal to unity. The present paper assesses the effectiveness of three different trial wavefunctions on three isotopes of hydrogen for cluster sizes N = 4, 8, and 13. The trial wavefunctions of interest are the unity trial wavefunction used in the original LePIGS work, a Jastrow trial wavefunction that includes correlations due to hard-core repulsions, and a normal mode trial wavefunction that includes information on the equilibrium geometry. Based on this analysis, we opt for the Jastrow wavefunction to calculate energetic and structural properties for parahydrogen, orthodeuterium, and paratritium clusters of size N = 4 − 19, 33. Energetic and structural properties are obtained and compared to earlier work based on Monte Carlo PIGS simulations to study the accuracy of the proposed approach. The new results for paratritium clusters will serve as benchmark for future studies. This paper provides a detailed, yet general method for optimizing the necessary parameters required for the study of the ground state of a large variety of systems.
Computational attributes of the integral form of the equation of transfer
NASA Technical Reports Server (NTRS)
Frankel, J. I.
1991-01-01
Difficulties can arise in radiative and neutron transport calculations when a highly anisotropic scattering phase function is present. In the presence of anisotropy, currently used numerical solutions are based on the integro-differential form of the linearized Boltzmann transport equation. This paper, departs from classical thought and presents an alternative numerical approach based on application of the integral form of the transport equation. Use of the integral formalism facilitates the following steps: a reduction in dimensionality of the system prior to discretization, the use of symbolic manipulation to augment the computational procedure, and the direct determination of key physical quantities which are derivable through the various Legendre moments of the intensity. The approach is developed in the context of radiative heat transfer in a plane-parallel geometry, and results are presented and compared with existing benchmark solutions. Encouraging results are presented to illustrate the potential of the integral formalism for computation. The integral formalism appears to possess several computational attributes which are well-suited to radiative and neutron transport calculations.
Classical integrability for beta-ensembles and general Fokker-Planck equations
Rumanov, Igor
2015-01-15
Beta-ensembles of random matrices are naturally considered as quantum integrable systems, in particular, due to their relation with conformal field theory, and more recently appeared connection with quantized Painlevé Hamiltonians. Here, we demonstrate that, at least for even integer beta, these systems are classically integrable, e.g., there are Lax pairs associated with them, which we explicitly construct. To come to the result, we show that a solution of every Fokker-Planck equation in one space (and one time) dimensions can be considered as a component of an eigenvector of a Lax pair. The explicit finding of the Lax pair depends on finding a solution of a governing system–a closed system of two nonlinear partial differential equations (PDEs) of hydrodynamic type. This result suggests that there must be a solution for all values of beta. We find the solution of this system for even integer beta in the particular case of quantum Painlevé II related to the soft edge of the spectrum for beta-ensembles. The solution is given in terms of Calogero system of β/2 particles in an additional time-dependent potential. Thus, we find another situation where quantum integrability is reduced to classical integrability.
NASA Astrophysics Data System (ADS)
Dault, Daniel Lawrence
The moment method is the predominant approach for the solution of electromagnetic boundary integral equations. Traditional moment method discretizations rely on the projection of solution currents onto basis sets that must satisfy strict continuity properties to model physical currents. The choice of basis sets is further restricted by the tight coupling of traditional functional descriptions to the underlying geometrical approximation of the scattering or radiating body. As a result, the choice of approximation function spaces and geometry discretizations for a given boundary integral equation is significantly limited. A quasi-meshless partition of unity based method called the Generalized Method of Moments (GMM) was recently introduced to overcome some of these limitations. The GMM partition of unity scheme affords automatic continuity of solution currents, and therefore permits the use of a much wider range of basis functions than traditional moment methods. However, prior to the work in this thesis, GMM was limited in practical applicability because it was only formulated for a few geometry types, could not be accurately applied to arbitrary scatterers, e.g. those with mixtures of geometrical features, and was not amenable to traditional acceleration methodologies that would permit its application to electrically large problems. The primary contribution of this thesis is to introduce several new GMM formulations that significantly expand the capabilities of the method to make it a practical, broadly applicable approach for solving boundary integral equations and overcoming the limitations inherent in traditional moment method discretizations. Additionally, several of the topics covered address continuing open problems in electromagnetic boundary integral equations with applicability beyond GMM. The work comprises five broad contributions. The first is a new GMM formulation capable of mixing both GMM-type basis sets and traditional basis sets in the same
A Course in Integrated Studies.
ERIC Educational Resources Information Center
Wasp, David
1980-01-01
The article describes how an integrated studies course at an English secondary school is helping to meet the multicultural needs of handicapped students. The 2-year course covers English, history, geography, religious education, and social studies and comprises about half of the students' day. (PHR)
Lee, Yong Woo; Lee, Duck Joo
2014-12-01
Kirchhoff's formula for the convective wave equation is derived using the generalized function theory. The generalized convective wave equation for a stationary surface is obtained, and the integral formulation, the convective Kirchhoff's formula, is derived. The formula has a similar form to the classical Kirchhoff's formula, but an additional term appears due to a moving medium effect. For convenience, the additional term is manipulated to a final form as the classical Kirchhoff's formula. The frequency domain boundary integral can be obtained from the current time domain boundary integral form. The derived formula is verified by comparison with the analytic solution of source in the uniform flow. The formula is also utilized as a boundary integral equation. Time domain boundary element method (BEM) analysis using the boundary integral equation is conducted, and the results show good agreement with the analytical solution. The formula derived here can be useful for sound radiation and scattering by arbitrary bodies in a moving medium in the time domain.
White, J.; Phillips, J.R.; Korsmeyer, T.
1994-12-31
Mixed first- and second-kind surface integral equations with (1/r) and {partial_derivative}/{partial_derivative} (1/r) kernels are generated by a variety of three-dimensional engineering problems. For such problems, Nystroem type algorithms can not be used directly, but an expansion for the unknown, rather than for the entire integrand, can be assumed and the product of the singular kernal and the unknown integrated analytically. Combining such an approach with a Galerkin or collocation scheme for computing the expansion coefficients is a general approach, but generates dense matrix problems. Recently developed fast algorithms for solving these dense matrix problems have been based on multipole-accelerated iterative methods, in which the fast multipole algorithm is used to rapidly compute the matrix-vector products in a Krylov-subspace based iterative method. Another approach to rapidly computing the dense matrix-vector products associated with discretized integral equations follows more along the lines of a multigrid algorithm, and involves projecting the surface unknowns onto a regular grid, then computing using the grid, and finally interpolating the results from the regular grid back to the surfaces. Here, the authors describe a precorrectted-FFT approach which can replace the fast multipole algorithm for accelerating the dense matrix-vector product associated with discretized potential integral equations. The precorrected-FFT method, described below, is an order n log(n) algorithm, and is asymptotically slower than the order n fast multipole algorithm. However, initial experimental results indicate the method may have a significant constant factor advantage for a variety of engineering problems.
Scattering by cometary dust using the volume-integral-equation method
NASA Astrophysics Data System (ADS)
Markkanen, J.; Penttilä, A.; Muinonen, K.
2014-07-01
Numerical computations of light scattering by nonspherical particles are of great importance when deriving physical properties of cometary dust, such as the size distribution, structure, and composition, from the characteristics of scattered light. Optical observations of scattering by cometary dust are most consistent with particle models composed of aggregates of submicron monomers [1] or irregularly shaped particles [2]. These models can reproduce, at least to an extent, the typical scattering features of cometary dust, e.g., the negative polarization near the backscattering direction and the weak increase of the backscattering intensity. To simulate light scattering by aggregates of particles with sizes comparable to the wavelength, a full numerical solution for the Maxwell equations is required. For the simulations of light scattering by cometary dust, the most commonly applied numerical tool is the superposition T-matrix method which is applicable to scattering problems involving aggregates of spherical particles. If constituents are inhomogeneous or non-spherical, the discrete-dipole-approximation (DDA) technique is usually applied. In the DDA solution, however, some accuracy problems have been reported, especially, near the backscattering direction. Here, we present a novel discretization scheme for the volume integral equation of electromagnetic scattering. The numerical method is based on the electric polarization current volume-integral-equation formulation (JVIE) [3]. The JVIE equation is bounded from L^2 (the vector space of the square integrable functions) to itself, hence the Galerkin method together with L^2-conforming basis functions provides an optimal convergence of the solution [4]. The integral equation is discretized with piecewise linear basis functions associated with tetrahedral elements, and an FFT-based fast algorithm is used for accelerating the matrix-vector multiplication in the solution process. Thus, the computational complexity is
Closed-form integrator for the quaternion (euler angle) kinematics equations
NASA Technical Reports Server (NTRS)
Whitmore, Stephen A. (Inventor)
2000-01-01
The invention is embodied in a method of integrating kinematics equations for updating a set of vehicle attitude angles of a vehicle using 3-dimensional angular velocities of the vehicle, which includes computing an integrating factor matrix from quantities corresponding to the 3-dimensional angular velocities, computing a total integrated angular rate from the quantities corresponding to a 3-dimensional angular velocities, computing a state transition matrix as a sum of (a) a first complementary function of the total integrated angular rate and (b) the integrating factor matrix multiplied by a second complementary function of the total integrated angular rate, and updating the set of vehicle attitude angles using the state transition matrix. Preferably, the method further includes computing a quanternion vector from the quantities corresponding to the 3-dimensional angular velocities, in which case the updating of the set of vehicle attitude angles using the state transition matrix is carried out by (a) updating the quanternion vector by multiplying the quanternion vector by the state transition matrix to produce an updated quanternion vector and (b) computing an updated set of vehicle attitude angles from the updated quanternion vector. The first and second trigonometric functions are complementary, such as a sine and a cosine. The quantities corresponding to the 3-dimensional angular velocities include respective averages of the 3-dimensional angular velocities over plural time frames. The updating of the quanternion vector preserves the norm of the vector, whereby the updated set of vehicle attitude angles are virtually error-free.
Solution of fractional kinetic equation by a class of integral transform of pathway type
NASA Astrophysics Data System (ADS)
Kumar, Dilip
2013-04-01
Solutions of fractional kinetic equations are obtained through an integral transform named Pα-transform introduced in this paper. The Pα-transform is a binomial type transform containing many class of transforms including the well known Laplace transform. The paper is motivated by the idea of pathway model introduced by Mathai [Linear Algebra Appl. 396, 317-328 (2005), 10.1016/j.laa.2004.09.022]. The composition of the transform with differential and integral operators are proved along with convolution theorem. As an illustration of applications to the general theory of differential equations, a simple differential equation is solved by the new transform. Being a new transform, the Pα-transform of some elementary functions as well as some generalized special functions such as H-function, G-function, Wright generalized hypergeometric function, generalized hypergeometric function, and Mittag-Leffler function are also obtained. The results for the classical Laplace transform is retrieved by letting α → 1.
Advances in numerical solutions to integral equations in liquid state theory
NASA Astrophysics Data System (ADS)
Howard, Jesse J.
Solvent effects play a vital role in the accurate description of the free energy profile for solution phase chemical and structural processes. The inclusion of solvent effects in any meaningful theoretical model however, has proven to be a formidable task. Generally, methods involving Poisson-Boltzmann (PB) theory and molecular dynamic (MD) simulations are used, but they either fail to accurately describe the solvent effects or require an exhaustive computation effort to overcome sampling problems. An alternative to these methods are the integral equations (IEs) of liquid state theory which have become more widely applicable due to recent advancements in the theory of interaction site fluids and the numerical methods to solve the equations. In this work a new numerical method is developed based on a Newton-type scheme coupled with Picard/MDIIS routines. To extend the range of these numerical methods to large-scale data systems, the size of the Jacobian is reduced using basis functions, and the Newton steps are calculated using a GMRes solver. The method is then applied to calculate solutions to the 3D reference interaction site model (RISM) IEs of statistical mechanics, which are derived from first principles, for a solute model of a pair of parallel graphene plates at various separations in pure water. The 3D IEs are then extended to electrostatic models using an exact treatment of the long-range Coulomb interactions for negatively charged walls and DNA duplexes in aqueous electrolyte solutions to calculate the density profiles and solution thermodynamics. It is found that the 3D-IEs provide a qualitative description of the density distributions of the solvent species when compared to MD results, but at a much reduced computational effort in comparison to MD simulations. The thermodynamics of the solvated systems are also qualitatively reproduced by the IE results. The findings of this work show the IEs to be a valuable tool for the study and prediction of
Coherent optical feedback for the analog solution of partial differential and integral equations
NASA Astrophysics Data System (ADS)
Cederquist, J. N.
1980-12-01
To extend and improve the capabilities of optical information processing systems, the use of coherent optical feedback was investigated. A confocal feedback system based on the confocal Fabry-Perot interferometer was developed and shown to have a very flexible, complex-valued coherent transfer function unattainable without feedback. This system was used to solve the three types of second order linear partial differential equations in two dimensions-elliptic, hyperbolic, and parabolic- for a variety of inhomogeneous terms and boundary and initial conditions. Space-variant image plane filters were used to allow the solution of partial differential equations with variable coefficients. An optical flat with a small wedge angle was added to perform time sampling of the feedback signal. The resulting system can then solve partial differential equations in three dimensions. A second confocal Fabry-Perot interferometer was placed inside the first to produce multiple feedback. Time sampling was also combined with multiple feedback to create a system capable of solving four dimensional problems. Methods for the solution of Fredholm and Voltera integral equations are also discussed.
NASA Technical Reports Server (NTRS)
Clarke, R.; Lintereur, L.; Bahm, C.
2016-01-01
A desire for more complete documentation of the National Aeronautics and Space Administration (NASA) Armstrong Flight Research Center (AFRC), Edwards, California legacy code used in the core simulation has led to this e ort to fully document the oblate Earth six-degree-of-freedom equations of motion and integration algorithm. The authors of this report have taken much of the earlier work of the simulation engineering group and used it as a jumping-o point for this report. The largest addition this report makes is that each element of the equations of motion is traced back to first principles and at no point is the reader forced to take an equation on faith alone. There are no discoveries of previously unknown principles contained in this report; this report is a collection and presentation of textbook principles. The value of this report is that those textbook principles are herein documented in standard nomenclature that matches the form of the computer code DERIVC. Previous handwritten notes are much of the backbone of this work, however, in almost every area, derivations are explicitly shown to assure the reader that the equations which make up the oblate Earth version of the computer routine, DERIVC, are correct.
New integration techniques for chemical kinetic rate equations. I - Efficiency comparison
NASA Technical Reports Server (NTRS)
Radhakrishnan, K.
1986-01-01
A comparison of the efficiency of several recently developed numerical techniques for solving chemical kinetic rate equations is presented. The solution procedures examined include two general-purpose codes, EPISODE and LSODE, developed as multipurpose differential equation solvers, and three specialzed codes, CHEMEQ, CREK1D, and GCKP84, developed specifically for chemical kinetics. The efficiency comparison is made by applying these codes to two practical combustion kinetics problems. Both problems describe adiabatic, constant-pressure, gas-phase chemical reactions and include all three combustion regimes: induction, heat release, and equilibration. The comparison shows that LSODE is the fastest routine currently available for solving chemical kinetic rate equations. An important finding is that an iterative solution of the algebraic enthalpy conservation equation for temperature can be significantly faster than evaluation of the temperature by integration of its time derivative. Significant increases in computational speed are realized by updating the reaction rate constants only when the temperature change exceeds an amount Delta-T that is problem dependent. An approximate expression for the automatic evaluation of Delta-T is presented and is shown to result in increased computational speed.
Du, Qi-Shi; Liu, Peng-Jun; Huang, Ri-Bo
2008-02-01
In this study the excess chemical potential of the integral equation theory, 3D-RISM-HNC [Q. Du, Q. Wei, J. Phys. Chem. B 107 (2003) 13463-13470], is visualized in three-dimensional form and localized at interaction sites of solute molecule. Taking the advantage of reference interaction site model (RISM), the calculation equations of chemical excess potential are reformulized according to the solute interaction sites s in molecular space. Consequently the solvation free energy is localized at every interaction site of solute molecule. For visualization of the 3D-RISM-HNC calculation results, the excess chemical potentials are described using radial and three-dimensional diagrams. It is found that the radial diagrams of the excess chemical potentials are more sensitive to the bridge functions than the radial diagrams of solvent site density distributions. The diagrams of average excess chemical potential provide useful information of solute-solvent electrostatic and van der Waals interactions. The local description of solvation free energy at active sites of solute in 3D-RISM-HNC may broaden the application scope of statistical mechanical integral equation theory in solution chemistry and life science.
Lin, Guang; Tartakovsky, Alexandre M.
2010-04-01
In this study, we solve the three-dimensional stochastic Darcy's equation and stochastic advection-diffusion-dispersion equation using a probabilistic collocation method (PCM) on sparse grids. Karhunen-Lo\\`{e}ve (KL) decomposition is employed to represent the three-dimensional log hydraulic conductivity $Y=\\ln K_s$. The numerical examples which demonstrate the convergence of PCM are presented. It appears that the faster convergence rate in the variance can be obtained by using the Jacobi-chaos representing the truncated Gaussian distributions than using the Hermite-chaos for the Gaussian distribution. The effect of dispersion coefficient on the mean and standard deviation of the hydraulic head and solute concentration is investigated. Additionally, we also study how the statistical properties of the hydraulic head and solute concentration vary while using different types of random distributions and different standard deviations of random hydraulic conductivity.
Integrability: mathematical methods for studying solitary waves theory
NASA Astrophysics Data System (ADS)
Wazwaz, Abdul-Majid
2014-03-01
In recent decades, substantial experimental research efforts have been devoted to linear and nonlinear physical phenomena. In particular, studies of integrable nonlinear equations in solitary waves theory have attracted intensive interest from mathematicians, with the principal goal of fostering the development of new methods, and physicists, who are seeking solutions that represent physical phenomena and to form a bridge between mathematical results and scientific structures. The aim for both groups is to build up our current understanding and facilitate future developments, develop more creative results and create new trends in the rapidly developing field of solitary waves. The notion of the integrability of certain partial differential equations occupies an important role in current and future trends, but a unified rigorous definition of the integrability of differential equations still does not exist. For example, an integrable model in the Painlevé sense may not be integrable in the Lax sense. The Painlevé sense indicates that the solution can be represented as a Laurent series in powers of some function that vanishes on an arbitrary surface with the possibility of truncating the Laurent series at finite powers of this function. The concept of Lax pairs introduces another meaning of the notion of integrability. The Lax pair formulates the integrability of nonlinear equation as the compatibility condition of two linear equations. However, it was shown by many researchers that the necessary integrability conditions are the existence of an infinite series of generalized symmetries or conservation laws for the given equation. The existence of multiple soliton solutions often indicates the integrability of the equation but other tests, such as the Painlevé test or the Lax pair, are necessary to confirm the integrability for any equation. In the context of completely integrable equations, studies are flourishing because these equations are able to describe the
Fedorov, Yuri E-mail: Chara.Pantazi@upc.edu; Pantazi, Chara E-mail: Chara.Pantazi@upc.edu
2014-03-15
We consider a family of genus 2 hyperelliptic curves of even order and obtain explicitly the systems of 5 linear ordinary differential equations for periods of the corresponding Abelian integrals of first, second, and third kind, as functions of some parameters of the curves. The systems can be regarded as extensions of the well-studied Picard–Fuchs equations for periods of complete integrals of first and second kind on odd hyperelliptic curves. The periods we consider are linear combinations of the action variables of several integrable systems, in particular the generalized Neumann system with polynomial separable potentials. Thus the solutions of the extended Picard–Fuchs equations can be used to study various properties of the actions.
Erguel, Ozguer; Guerel, Levent
2008-12-01
We present a novel stabilization procedure for accurate surface formulations of electromagnetic scattering problems involving three-dimensional dielectric objects with arbitrarily low contrasts. Conventional surface integral equations provide inaccurate results for the scattered fields when the contrast of the object is low, i.e., when the electromagnetic material parameters of the scatterer and the host medium are close to each other. We propose a stabilization procedure involving the extraction of nonradiating currents and rearrangement of the right-hand side of the equations using fictitious incident fields. Then, only the radiating currents are solved to calculate the scattered fields accurately. This technique can easily be applied to the existing implementations of conventional formulations, it requires negligible extra computational cost, and it is also appropriate for the solution of large problems with the multilevel fast multipole algorithm. We show that the stabilization leads to robust formulations that are valid even for the solutions of extremely low-contrast objects.
Matrix equilibration in method of moment solutions of surface integral equations
NASA Astrophysics Data System (ADS)
Kolundzija, Branko M.; Kostic, Milan M.
2014-12-01
Basic theory of matrix equilibration is presented, relating it to other techniques for decreasing the condition number of matrix equations obtained by the method of moments (MOM) applied to surface integral equations (SIEs). It is shown that matrix equilibration is a general technique that can be used for both (1) balancing field and source quantities in SIEs, which is used to decrease the condition number in the case of SIEs of mixed type and high contrast in material properties, and (2) scaling basis and test functions in MOM, which is used to decrease the condition number in the case of higher-order bases and patches of different sizes. In particular, it is demonstrated that a combination of such balancing and scaling can be performed using simple matrix equilibration based on magnitudes of diagonal elements and 2-norms of rows/columns of the MOM matrix.
Direct Solve of Electrically Large Integral Equations for Problem Sizes to 1M Unknowns
NASA Technical Reports Server (NTRS)
Shaeffer, John
2008-01-01
Matrix methods for solving integral equations via direct solve LU factorization are presently limited to weeks to months of very expensive supercomputer time for problems sizes of several hundred thousand unknowns. This report presents matrix LU factor solutions for electromagnetic scattering problems for problem sizes to one million unknowns with thousands of right hand sides that run in mere days on PC level hardware. This EM solution is accomplished by utilizing the numerical low rank nature of spatially blocked unknowns using the Adaptive Cross Approximation for compressing the rank deficient blocks of the system Z matrix, the L and U factors, the right hand side forcing function and the final current solution. This compressed matrix solution is applied to a frequency domain EM solution of Maxwell's equations using standard Method of Moments approach. Compressed matrix storage and operations count leads to orders of magnitude reduction in memory and run time.
NASA Astrophysics Data System (ADS)
Chakraborty, Rumpa; Mondal, Arpita; Gayen, R.
2016-10-01
In this paper, we present an alternative method to investigate scattering of water waves by a submerged thin vertical elastic plate in the context of linear theory. The plate is submerged either in deep water or in the water of uniform finite depth. Using the condition on the plate, together with the end conditions, the derivative of the velocity potential in the direction of normal to the plate is expressed in terms of a Green's function. This expression is compared with that obtained by employing Green's integral theorem to the scattered velocity potential and the Green's function for the fluid region. This produces a hypersingular integral equation of the first kind in the difference in potential across the plate. The reflection coefficients are computed using the solution of the hypersingular integral equation. We find good agreement when the results for these quantities are compared with those for a vertical elastic plate and submerged and partially immersed rigid plates. New results for the hydrodynamic force on the plate, the shear stress and the shear strain of the vertical elastic plate are also evaluated and represented graphically.
Phase integral approximation for coupled ordinary differential equations of the Schroedinger type
Skorupski, Andrzej A.
2008-05-15
Four generalizations of the phase integral approximation (PIA) to sets of ordinary differential equations of Schroedinger type [u{sub j}{sup ''}(x)+{sigma}{sub k=1}{sup N}R{sub jk}(x)u{sub k}(x)=0, j=1,2,...,N] are described. The recurrence relations for higher order corrections are given in a form valid to arbitrary order and for the matrix R(x)[{identical_to}(R{sub jk}(x))] either Hermitian or non-Hermitian. For Hermitian and negative definite R(x) matrices, a Wronskian conserving PIA theory is formulated, which generalizes Fulling's current conserving theory pertinent to positive definite R(x) matrices. The idea of a modification of the PIA, which is well known for one equation [u{sup ''}(x)+R(x)u(x)=0], is generalized to sets. A simplification of Wronskian or current conserving theories is proposed which in each order eliminates one integration from the formulas for higher order corrections. If the PIA is generated by a nondegenerate eigenvalue of the R(x) matrix, the eliminated integration is the only one present. In that case, the simplified theory becomes fully algorithmic and is generalized to non-Hermitian R(x) matrices. The general theory is illustrated by a few examples automatically generated by using the author's program in MATHEMATICA published in e-print arXiv:0710.5406 [math-ph].
Modeling AC ripple currents in HTS coated conductors by integral equations
NASA Astrophysics Data System (ADS)
Grilli, Francesco; Xu, Zhihan
2016-12-01
In several HTS applications, the superconducting tapes experience the simultaneous presence of DC and AC excitations. For example in high-current DC cables, where the transport current is not perfectly constant, but it exhibits some ripples at different frequencies introduced by the rectification process (AC-DC conversion). These ripples give rise to dissipation, whose magnitude and possible influence on the device's cooling requirements need to be evaluated. Here we report a study of the AC losses in a HTS coated conductor subjected to DC currents and AC ripples simultaneously. The modeling approach is based on an integral equation method for thin superconductors: the superconducting tape is modeled as a 1-D object with a non-linear resistivity, which includes the dependence of the critical current density Jc on the magnetic field. The model, implemented in a commercial finite-element program, runs very fast (the simulation of one AC cycle typically takes a few seconds on standard desktop workstation): this allows simulating a large number of cycles and estimating when the AC ripple losses stabilize to a constant value. The model is used to study the influence of the flux creep power index n on the stabilization speed and on the AC loss values, as well as the effect of using a field-dependent Jc instead of a constant one. The simulations confirm that the dissipation level should not be a practical concern in HTS DC cables. At the same time, however, they reveal a strong dependence of the results upon the power index n and the form of Jc , which spurs the question whether the power-law is the most suitable description of the superconductor's electrical behavior for this kind of analysis.
Orbit determination based on meteor observations using numerical integration of equations of motion
NASA Astrophysics Data System (ADS)
Dmitriev, Vasily; Lupovka, Valery; Gritsevich, Maria
2015-11-01
Recently, there has been a worldwide proliferation of instruments and networks dedicated to observing meteors, including airborne and future space-based monitoring systems . There has been a corresponding rapid rise in high quality data accumulating annually. In this paper, we present a method embodied in the open-source software program "Meteor Toolkit", which can effectively and accurately process these data in an automated mode and discover the pre-impact orbit and possibly the origin or parent body of a meteoroid or asteroid. The required input parameters are the topocentric pre-atmospheric velocity vector and the coordinates of the atmospheric entry point of the meteoroid, i.e. the beginning point of visual path of a meteor, in an Earth centered-Earth fixed coordinate system, the International Terrestrial Reference Frame (ITRF). Our method is based on strict coordinate transformation from the ITRF to an inertial reference frame and on numerical integration of the equations of motion for a perturbed two-body problem. Basic accelerations perturbing a meteoroid's orbit and their influence on the orbital elements are also studied and demonstrated. Our method is then compared with several published studies that utilized variations of a traditional analytical technique, the zenith attraction method, which corrects for the direction of the meteor's trajectory and its apparent velocity due to Earth's gravity. We then demonstrate the proposed technique on new observational data obtained from the Finnish Fireball Network (FFN) as well as on simulated data. In addition, we propose a method of analysis of error propagation, based on general rule of covariance transformation.
Orbit determination based on meteor observations using numerical integration of equations of motion
NASA Astrophysics Data System (ADS)
Dmitriev, V.; Lupovka, V.; Gritsevich, M.
2014-07-01
We review the definitions and approaches to orbital-characteristics analysis applied to photographic or video ground-based observations of meteors. A number of camera networks dedicated to meteors registration were established all over the word, including USA, Canada, Central Europe, Australia, Spain, Finland and Poland. Many of these networks are currently operational. The meteor observations are conducted from different locations hosting the network stations. Each station is equipped with at least one camera for continuous monitoring of the firmament (except possible weather restrictions). For registered multi-station meteors, it is possible to accurately determine the direction and absolute value for the meteor velocity and thus obtain the topocentric radiant. Based on topocentric radiant one further determines the heliocentric meteor orbit. We aim to reduce total uncertainty in our orbit-determination technique, keeping it even less than the accuracy of observations. The additional corrections for the zenith attraction are widely in use and are implemented, for example, here [1]. We propose a technique for meteor-orbit determination with higher accuracy. We transform the topocentric radiant in inertial (J2000) coordinate system using the model recommended by IAU [2]. The main difference if compared to the existing orbit-determination techniques is integration of ordinary differential equations of motion instead of addition correction in visible velocity for zenith attraction. The attraction of the central body (the Sun), the perturbations by Earth, Moon and other planets of the Solar System, the Earth's flattening (important in the initial moment of integration, i.e. at the moment when a meteoroid enters the atmosphere), atmospheric drag may be optionally included in the equations. In addition, reverse integration of the same equations can be performed to analyze orbital evolution preceding to meteoroid's collision with Earth. To demonstrate the developed
NASA Astrophysics Data System (ADS)
Castro-Ramos, Jorge; Alejandro Juárez-Reyes, Salvador; Marcelino-Aranda, Mariana; Ortega-Vidals, Paula; Silva-Ortigoza, Gilberto; Silva-Ortigoza, Ramón; Suárez-Xique, Román
2015-01-01
In this work we assume that we have two given optical media with constant refraction indexes, which are separated by an arbitrary refracting surface. In one of the optical media we place a point light source at an arbitrary position. The aim of this work is to use a particular complete integral of the eikonal equation and Huygens’ principle to obtain the refraction and reflection laws. We remark that this complete integral associates a new point light source with each light ray that arrives at the refracting surface. This means that by using only this complete integral it is not possible to determine the direction of propagation of the refracted light rays; the direction of propagation is obtained by imposing two extra conditions on the complete integral which are equivalent to Huygens’ principle (in two dimensions, only one condition is needed). Finally, we establish the connection between the complete integral used here and that derived by using the k-function procedure introduced by Stavroudis, which works with plane wavefronts instead of spherical ones.
Martingale integrals over Poissonian processes and the Ito-type equations with white shot noise
NASA Astrophysics Data System (ADS)
Zygadło, Ryszard
2003-10-01
The construction of the Ito-type stochastic integrals and differential equations for compound Poisson processes is provided. The general martingale and nonanticipating properties of the ordinary (Gaussian) Ito theory are conserved. These properties appear particularly important if the stochastic description has to be proposed according to game theory or the linear relaxation (or the exponential growth) requirements. In contrast to the ordinary Ito theory the (uncorrelated) parametric fluctuation of a definite sign can be still modeled by asymmetric white shot noise, so the general scope of applications is not restricted by the positivity requirements. The possible use of the developed formalism in econophysics is addressed.
Steady and unsteady three-dimensional transonic flow computations by integral equation method
NASA Technical Reports Server (NTRS)
Hu, Hong
1994-01-01
This is the final technical report of the research performed under the grant: NAG1-1170, from the National Aeronautics and Space Administration. The report consists of three parts. The first part presents the work on unsteady flows around a zero-thickness wing. The second part presents the work on steady flows around non-zero thickness wings. The third part presents the massively parallel processing implementation and performance analysis of integral equation computations. At the end of the report, publications resulting from this grant are listed and attached.
NASA Astrophysics Data System (ADS)
Artoun, Ojenie; David-Rus, Diana; Emmett, Matthew; Fishman, Lou; Fital, Sandra; Hogan, Chad; Lim, Jisun; Lushi, Enkeleida; Marinov, Vesselin
2006-05-01
In this report we summarize an extension of Fourier analysis for the solution of the wave equation with a non-constant coefficient corresponding to an inhomogeneous medium. The underlying physics of the problem is exploited to link pseudodifferential operators and phase space path integrals to obtain a marching algorithm that incorporates the backward scattering into the evolution of the wave. This allows us to successfully apply single-sweep, one-way marching methods in inherently two-way environments, which was not achieved before through other methods for this problem.
NASA Technical Reports Server (NTRS)
Yates, E. Carson, Jr.
1990-01-01
Progress in the development of computational methods for steady and unsteady aerodynamics has perennially paced advancements in aeroelastic analysis and design capabilities. Since these capabilities are of growing importance in the analysis and design of high-performance aircraft, considerable effort has been directed toward the development of appropriate aerodynamic methodology. The contributions to those efforts from the integral-equations research program at the NASA Langley Research Center is reviewed. Specifically, the current scope, progress, and plans for research and development for inviscid and viscous flows are discussed, and example applications are shown in order to highlight the generality, versatility, and attractive features of this methodology.
Ayadim, A; Malherbe, J G; Amokrane, S
2005-06-15
The potential of mean force for uncharged macroparticles suspended in a fluid confined by a wall or a narrow pore is computed for solvent-wall and solvent-macroparticle interactions with attractive forces. Bridge functions taken from Rosenfeld's density-functional theory are used in the reference hypernetted chain closure of the Ornstein-Zernike integral equations. The quality of this closure is assessed by comparison with simulation. As an illustration, the role of solvation forces is investigated. When the "residual" attractive tails are given a range appropriate to "hard sphere-like" colloids, the unexpected role of solvation forces previously observed in bulk colloids is confirmed in the confinement situation.
NASA Technical Reports Server (NTRS)
Madsen, Niel K.
1992-01-01
Several new discrete surface integral (DSI) methods for solving Maxwell's equations in the time-domain are presented. These methods, which allow the use of general nonorthogonal mixed-polyhedral unstructured grids, are direct generalizations of the canonical staggered-grid finite difference method. These methods are conservative in that they locally preserve divergence or charge. Employing mixed polyhedral cells, (hexahedral, tetrahedral, etc.) these methods allow more accurate modeling of non-rectangular structures and objects because the traditional stair-stepped boundary approximations associated with the orthogonal grid based finite difference methods can be avoided. Numerical results demonstrating the accuracy of these new methods are presented.
Communication: An exact bound on the bridge function in integral equation theories
NASA Astrophysics Data System (ADS)
Kast, Stefan M.; Tomazic, Daniel
2012-11-01
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.
Communication: An exact bound on the bridge function in integral equation theories.
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.
NASA Astrophysics Data System (ADS)
Mesa, F.; Medina, F.
2006-12-01
This work presents a new implementation of the mixed potential integral equation (MPIE) for planar structures that can include ferrite layers arbitrarily magnetized. The implementation of the MPIE here reported is carried out in the space domain. Thus it will combine the well-known numerical advantages of working with potentials as well as the flexibility for analyzing nonrectangular shape conductors with the additional ability of including anisotropic layers of arbitrarily magnetized ferrites. In this way, our approach widens the scope of the space domain MPIE and sets this method as a very efficient and versatile numerical tool to deal with a wide class of planar microwave circuits and antennas.
NASA Technical Reports Server (NTRS)
Fymat, A. L.
1975-01-01
The determination of the microstructure, chemical nature, and dynamical evolution of scattering particulates in the atmosphere is considered. A description is given of indirect sampling techniques which can circumvent most of the difficulties associated with direct sampling techniques, taking into account methods based on scattering, extinction, and diffraction of an incident light beam. Approaches for reconstructing the particulate size distribution from the direct and the scattered radiation are discussed. A new method is proposed for determining the chemical composition of the particulates and attention is given to the relevance of methods of solution involving first kind Fredholm integral equations.
Solitons and dynamics for a general integrable nonlocal coupled nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Song, Cai-Qin; Xiao, Dong-Mei; Zhu, Zuo-Nong
2017-04-01
In this paper, we investigate a general integrable nonlocal coupled nonlinear Schrödinger (NLS) system with the parity-time (PT) symmetry, which contains not only the nonlocal self-phase modulation and the nonlocal cross-phase modulation, but also the nonlocal four-wave mixing terms. This nonlocal coupled NLS system is a nonlocal version of a coupled NLS system. The general N-th Darboux transformation for the nonlocal coupled NLS equation is constructed. By using the Darboux transformation, its soliton solutions are obtained. Dynamics and interactions of different kinds of soliton solutions are discussed.
A Fast Spectral Galerkin Method for Hypersingular Boundary Integral Equations in Potential Theory
Nintcheu Fata, Sylvain; Gray, Leonard J
2009-01-01
This research is focused on the development of a fast spectral method to accelerate the solution of three-dimensional hypersingular boundary integral equations of potential theory. Based on a Galerkin approximation, the Fast Fourier Transform and local interpolation operators, the proposed method is a generalization of the Precorrected-FFT technique to deal with double-layer potential kernels, hypersingular kernels and higher-order basis functions. Numerical examples utilizing piecewise linear shape functions are included to illustrate the performance of the method.
NASA Technical Reports Server (NTRS)
Bednarcyk, Brett A.; Aboudi, Jacob; Arnold, Steven M.
2006-01-01
The radial return and Mendelson methods for integrating the equations of classical plasticity, which appear independently in the literature, are shown to be identical. Both methods are presented in detail as are the specifics of their algorithmic implementation. Results illustrate the methods' equivalence across a range of conditions and address the question of when the methods require iteration in order for the plastic state to remain on the yield surface. FORTRAN code implementations of the radial return and Mendelson methods are provided in the appendix.
Martingale integrals over Poissonian processes and the Ito-type equations with white shot noise.
Zygadło, Ryszard
2003-10-01
The construction of the Ito-type stochastic integrals and differential equations for compound Poisson processes is provided. The general martingale and nonanticipating properties of the ordinary (Gaussian) Ito theory are conserved. These properties appear particularly important if the stochastic description has to be proposed according to game theory or the linear relaxation (or the exponential growth) requirements. In contrast to the ordinary Ito theory the (uncorrelated) parametric fluctuation of a definite sign can be still modeled by asymmetric white shot noise, so the general scope of applications is not restricted by the positivity requirements. The possible use of the developed formalism in econophysics is addressed.
Percolation of clusters with a residence time in the bond definition: Integral equation theory.
Zarragoicoechea, Guillermo J; Pugnaloni, Luis A; Lado, Fred; Lomba, Enrique; Vericat, Fernando
2005-03-01
We consider the clustering and percolation of continuum systems whose particles interact via the Lennard-Jones pair potential. A cluster definition is used according to which two particles are considered directly connected (bonded) at time t if they remain within a distance d, the connectivity distance, during at least a time of duration tau, the residence time. An integral equation for the corresponding pair connectedness function, recently proposed by two of the authors [Phys. Rev. E 61, R6067 (2000)], is solved using the orthogonal polynomial approach developed by another of the authors [Phys. Rev. E 55, 426 (1997)]. We compare our results with those obtained by molecular dynamics simulations.
Kamon, M.; Phillips, J.R.
1994-12-31
In this paper techniques are presented for preconditioning equations generated by discretizing constrained vector integral equations associated with magnetoquasistatic analysis. Standard preconditioning approaches often fail on these problems. The authors present a specialized preconditioning technique and prove convergence bounds independent of the constraint equations and electromagnetic excitation frequency. Computational results from analyzing several electronic packaging examples are given to demonstrate that the new preconditioning approach can sometimes reduce the number of GMRES iterations by more than an order of magnitude.
2014-06-01
ARKode is part of a software family called SUNDIALS: SUite of Nonlinear and Differential/ALgebraic equation Solvers [1]. The ARKode solver library provides an adaptive-step time integration package for stiff, nonstiff and multi-rate systems of ordinary differential equations (ODEs) using Runge Kutta methods [2].
Benchmark values for molecular three-center integrals arising in the Dirac equation
NASA Astrophysics Data System (ADS)
Baǧcı, A.; Hoggan, P. E.
2015-10-01
Previous papers by the authors report that they obtained compact, arbitrarily accurate expressions for two-center, one- and two-electron relativistic molecular integrals expressed over Slater-type orbitals. In the present study, accuracy limits of expressions given are examined for three-center nuclear attraction integrals, which are one-electron, three-center integrals with no analytically closed-form expression. In this work new molecular auxiliary functions are used. They are obtained via Neumann expansion of the Coulomb interaction. The numerical global adaptive method is used to evaluate these integrals for arbitrary values of orbital parameters and quantum numbers. Several methods, such as Laplace expansion of Coulomb interaction, single-center expansion, and the Fourier transformation method, have previously been used to evaluate these integrals considering the values of principal quantum numbers in the set of positive integer numbers. This study of three-center integrals places no restrictions on quantum numbers in all ranges of orbital parameters.
Benchmark values for molecular three-center integrals arising in the Dirac equation.
Bağcı, A; Hoggan, P E
2015-10-01
Previous papers by the authors report that they obtained compact, arbitrarily accurate expressions for two-center, one- and two-electron relativistic molecular integrals expressed over Slater-type orbitals. In the present study, accuracy limits of expressions given are examined for three-center nuclear attraction integrals, which are one-electron, three-center integrals with no analytically closed-form expression. In this work new molecular auxiliary functions are used. They are obtained via Neumann expansion of the Coulomb interaction. The numerical global adaptive method is used to evaluate these integrals for arbitrary values of orbital parameters and quantum numbers. Several methods, such as Laplace expansion of Coulomb interaction, single-center expansion, and the Fourier transformation method, have previously been used to evaluate these integrals considering the values of principal quantum numbers in the set of positive integer numbers. This study of three-center integrals places no restrictions on quantum numbers in all ranges of orbital parameters.
NASA Technical Reports Server (NTRS)
Goorjian, Peter M.; Taflove, Allen
1992-01-01
The initial results for femtosecond electromagnetic soliton propagation and collision obtained from first principles, i.e., by a direct time integration of Maxwell's equations are reported. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization and can take into account such quantum effects as Kerr and Raman interactions. The present approach is robust and should permit the modeling of 2D and 3D optical soliton propagation, scattering, and switching from the full-vector Maxwell's equations.
The anisotropic Ising correlations as elliptic integrals: duality and differential equations
NASA Astrophysics Data System (ADS)
McCoy, B. M.; Maillard, J.-M.
2016-10-01
We present the reduction of the correlation functions of the Ising model on the anisotropic square lattice to complete elliptic integrals of the first, second and third kind, the extension of Kramers-Wannier duality to anisotropic correlation functions, and the linear differential equations for these anisotropic correlations. More precisely, we show that the anisotropic correlation functions are homogeneous polynomials of the complete elliptic integrals of the first, second and third kind. We give the exact dual transformation matching the correlation functions and the dual correlation functions. We show that the linear differential operators annihilating the general two-point correlation functions are factorized in a very simple way, in operators of decreasing orders. Dedicated to A J Guttmann, for his 70th birthday.
Yangian symmetries and integrability of the Dirac equation with spin symmetry
Xu, Lei; Jing, Jian; Yuan, Zi-Gang; Kong, Ling-Bao; Long, Zheng-Wen
2013-02-15
We show that a Yangian symmetry, namely, Y(su(2)), exists in the Dirac equation with spin symmetry when the potential term takes a Coulomb form. We construct the generators of Y(su(2)) explicitly and get the energy spectrum of this model from the representation theory for Y(su(2)). We also show that this model is integrable, from RTT relations. - Highlights: Black-Right-Pointing-Pointer The Y(sl(2)) symmetry is found in a model, and the generators of Y(sl(2)) are constructed from the so(4) Lie algebra. Black-Right-Pointing-Pointer The energy spectrum is derived on the basis of the representation theory for Y(sl(2)). Black-Right-Pointing-Pointer The integrability of this model is proved from the RTT relation.
Fast Spectral Collocation Method for Surface Integral Equations of Potential Problems in a Spheroid
Xu, Zhenli; Cai, Wei
2009-01-01
This paper proposes a new technique to speed up the computation of the matrix of spectral collocation discretizations of surface single and double layer operators over a spheroid. The layer densities are approximated by a spectral expansion of spherical harmonics and the spectral collocation method is then used to solve surface integral equations of potential problems in a spheroid. With the proposed technique, the computation cost of collocation matrix entries is reduced from 𝒪(M2N4) to 𝒪(MN4), where N2 is the number of spherical harmonics (i.e., size of the matrix) and M is the number of one-dimensional integration quadrature points. Numerical results demonstrate the spectral accuracy of the method. PMID:20414359
Variational volume integral equation method for electromagnetic scattering by irregular grains.
NASA Astrophysics Data System (ADS)
Peltoniemi, J. I.
1996-05-01
An improved volume integral equation method for computing electromagnetic scattering by small particles is introduced. The features include: use of a variational technique (least squaring) instead of a collocation (point-matching) or Galerkin-method; use of vector spherical waves instead of pulse or linear ones; analytical second order treatment of the singularity and its immediate surroundings; and precise and effective quadratures. Comparisons to Lorenz-Mie theory, T-matrix method, DDA-code and experimental results are made. The new code is found to be significantly more accurate than other commonly used volume integral and DDA codes and does not require particle homogeneity and/or symmetry as some T-matrix codes do.
Integrated nonthermal treatment system study
Biagi, C.; Bahar, D.; Teheranian, B.; Vetromile, J.; Quapp, W.J.; Bechtold, T.; Brown, B.; Schwinkendorf, W.; Swartz, G.
1997-01-01
This report presents the results of a study of nonthermal treatment technologies. The study consisted of a systematic assessment of five nonthermal treatment alternatives. The treatment alternatives consist of widely varying technologies for safely destroying the hazardous organic components, reducing the volume, and preparing for final disposal of the contact-handled mixed low-level waste (MLLW) currently stored in the US Department of Energy complex. The alternatives considered were innovative nonthermal treatments for organic liquids and sludges, process residue, soil and debris. Vacuum desorption or various washing approaches are considered for treatment of soil, residue and debris. Organic destruction methods include mediated electrochemical oxidation, catalytic wet oxidation, and acid digestion. Other methods studied included stabilization technologies and mercury separation of treatment residues. This study is a companion to the integrated thermal treatment study which examined 19 alternatives for thermal treatment of MLLW waste. The quantities and physical and chemical compositions of the input waste are based on the inventory database developed by the US Department of Energy. The Integrated Nonthermal Treatment Systems (INTS) systems were evaluated using the same waste input (2,927 pounds per hour) as the Integrated Thermal Treatment Systems (ITTS). 48 refs., 68 figs., 37 tabs.
Higher-order time integration of Coulomb collisions in a plasma using Langevin equations
Dimits, A. M.; Cohen, B. I.; Caflisch, R. E.; ...
2013-02-08
The extension of Langevin-equation Monte-Carlo algorithms for Coulomb collisions from the conventional Euler-Maruyama time integration to the next higher order of accuracy, the Milstein scheme, has been developed, implemented, and tested. This extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the two fixed-frame spherical-coordinate velocity variables. Results from the numerical implementation show the expected improvement [O(Δt) vs. O(Δt1/2)] in the strong convergence rate both for the speed |v| and angular components of the scattering. An important result is that this improved convergence is achieved for the angular component of the scattering if andmore » only if the “area-integral” terms in the Milstein scheme are included. The resulting Milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency. These include both algorithms with improved convergence in the averages (weak convergence) and multi-time-level schemes. The latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional Monte-Carlo schemes, and their performance is improved considerably when the Milstein algorithm is used for the underlying time advance versus the Euler-Maruyama algorithm. A new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy. Lastly, this method, while being useful in its own right because of its relative simplicity, is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration.« less
Higher-order time integration of Coulomb collisions in a plasma using Langevin equations
Dimits, A. M.; Cohen, B. I.; Caflisch, R. E.; Rosin, M. S.; Ricketson, L. F.
2013-02-08
The extension of Langevin-equation Monte-Carlo algorithms for Coulomb collisions from the conventional Euler-Maruyama time integration to the next higher order of accuracy, the Milstein scheme, has been developed, implemented, and tested. This extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the two fixed-frame spherical-coordinate velocity variables. Results from the numerical implementation show the expected improvement [O(Δt) vs. O(Δt^{1/2})] in the strong convergence rate both for the speed |v| and angular components of the scattering. An important result is that this improved convergence is achieved for the angular component of the scattering if and only if the “area-integral” terms in the Milstein scheme are included. The resulting Milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency. These include both algorithms with improved convergence in the averages (weak convergence) and multi-time-level schemes. The latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional Monte-Carlo schemes, and their performance is improved considerably when the Milstein algorithm is used for the underlying time advance versus the Euler-Maruyama algorithm. A new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy. Lastly, this method, while being useful in its own right because of its relative simplicity, is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration.
Numerical study of fractional nonlinear Schrödinger equations.
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-12-08
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation.
NASA Astrophysics Data System (ADS)
Jiang, Tian; Zhang, Yong-Tao
2013-11-01
Implicit integration factor (IIF) methods are originally a class of efficient “exactly linear part” time discretization methods for solving time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. For complex systems (e.g. advection-diffusion-reaction (ADR) systems), the highest order derivative term can be nonlinear, and nonlinear nonstiff terms and nonlinear stiff terms are often mixed together. High order weighted essentially non-oscillatory (WENO) methods are often used to discretize the hyperbolic part in ADR systems. There are two open problems on IIF methods for solving ADR systems: (1) how to obtain higher than the second order global time discretization accuracy; (2) how to design IIF methods for solving fully nonlinear PDEs, i.e., the highest order terms are nonlinear. In this paper, we solve these two problems by developing new Krylov IIF-WENO methods to deal with both semilinear and fully nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the nonstiff hyperbolic part of the system. Large time step size computations are obtained. We analyze the stability and truncation errors of the schemes. Numerical examples of both scalar equations and systems in two and three spatial dimensions are shown to demonstrate the accuracy, efficiency and robustness of the methods.
NASA Astrophysics Data System (ADS)
Li, H.; Farthing, M. W.; Dawson, C. N.; Miller, C. T.
2004-12-01
Numerical simulation of Richards' equation continues to be difficult. It is highly nonlinear under common constitutive relations and exhibits sharp fronts in both the pressure head and volume fraction for many problems of interest. For a number of multiphase flow problems, the use of variable order and variable step size temporal discretizations has shown some advantages. However, the spatial discretizations commonly used for variably saturated flow are dominated by nonadaptive, low-order finite difference and finite element methods. Discontinuous Galerkin (DG) finite element methods have received significant attention in a number of fields for hyperbolic PDE's and, more recently, for elliptic and parabolic problems. DG approaches like the local discontinuous Galerkin (LDG) method are appealing for modeling subsurface flow since they can lead to velocity fields that are locally mass-conservative without the need for auxiliary variables or alternative meshes. DG discretizations are also inherently local and so better-suited for unstructured meshes and h-p adaption strategies than traditional methods. While some work has been done recently for multiphase subsurface flow, there are a range of issues related to the performance of DG methods for highly nonlinear parabolic problems like Richards' equation that have not been investigated fully. In this work, we consider the combination of higher order adaptive time integration with an LDG spatial discretization for Richards' equation. We compare this approach to standard low-order methods for a series of test problems and consider a number of issues including the methods' relative accuracy and computational efficiency.
Active flow control insight gained from a modified integral boundary layer equation
NASA Astrophysics Data System (ADS)
Seifert, Avraham
2016-11-01
Active Flow Control (AFC) can alter the development of boundary layers with applications (e.g., reducing drag by separation delay or separating the boundary layers and enhancing vortex shedding to increase drag). Historically, significant effects of steady AFC methods were observed. Unsteady actuation is significantly more efficient than steady. Full-scale AFC tests were conducted with varying levels of success. While clearly relevant to industry, AFC implementation relies on expert knowledge with proven intuition and or costly and lengthy computational efforts. This situation hinders the use of AFC while simple, quick and reliable design method is absent. An updated form of the unsteady integral boundary layer (UIBL) equations, that include AFC terms (unsteady wall transpiration and body forces) can be used to assist in AFC analysis and design. With these equations and given a family of suitable velocity profiles, the momentum thickness can be calculated and matched with an outer, potential flow solution in 2D and 3D manner to create an AFC design tool, parallel to proven tools for airfoil design. Limiting cases of the UIBL equation can be used to analyze candidate AFC concepts in terms of their capability to modify the boundary layers development and system performance.
NASA Astrophysics Data System (ADS)
Duh, Der-Ming; Henderson, Douglas
1996-05-01
The pure Lennard-Jones fluid and various binary mixtures of Lennard-Jones fluids are studied by both molecular dynamics simulation and with a new integral equation which is based on that proposed by Duh and Haymet recently [J. Chem. Phys. 103, 2625 (1995)]. The structural and thermodynamic properties calculated from this integral equation show excellent agreement with simulations for both pure fluids and mixtures under the conditions which we have studied. For mixtures, the effect of deviations from the Lorentz-Berthelot (LB) mixing rules for the interaction parameters between unlike species is studied. Positive deviations from the nonadditivity of the molecular cores leads to an entropy driven tendency for the species to separate. This tendency persists even in the presence of a deviation from the LB rule for the energy parameter which enhances the attraction of the unlike species. On the other hand, in the case of negative deviations from nonadditivity, the tendency for association may be either energy or entropy driven, depending on the size ratio.
Color path integral equation of state of the quark-gluon plasma at nonzero chemical potential
NASA Astrophysics Data System (ADS)
Filinov, V. S.; Bonitz, M.; Ivanov, Yu B.; Ilgenfritz, E.-M.; Fortov, V. E.
2015-04-01
Based on the constituent quasiparticle model of the quark-gluon plasma (QGP), a color quantum path-integral Monte-Carlo (PIMC) method for calculation of the thermodynamic properties of the QGP is developed. We show that the PIMC method can be used for calculations of the equation of state at zero and non-zero baryon chemical potential not only above but also below the QCD critical temperature. Our results agree with lattice QCD calculations based on a Taylor expansion around zero baryon chemical potential. In our approach the QGP partition function is presented in the form of a color path integral with a relativistic measure replacing the Gaussian one traditionally used in the Feynman-Wiener path integrals. A procedure of sampling color variables according to the SU(3) group Haar measure is used for integration over the color variables. We expect that this approach will be useful to predict additional properties of the QGP that are still unaccesible in lattice QCD.
NASA Astrophysics Data System (ADS)
Li, Jie; Dault, Daniel; Liu, Beibei; Tong, Yiying; Shanker, Balasubramaniam
2016-08-01
The analysis of electromagnetic scattering has long been performed on a discrete representation of the geometry. This representation is typically continuous but not differentiable. The need to define physical quantities on this geometric representation has led to development of sets of basis functions that need to satisfy constraints at the boundaries of the elements/tessellations (viz., continuity of normal or tangential components across element boundaries). For electromagnetics, these result in either curl/div-conforming basis sets. The geometric representation used for analysis is in stark contrast with that used for design, wherein the surface representation is higher order differentiable. Using this representation for both geometry and physics on geometry has several advantages, and is elucidated in Hughes et al. (2005) [7]. Until now, a bulk of the literature on isogeometric methods have been limited to solid mechanics, with some effort to create NURBS based basis functions for electromagnetic analysis. In this paper, we present the first complete isogeometry solution methodology for the electric field integral equation as applied to simply connected structures. This paper systematically proceeds through surface representation using subdivision, definition of vector basis functions on this surface, to fidelity in the solution of integral equations. We also present techniques to stabilize the solution at low frequencies, and impose a Calderón preconditioner. Several results presented serve to validate the proposed approach as well as demonstrate some of its capabilities.
An integral equation formulation for the diffraction from convex plates and polyhedra.
Asheim, Andreas; Svensson, U Peter
2013-06-01
A formulation of the problem of scattering from obstacles with edges is presented. The formulation is based on decomposing the field into geometrical acoustics, first-order, and multiple-order edge diffraction components. An existing secondary-source model for edge diffraction from finite edges is extended to handle multiple diffraction of all orders. It is shown that the multiple-order diffraction component can be found via the solution to an integral equation formulated on pairs of edge points. This gives what can be called an edge source signal. In a subsequent step, this edge source signal is propagated to yield a multiple-order diffracted field, taking all diffraction orders into account. Numerical experiments demonstrate accurate response for frequencies down to 0 for thin plates and a cube. No problems with irregular frequencies, as happen with the Kirchhoff-Helmholtz integral equation, are observed for this formulation. For the axisymmetric scattering from a circular disc, a highly effective symmetric formulation results, and results agree with reference solutions across the entire frequency range.
NASA Astrophysics Data System (ADS)
Chang, Ruinan
The ability to have a good understanding of and to manipulate electromagnetic elds has been increasingly important for many hardware technologies. There is a strong need for advanced numeric algorithms that yield fast and accuracy controllable solvers for electromagnetic and micromagnetic simulations. The first part of the dissertation presents methods constituting the core of the high-performance simulator FastMag. FastMag derives its high speed from three aspects. First, it leverages the state-of-the-art graphics processing unit computational architectures, which can be hundreds of times faster than a single central processing unit. Moreover, ecient and and accurate implementations of numeric quadrature was invoked. Thirdly, we provide an analytic method for Jacobian vector products. Some advanced features are provided in FastMag. Quadratic basis functions are used to provide better accuracy. Hexahedral elements were also implemented because they are more accurate, consume less memory. The second part of the dissertation is devoted to electromagnetic scattering problems. We developed new algorithms that signicantly improved the traditional methods. First of all, potential volume integral equations were implemented, where the potential quantities (vector and scalar potential). Another important contribution of this disertation is quadrilateral barycentric basis functions (QBBFs). The QBBFs can serve as a fundamental block for primary basis functions (PBFs) and dual basis functions (DBFs). The PBFs and DBFs, when applied in combination into traditional electric and magnetic eld integral equations (EFIE and MFIE), give rise to accurate and robust results. Moreover, the DBFs make the famous Calderon preconditioner multiplicative.
NASA Astrophysics Data System (ADS)
Vaneeva, Olena; Sophocleous, Christodoulos; Popovych, Roman; Boyko, Vyacheslav; Damianou, Pantelis
2015-06-01
The Seventh International Workshop "Group Analysis of Differential Equations and Integrable Systems" (GADEIS-VII) took place at Flamingo Beach Hotel, Larnaca, Cyprus during the period June 15-19, 2014. Fifty nine scientists from nineteen countries participated in the Workshop, and forty one lectures were presented. The Workshop topics ranged from theoretical developments of group analysis of differential equations, hypersymplectic structures, theory of Lie algebras, integrability and superintegrability to their applications in various fields. The Series of Workshops is a joint initiative by the Department of Mathematics and Statistics, University of Cyprus, and the Department of Applied Research of the Institute of Mathematics, National Academy of Sciences, Ukraine. The Workshops evolved from close collaboration among Cypriot and Ukrainian scientists. The first three meetings were held at the Athalassa campus of the University of Cyprus (October 27, 2005, September 25-28, 2006, and October 4-5, 2007). The fourth (October 26-30, 2008), the fifth (June 6-10, 2010) and the sixth (June 17-21, 2012) meetings were held at the coastal resort of Protaras. We would like to thank all the authors who have published papers in the Proceedings. All of the papers have been reviewed by at least two independent referees. We express our appreciation of the care taken by the referees. Their constructive suggestions have improved most of the papers. The importance of peer review in the maintenance of high standards of scientific research can never be overstated. Olena Vaneeva, Christodoulos Sophocleous, Roman Popovych, Vyacheslav Boyko, Pantelis Damianou
Integral equation analysis of single-site coarse-grained models for polymer-colloid mixtures
NASA Astrophysics Data System (ADS)
Menichetti, Roberto; D'Adamo, Giuseppe; Pelissetto, Andrea; Pierleoni, Carlo
2015-09-01
We discuss the reliability of integral equation methods based on several commonly used closure relations in determining the phase diagram of coarse-grained models of soft-matter systems characterised by mutually interacting soft- and hard-core particles. Specifically, we consider a set of potentials appropriate to describe a system of hard-sphere colloids and linear homopolymers in good solvent, and investigate the behaviour when the soft particles are smaller than the colloids, which is the regime of validity of the coarse-grained models. Using computer-simulation results as a benchmark, we find that the hypernetted-chain approximation provides accurate estimates of thermodynamics and structure in the colloid-gas phase in which the density of colloids is small. On the other hand, all closures considered appear to be unable to describe the behaviour of the mixture in the colloid-liquid phase, as they cease to converge at polymer densities significantly smaller than those at the binodal. As a consequence, integral equations appear to be unable to predict a quantitatively correct phase diagram.
NASA Technical Reports Server (NTRS)
Young, D. P.; Woo, A. C.; Bussoletti, J. E.; Johnson, F. T.
1986-01-01
A general method is developed combining fast direct methods and boundary integral equation methods to solve Poisson's equation on irregular exterior regions. The method requires O(N log N) operations where N is the number of grid points. Error estimates are given that hold for regions with corners and other boundary irregularities. Computational results are given in the context of computational aerodynamics for a two-dimensional lifting airfoil. Solutions of boundary integral equations for lifting and nonlifting aerodynamic configurations using preconditioned conjugate gradient are examined for varying degrees of thinness.
Theoretical study of the incompressible Navier-Stokes equations by the least-squares method
NASA Technical Reports Server (NTRS)
Jiang, Bo-Nan; Loh, Ching Y.; Povinelli, Louis A.
1994-01-01
Usually the theoretical analysis of the Navier-Stokes equations is conducted via the Galerkin method which leads to difficult saddle-point problems. This paper demonstrates that the least-squares method is a useful alternative tool for the theoretical study of partial differential equations since it leads to minimization problems which can often be treated by an elementary technique. The principal part of the Navier-Stokes equations in the first-order velocity-pressure-vorticity formulation consists of two div-curl systems, so the three-dimensional div-curl system is thoroughly studied at first. By introducing a dummy variable and by using the least-squares method, this paper shows that the div-curl system is properly determined and elliptic, and has a unique solution. The same technique then is employed to prove that the Stokes equations are properly determined and elliptic, and that four boundary conditions on a fixed boundary are required for three-dimensional problems. This paper also shows that under four combinations of non-standard boundary conditions the solution of the Stokes equations is unique. This paper emphasizes the application of the least-squares method and the div-curl method to derive a high-order version of differential equations and additional boundary conditions. In this paper, an elementary method (integration by parts) is used to prove Friedrichs' inequalities related to the div and curl operators which play an essential role in the analysis.
NASA Astrophysics Data System (ADS)
Liu, F.; Schaller, E.; Mott, D. R.
2005-08-01
A major task in many applications of atmospheric chemistry transport problems is the numerical integration of stiff systems of Ordinary Differential Equations (ODEs) describing the chemical transformations. A faster solver that is easier to couple to the other physics in the problem is still needed. The integration method, α-QSS, corresponding to the solver CHEMEQ2 aims at meeting the demands of a process-split, reacting-flow simulation (Mott 2000; Mott and Oran, 2001). However, this integrator has yet to be applied to the numerical integration of kinetic equations in tropospheric chemistry. A zero-dimensional (box) model is developed to test how well CHEMEQ2 works on the tropospheric chemistry equations. This paper presents the testing results. The reference chemical mechanisms herein used are Regional Atmospheric Chemistry Mechanism (RACM) (Stockwell et al., 1997) and its secondary lumped successor Regional Lumped Atmospheric Chemical Scheme (ReLACS) (Crassier et al., 2000). The box model is forced and initialized by the DRY scenarios of Protocol Ver. 2 developed by EUROTRAC (Poppe et al., 2001). The accuracy of CHEMEQ2 is evaluated by comparing the results to solutions obtained with VODE. This comparison is made with parameters of the error tolerance, relative difference with respect to VODE scheme, trade off between accuracy and efficiency, global time step for integration etc. The study based on the comparison concludes that the single-point α-QSS approach is fast and moderately accurate as well as easy to couple to reacting flow simulation models, which makes CHEMEQ2 one of the best candidates for three-dimensional atmospheric Chemistry Transport Modelling (CTM) studies. In addition the RACM mechanism may be replaced by ReLACS mechanism for tropospheric chemistry transport modelling. The testing results also imply that the accuracy for chemistry numerical simulations is highly different from species to species. Therefore ozone is not the good choice for
NASA Astrophysics Data System (ADS)
Singer, B. Sh.
2008-12-01
The paper presents a new code for modelling electromagnetic fields in complicated 3-D environments and provides examples of the code application. The code is based on an integral equation (IE) for the scattered electromagnetic field, presented in the form used by the Modified Iterative Dissipative Method (MIDM). This IE possesses contraction properties that allow it to be solved iteratively. As a result, for an arbitrary earth model and any source of the electromagnetic field, the sequence of approximations converges to the solution at any frequency. The system of linear equations that represents a finite-dimensional counterpart of the continuous IE is derived using a projection definition of the system matrix. According to this definition, the matrix is calculated by integrating the Green's function over the `source' and `receiver' cells of the numerical grid. Such a system preserves contraction properties of the continuous equation and can be solved using the same iterative technique. The condition number of the system matrix and, therefore, the convergence rate depends only on the physical properties of the model under consideration. In particular, these parameters remain independent of the numerical grid used for numerical simulation. Applied to the system of linear equations, the iterative perturbation approach generates a sequence of approximations, converging to the solution. The number of iterations is significantly reduced by finding the best possible approximant inside the Krylov subspace, which spans either all accumulated iterates or, if it is necessary to save the memory, only a limited number of the latest iterates. Optimization significantly reduces the number of iterates and weakens its dependence on the lateral contrast of the model. Unlike more traditional conjugate gradient approaches, the iterations are terminated when the approximate solution reaches the requested relative accuracy. The number of the required iterates, which for simple
NASA Astrophysics Data System (ADS)
Zhang, Xiang
2013-05-01
Llibre and Valls, in [Physica D, 241(2012) 1417-1420], proved that, if the Kirchoff equations have a proper Darboux polynomial with its cofactor satisfying some symmetry, they have a polynomial first integral. In this note we will improve this last result, and obtain that, if the Kirchoff equations have a proper Darboux polynomial, they always have a polynomial first integral functionally independent of the three known ones. Our result improves that of Llibre and Valls in two aspects: we drop the symmetric condition, and prove that the obtained first integral is functionally independent of the known ones.
Eshkuvatov, Z K; Zulkarnain, F S; Nik Long, N M A; Muminov, Z
2016-01-01
Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [-1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707-1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265-274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636-641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.
Al Khawaja, U.
2010-05-15
We derive the integrability conditions of nonautonomous nonlinear Schroedinger equations using the Lax pair and similarity transformation methods. We present a comparative analysis of these integrability conditions with those of the Painleve method. We show that while the Painleve integrability conditions restrict the dispersion, nonlinearity, and dissipation/gain coefficients to be space independent and the external potential to be only a quadratic function of position, the Lax Pair and the similarity transformation methods allow for space-dependent coefficients and an external potential that is not restricted to the quadratic form. The integrability conditions of the Painleve method are retrieved as a special case of our general integrability conditions. We also derive the integrability conditions of nonautonomous nonlinear Schroedinger equations for two- and three-spacial dimensions.
NASA Technical Reports Server (NTRS)
Mager, Arthur
1952-01-01
The Navier-Stokes equations of motion and the equation of continuity are transformed so as to apply to an orthogonal curvilinear coordinate system rotating with a uniform angular velocity about an arbitrary axis in space. A usual simplification of these equations as consistent with the accepted boundary-layer theory and an integration of these equations through the boundary layer result in boundary-layer momentum-integral equations for three-dimensional flows that are applicable to either rotating or nonrotating fluid boundaries. These equations are simplified and an approximate solution in closed integral form is obtained for a generalized boundary-layer momentum-loss thickness and flow deflection at the wall in the turbulent case. A numerical evaluation of this solution carried out for data obtained in a curving nonrotating duct shows a fair quantitative agreement with the measures values. The form in which the equations are presented is readily adaptable to cases of steady, three-dimensional, incompressible boundary-layer flow like that over curved ducts or yawed wings; and it also may be used to describe the boundary-layer flow over various rotating surfaces, thus applying to turbomachinery, propellers, and helicopter blades.
Using structural equation modeling to link human activities to wetland ecological integrity
Schweiger, E. William; Grace, James B.; Cooper, David; Bobowski, Ben; Britten, Mike
2016-01-01
The integrity of wetlands is of global concern. A common approach to evaluating ecological integrity involves bioassessment procedures that quantify the degree to which communities deviate from historical norms. While helpful, bioassessment provides little information about how altered conditions connect to community response. More detailed information is needed for conservation and restoration. We have illustrated an approach to addressing this challenge using structural equation modeling (SEM) and long-term monitoring data from Rocky Mountain National Park (RMNP). Wetlands in RMNP are threatened by a complex history of anthropogenic disturbance including direct alteration of hydrologic regimes; elimination of elk, wolves, and grizzly bears; reintroduction of elk (absent their primary predators); and the extirpation of beaver. More recently, nonnative moose were introduced to the region and have expanded into the park. Bioassessment suggests that up to half of the park's wetlands are not in reference condition. We developed and evaluated a general hypothesis about how human alterations influence wetland integrity and then develop a specific model using RMNP wetlands. Bioassessment revealed three bioindicators that appear to be highly sensitive to human disturbance (HD): (1) conservatism, (2) degree of invasion, and (3) cover of native forbs. SEM analyses suggest several ways human activities have impacted wetland integrity and the landscape of RMNP. First, degradation is highest where the combined effects of all types of direct HD have been the greatest (i.e., there is a general, overall effect). Second, specific HDs appear to create a “mixed-bag” of complex indirect effects, including reduced invasion and increased conservatism, but also reduced native forb cover. Some of these effects are associated with alterations to hydrologic regimes, while others are associated with altered shrub production. Third, landscape features created by historical beaver
Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps
Yuan Jianhua; Lu Yayan Antoine, Xavier
2008-04-20
Efficient numerical methods for analyzing photonic crystals (PhCs) can be developed using the Dirichlet-to-Neumann (DtN) maps of the unit cells. The DtN map is an operator that takes the wave field on the boundary of a unit cell to its normal derivative. In frequency domain calculations for band structures and transmission spectra of finite PhCs, the DtN maps allow us to reduce the computation to the boundaries of the unit cells. For two-dimensional (2D) PhCs with unit cells containing circular cylinders, the DtN maps can be constructed from analytic solutions (the cylindrical waves). In this paper, we develop a boundary integral equation method for computing DtN maps of general unit cells containing cylinders with arbitrary cross sections. The DtN map method is used to analyze band structures for 2D PhCs with elliptic and other cylinders.
Efficient solution of liquid state integral equations using the Newton-GMRES algorithm
NASA Astrophysics Data System (ADS)
Booth, Michael J.; Schlijper, A. G.; Scales, L. E.; Haymet, A. D. J.
1999-06-01
We present examples of the accurate, robust and efficient solution of Ornstein-Zernike type integral equations which describe the structure of both homogeneous and inhomogeneous fluids. In this work we use the Newton-GMRES algorithm as implemented in the public-domain nonlinear Krylov solvers NKSOL [ P. Brown, Y. Saad, SIAM J. Sci. Stat. Comput. 11 (1990) 450] and NITSOL [ M. Pernice, H.F. Walker, SIAM J. Sci. Comput. 19 (1998) 302]. We compare and contrast this method with more traditional approaches in the literature, using Picard iteration (successive-substitution) and hybrid Newton-Raphson and Picard methods, and a recent vector extrapolation method [ H.H.H. Homeier, S. Rast, H. Krienke, Comput. Phys. Commun. 92 (1995) 188]. We find that both the performance and ease of implementation of these nonlinear solvers recommend them for the solution of this class of problem.
Shafii, Mohammad Ali Meidianti, Rahma Wildian, Fitriyani, Dian; Tongkukut, Seni H. J.; Arkundato, Artoto
2014-09-30
Theoretical analysis of integral neutron transport equation using collision probability (CP) method with quadratic flux approach has been carried out. In general, the solution of the neutron transport using the CP method is performed with the flat flux approach. In this research, the CP method is implemented in the cylindrical nuclear fuel cell with the spatial of mesh being conducted into non flat flux approach. It means that the neutron flux at any point in the nuclear fuel cell are considered different each other followed the distribution pattern of quadratic flux. The result is presented here in the form of quadratic flux that is better understanding of the real condition in the cell calculation and as a starting point to be applied in computational calculation.
Utilization of integrated Michaelis-Menten equation to determine kinetic constants.
Bezerra, Rui M F; Dias, Albino A
2007-03-01
Students of biochemistry and related biosciences are urged to solve problems where kinetic parameters are calculated from initial rates obtained at different substrate concentrations. Troubles begin when they go to the laboratory to perform kinetic experiments and realize that usual laboratory instruments do not measure initial rates but only substrate or product concentrations as a function of reaction time. To overcome this problem we present a methodology which uses the integrated form of Michaelis-Menten equation. The method presented has a theoretical and pedagogic basis which is not as arbitrary as other approaches. Here we present and describe the methodology for analyzing time course data together with some examples of the essential computer procedures to implement these analyses. To simplify the understanding of this methodology the experimental examples are confined to linear inhibitions and experimental points utilized are the same from which the initial rates are determined.
NASA Technical Reports Server (NTRS)
Womble, M. E.; Potter, J. E.
1975-01-01
A prefiltering version of the Kalman filter is derived for both discrete and continuous measurements. The derivation consists of determining a single discrete measurement that is equivalent to either a time segment of continuous measurements or a set of discrete measurements. This prefiltering version of the Kalman filter easily handles numerical problems associated with rapid transients and ill-conditioned Riccati matrices. Therefore, the derived technique for extrapolating the Riccati matrix from one time to the next constitutes a new set of integration formulas which alleviate ill-conditioning problems associated with continuous Riccati equations. Furthermore, since a time segment of continuous measurements is converted into a single discrete measurement, Potter's square root formulas can be used to update the state estimate and its error covariance matrix. Therefore, if having the state estimate and its error covariance matrix at discrete times is acceptable, the prefilter extends square root filtering with all its advantages, to continuous measurement problems.
The lambda-scheme. [for numerical integration of Euler equation of compressible gas flow
NASA Technical Reports Server (NTRS)
Moretti, G.
1979-01-01
A method for integrating the Euler equations of gas dynamics for compressible flows in any hyperbolic case is presented. This method is applied to the Mach number distribution over a stretch of an infinite duct having a variable cross section, and to the distribution in a channel opening into a vacuum with the Mach number equalling 1.04. An example of the ability of this method to handle two-dimensional unsteady flows is shown using the steady shock-and-isobars pattern reached asymptotically about an ablated blunt body with a free stream Mach number equalling 12. A final example is presented where the technique is applied to a three-dimensional steady supersonic flow, with a Mach number of 2 and an angle of attack of 5 deg.
Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation
NASA Astrophysics Data System (ADS)
Bokhari, Ashfaque H.; Mahomed, F. M.; Zaman, F. D.
2010-05-01
The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.
NASA Astrophysics Data System (ADS)
Bomont, Jean-Marc; Pastore, Giorgio; Hansen, Jean-Pierre
2017-03-01
The pair structure, free energy, and configurational overlap order parameter Q of an annealed system of two weakly coupled replicas of a supercooled "soft sphere" fluid are determined by solving the hypernetted-chain (HNC) and self-consistent Rogers-Young (RY) integral equations over a wide range of thermodynamic conditions ρ (number-density), T (temperature), and inter-replicas couplings ɛ12. Analysis of the resulting effective (or Landau) potential W(ρ,T; Q) and of its derivative with respect to Q confirms the existence of a "precursor transition" between weak and strong overlap phases below a critical temperature Tc well above the temperature To of the "ideal glass" transition observed in the limit ɛ12→0 . The precursor transition is signalled by a loss of convexity of the potential W(Q) and by a concomitant discontinuity of the order parameter Q just below Tc, which crosses over to a mean-field-like van der Waals loop at lower temperatures. The HNC and RY equations lead to the same phase transition scenario, with quantitative differences in the predicted temperatures Tc and To.
Symmetries and integrability of a fourth-order Euler-Bernoulli beam equation
Bokhari, Ashfaque H.; Zaman, F. D.; Mahomed, F. M.
2010-05-15
The complete symmetry group classification of the fourth-order Euler-Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry.
NASA Technical Reports Server (NTRS)
Jothiprasad, Giridhar; Mavriplis, Dimitri J.; Caughey, David A.; Bushnell, Dennis M. (Technical Monitor)
2002-01-01
The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.
Solutions to Kuessner's integral equation in unsteady flow using local basis functions
NASA Technical Reports Server (NTRS)
Fromme, J. A.; Halstead, D. W.
1975-01-01
The computational procedure and numerical results are presented for a new method to solve Kuessner's integral equation in the case of subsonic compressible flow about harmonically oscillating planar surfaces with controls. Kuessner's equation is a linear transformation from pressure to normalwash. The unknown pressure is expanded in terms of prescribed basis functions and the unknown basis function coefficients are determined in the usual manner by satisfying the given normalwash distribution either collocationally or in the complex least squares sense. The present method of solution differs from previous ones in that the basis functions are defined in a continuous fashion over a relatively small portion of the aerodynamic surface and are zero elsewhere. This method, termed the local basis function method, combines the smoothness and accuracy of distribution methods with the simplicity and versatility of panel methods. Predictions by the local basis function method for unsteady flow are shown to be in excellent agreement with other methods. Also, potential improvements to the present method and extensions to more general classes of solutions are discussed.
Flatté, Stanley M; Vera, Michael D
2003-08-01
Line-integral approximations to the acoustic path integral have been used to estimate the magnitude of the fluctuations in an acoustic signal traveling through an ocean filled with internal waves. These approximations for the root-mean-square (rms) fluctuation and the bias of travel time, rms fluctuation in a vertical arrival angle, and the spreading of the acoustic pulse are compared here to estimates from simulations that use the parabolic equation (PE). PE propagations at 250 Hz with a maximum range of 1000 km were performed. The model environment consisted of one of two sound-speed profiles perturbed by internal waves conforming to the Garrett-Munk (GM) spectral model with strengths of 0.5, 1, and 2 times the GM reference energy level. Integral-approximation (IA) estimates of rms travel-time fluctuations were within statistical uncertainty at 1000 km for the SLICE89 profile, and in disagreement by between 20% and 60% for the Canonical profile. Bias estimates were accurate for the first few hundred kilometers of propagation, but became a strong function of time front ID beyond, with some agreeing with the PE results and others very much larger. The IA structure functions of travel time with depth are predicted to be quadratic with the form theta(2)vc0(-2)deltaz(2), where deltaz is vertical separation, c0 is a reference sound speed, and thetav is the rms fluctuation in an arrival angle. At 1000 km, the PE results were close to quadratic at small deltaz, with values of thetav in disagreement with those of the integral approximation by factors of order 2. Pulse spreads in the PE results were much smaller than predicted by the IA estimates. Results imply that acoustic tomography of internal waves at ranges up to 1000 km can use the IA estimate of travel-time variance with reasonable reliability.
Wavelets in the solution of the volume integral equation: Application to eddy current modeling
Wang, B.; Moulder, J.C.; Basart, J.P.
1997-05-01
There is growing interest in the applications of wavelets as basis functions in solutions of integral equations, especially in the area of electromagnetic field problems. In this article we apply a wavelet expansion to the solution of the three-dimensional eddy current modeling problem based on the volume integral method. Although this method shows promise for eddy current modeling of three-dimensional flaws, it is restricted by the computing power required to solve a large linear system. In this article we show that applying a wavelet basis to the volume integral method can dramatically reduce the size of the linear system to be solved. In our approach, the unknown total field is expressed as a twofold summation of shifted and dilated forms of a properly chosen basis function that is often referred to as the mother wavelet. The wavelet expansion can adaptively fit itself to the total field distribution by distributing the localized functions near the flaw boundary, where the field change is large, and the more spatially diffused functions over the interior of the flaw where the total field tends to be smooth. The approach is thus best suited to modeling large three-dimensional flaws where the large number of elements used in the volume integral method requires extremely large memory space and computational capacity. The feasibility of the wavelet method is discussed in the context of the physical nature of eddy-current modeling problems. Numerical examples using both Haar wavelets and Daubechies compactly supported wavelets with periodic extension are given. The results of the wavelet method are also compared with experimental results from a cylindrical flat-bottom hole in an aluminum plate. These numerical examples and comparisons indicate that the wavelet method can greatly reduce the numerical complexity of the problem with negligible loss in accuracy. {copyright} {ital 1997 American Institute of Physics.}
Wavelets in the solution of the volume integral equation: Application to eddy current modeling
NASA Astrophysics Data System (ADS)
Wang, Bing; Moulder, John C.; Basart, John P.
1997-05-01
There is growing interest in the applications of wavelets as basis functions in solutions of integral equations, especially in the area of electromagnetic field problems. In this article we apply a wavelet expansion to the solution of the three-dimensional eddy current modeling problem based on the volume integral method. Although this method shows promise for eddy current modeling of three-dimensional flaws, it is restricted by the computing power required to solve a large linear system. In this article we show that applying a wavelet basis to the volume integral method can dramatically reduce the size of the linear system to be solved. In our approach, the unknown total field is expressed as a twofold summation of shifted and dilated forms of a properly chosen basis function that is often referred to as the mother wavelet. The wavelet expansion can adaptively fit itself to the total field distribution by distributing the localized functions near the flaw boundary, where the field change is large, and the more spatially diffused functions over the interior of the flaw where the total field tends to be smooth. The approach is thus best suited to modeling large three-dimensional flaws where the large number of elements used in the volume integral method requires extremely large memory space and computational capacity. The feasibility of the wavelet method is discussed in the context of the physical nature of eddy-current modeling problems. Numerical examples using both Haar wavelets and Daubechies compactly supported wavelets with periodic extension are given. The results of the wavelet method are also compared with experimental results from a cylindrical flat-bottom hole in an aluminum plate. These numerical examples and comparisons indicate that the wavelet method can greatly reduce the numerical complexity of the problem with negligible loss in accuracy.
Liao, Fei; Tian, Kao-Cong; Yang, Xiao; Zhou, Qi-Xin; Zeng, Zhao-Chun; Zuo, Yu-Ping
2003-03-01
The reliability of kinetic substrate quantification by nonlinear fitting of the enzyme reaction curve to the integrated Michaelis-Menten equation was investigated by both simulation and preliminary experimentation. For simulation, product absorptivity epsilon was 3.00 mmol(-1) L cm(-1) and K(m) was 0.10 mmol L(-1), and uniform absorbance error sigma was randomly inserted into the error-free reaction curve of product absorbance A(i) versus reaction time t(i) calculated according to the integrated Michaelis-Menten equation. The experimental reaction curve of arylesterase acting on phenyl acetate was monitored by phenol absorbance at 270 nm. Maximal product absorbance A(m) was predicted by nonlinear fitting of the reaction curve to Eq. (1) with K(m) as constant. There were unique A(m) for best fitting of both the simulated and experimental reaction curves. Neither the error in reaction origin nor the variation of enzyme activity changed the background-corrected value of A(m). But the range of data under analysis, the background absorbance, and absorbance error sigma had an effect. By simulation, A(m) from 0.150 to 3.600 was predicted with reliability and linear response to substrate concentration when there was 80% consumption of substrate at sigma of 0.001. Restriction of absorbance to 0.700 enabled A(m) up to 1.800 to be predicted at sigma of 0.001. Detection limit reached A(m) of 0.090 at sigma of 0.001. By experimentation, the reproducibility was 4.6% at substrate concentration twice the K(m), and A(m) linearly responded to phenyl acetate with consistent absorptivity for phenol, and upper limit about twice the maximum of experimental absorbance. These results supported the reliability of this new kinetic method for enzymatic analysis with enhanced upper limit and precision.
Time-filtered leapfrog integration of Maxwell equations using unstaggered temporal grids
NASA Astrophysics Data System (ADS)
Mahalov, A.; Moustaoui, M.
2016-11-01
A finite-difference time-domain method for integration of Maxwell equations is presented. The computational algorithm is based on the leapfrog time stepping scheme with unstaggered temporal grids. It uses a fourth-order implicit time filter that reduces computational modes and fourth-order finite difference approximations for spatial derivatives. The method can be applied within both staggered and collocated spatial grids. It has the advantage of allowing explicit treatment of terms involving electric current density and application of selective numerical smoothing which can be used to smooth out errors generated by finite differencing. In addition, the method does not require iteration of the electric constitutive relation in nonlinear electromagnetic propagation problems. The numerical method is shown to be effective and stable when employed within Perfectly Matched Layers (PML). Stability analysis demonstrates that the proposed method is effective in stabilizing and controlling numerical instabilities of computational modes arising in wave propagation problems with physical damping and artificial smoothing terms while maintaining higher accuracy for the physical modes. Comparison of simulation results obtained from the proposed method and those computed by the classical time filtered leapfrog, where Maxwell equations are integrated for a lossy medium, within PML regions and for Kerr-nonlinear media show that the proposed method is robust and accurate. The performance of the computational algorithm is also verified by analyzing parametric four wave mixing in an optical nonlinear Kerr medium. The algorithm is found to accurately predict frequencies and amplitudes of nonlinearly converted waves under realistic conditions proposed in the literature.
Numerical study of fractional nonlinear Schrödinger equations
Klein, Christian; Sparber, Christof; Markowich, Peter
2014-01-01
Using a Fourier spectral method, we provide a detailed numerical investigation of dispersive Schrödinger-type equations involving a fractional Laplacian in an one-dimensional case. By an appropriate choice of the dispersive exponent, both mass and energy sub- and supercritical regimes can be identified. This allows us to study the possibility of finite time blow-up versus global existence, the nature of the blow-up, the stability and instability of nonlinear ground states and the long-time dynamics of solutions. The latter is also studied in a semiclassical setting. Moreover, we numerically construct ground state solutions of the fractional nonlinear Schrödinger equation. PMID:25484604
NASA Technical Reports Server (NTRS)
Logan, Terry G.
1994-01-01
The purpose of this study is to investigate the performance of the integral equation computations using numerical source field-panel method in a massively parallel processing (MPP) environment. A comparative study of computational performance of the MPP CM-5 computer and conventional Cray-YMP supercomputer for a three-dimensional flow problem is made. A serial FORTRAN code is converted into a parallel CM-FORTRAN code. Some performance results are obtained on CM-5 with 32, 62, 128 nodes along with those on Cray-YMP with a single processor. The comparison of the performance indicates that the parallel CM-FORTRAN code near or out-performs the equivalent serial FORTRAN code for some cases.
NASA Astrophysics Data System (ADS)
Fishman, Louis
2000-11-01
The role of mathematical modeling in the physical sciences will be briefly addressed. Examples will focus on computational acoustics, with applications to underwater sound propagation, electromagnetic modeling, optics, and seismic inversion. Direct and inverse wave propagation problems in both the time and frequency domains will be considered. Focusing on fixed-frequency (elliptic) wave propagation problems, the usual, two-way, partial differential equation formulation will be exactly reformulated, in a well-posed manner, as a one-way (marching) problem. This is advantageous for both direct and inverse considerations, as well as stochastic modeling problems. The reformulation will require the introduction of pseudodifferential operators and their accompanying phase space analysis (calculus), in addition to path integral representations for the fundamental solutions and their subsequent computational algorithms. Unlike the more traditional, purely numerical applications of, for example, finite-difference and finite-element methods, this approach, in effect, writes the exact, or, more generally, the asymptotically correct, answer as a functional integral and, subsequently, computes it directly. The overall computational philosophy is to combine analysis, asymptotics, and numerical methods to attack complicated, real-world problems. Exact and asymptotic analysis will stress the complementary nature of the direct and inverse formulations, as well as indicating the explicit structural connections between the time- and frequency-domain solutions.
NASA Astrophysics Data System (ADS)
Tsuchida, Takayuki
2010-10-01
We propose a new method for discretizing the time variable in integrable lattice systems while maintaining the locality of the equations of motion. The method is based on the zero-curvature (Lax pair) representation and the lowest-order 'conservation laws'. In contrast to the pioneering work of Ablowitz and Ladik, our method allows the auxiliary dependent variables appearing in the stage of time discretization to be expressed locally in terms of the original dependent variables. The time-discretized lattice systems have the same set of conserved quantities and the same structures of the solutions as the continuous-time lattice systems; only the time evolution of the parameters in the solutions that correspond to the angle variables is discretized. The effectiveness of our method is illustrated using examples such as the Toda lattice, the Volterra lattice, the modified Volterra lattice, the Ablowitz-Ladik lattice (an integrable semi-discrete nonlinear Schrödinger system) and the lattice Heisenberg ferromagnet model. For the modified Volterra lattice, we also present its ultradiscrete analogue.
NASA Astrophysics Data System (ADS)
Wang, Shyh-Wei; Guo, Shuang-Fa
1998-01-01
New techniques for more accurate and efficient simulation of ion implantations by a stepwise numerical integration of the Boltzmann transport equation (BTE) have been developed in this work. Instead of using uniform energy grid, a non-uniform grid is employed to construct the momentum distribution matrix. A more accurate simulation result is obtained for heavy ions implanted into silicon. In the same time, rather than utilizing the conventional Lindhard, Nielsen and Schoitt (LNS) approximation, an exact evaluation of the integrals involving the nuclear differential scattering cross-section (dσn=2πp dp) is proposed. The impact parameter p as a function of ion energy E and scattering angle φ is obtained by solving the magic formula iteratively and an interpolation techniques is devised during the simulation process. The simulation time using exact evaluation is about 3.5 times faster than that using the Littmark and Ziegler (LZ) spline fitted cross-section function for phosphorus implantation into silicon.
NASA Astrophysics Data System (ADS)
Liu, Chongxuan; Szecsody, Jim E.; Zachara, John M.; Ball, William P.
The generalized integral transform technique (GITT) is applied to solve the one-dimensional advection-dispersion equation (ADE) in heterogeneous porous media coupled with either linear or nonlinear sorption and decay. When both sorption and decay are linear, analytical solutions are obtained using the GITT for one-dimensional ADEs with spatially and temporally variable flow and dispersion coefficient and arbitrary initial and boundary conditions. When either sorption or decay is nonlinear the solutions to ADEs with the GITT are hybrid analytical-numerical. In both linear and nonlinear cases, the forward and inverse integral transforms for the problems described in the paper are apparent and straightforward. Some illustrative examples with linear sorption and decay are presented to demonstrate the application and check the accuracy of the derived analytical solutions. The derived hybrid analytical-numerical solutions are checked against a numerical approach and demonstratively applied to a nonlinear transport example, which simulates a simplified system of iron oxide bioreduction with nonlinear sorption and nonlinear reaction kinetics.
High-Order Integral Equation Methods for Diffraction Problems Involving Screens and Apertures
NASA Astrophysics Data System (ADS)
Lintner, Stephane K.
This thesis presents a novel approach for the numerical solution of problems of diffraction by infinitely thin screens and apertures. The new methodology relies on combination of weighted versions of the classical operators associated with the Dirichlet and Neumann open-surface problems. In the two-dimensional case, a rigorous proof is presented, establishing that the new weighted formulations give rise to second-kind Fredholm integral equations, thus providing a generalization to open surfaces of the classical closed-surface Calderon formulae. High-order quadrature rules are introduced for the new weighted operators, both in the two-dimensional case as well as the scalar three-dimensional case. Used in conjunction with Krylov subspace iterative methods, these rules give rise to efficient and accurate numerical solvers which produce highly accurate solutions in small numbers of iterations, and whose performance is comparable to that arising from efficient high-order integral solvers recently introduced for closed-surface problems. Numerical results are presented for a wide range of frequencies and a variety of geometries in two- and three-dimensional space, including complex resonating structures as well as, for the first time, accurate numerical solutions of classical diffraction problems considered by the 19th-century pioneers: diffraction of high-frequency waves by the infinitely thin disc, the circular aperture, and the two-hole geometry inherent in Young's experiment.
NASA Technical Reports Server (NTRS)
Jothiprasad, Giridhar; Mavriplis, Dimitri J.; Caughey, David A.
2002-01-01
The rapid increase in available computational power over the last decade has enabled higher resolution flow simulations and more widespread use of unstructured grid methods for complex geometries. While much of this effort has been focused on steady-state calculations in the aerodynamics community, the need to accurately predict off-design conditions, which may involve substantial amounts of flow separation, points to the need to efficiently simulate unsteady flow fields. Accurate unsteady flow simulations can easily require several orders of magnitude more computational effort than a corresponding steady-state simulation. For this reason, techniques for improving the efficiency of unsteady flow simulations are required in order to make such calculations feasible in the foreseeable future. The purpose of this work is to investigate possible reductions in computer time due to the choice of an efficient time-integration scheme from a series of schemes differing in the order of time-accuracy, and by the use of more efficient techniques to solve the nonlinear equations which arise while using implicit time-integration schemes. This investigation is carried out in the context of a two-dimensional unstructured mesh laminar Navier-Stokes solver.
NASA Astrophysics Data System (ADS)
Lileg, Klemens
1990-12-01
The electric field integral equation is solved for a cylindrical antenna of arbitrary radius with flat endcaps using the method of moments. Trigonometric subdomain functions are used as basis functions; the weighting functions have the same shape as the basis functions (Galerkin's method). For the endcaps the approximation of the program NEC is used; the excitation is due to a homogeneous field in a gap in the center of the antenna. No analytical approximations are employed in the evaluation of the integrals needed for the computation of the impedance matrix. The admittance so obtained converges better than that found with the help of NEC, but in many cases it is not completely satisfactory. Therefore, the approximate condition for the endcaps are introduced, and trigonometric subdomain functions analogous to those used on the cylinder are used as basis functions. All additional evaluations are done without approximations. The results for the admittance converge in all cases even for a small number of segments. The impedance is measured for a number of monopoles of various radii above a conducting plane; for all frequencies good agreement with the calculation is obtained.
NASA Astrophysics Data System (ADS)
Tang, Min; Wang, Yihong
2017-02-01
In magnetized plasma, the magnetic field confines the particles around the field lines. The anisotropy intensity in the viscosity and heat conduction may reach the order of 1012. When the boundary conditions are periodic or Neumann, the strong diffusion leads to an ill-posed limiting problem. To remove the ill-conditionedness in the highly anisotropic diffusion equations, we introduce a simple but very efficient asymptotic preserving reformulation in this paper. The key idea is that, instead of discretizing the Neumann boundary conditions locally, we replace one of the Neumann boundary condition by the integration of the original problem along the field line, the singular 1 / ɛ terms can be replaced by O (1) terms after the integration, which yields a well-posed problem. Small modifications to the original code are required and no change of coordinates nor mesh adaptation are needed. Uniform convergence with respect to the anisotropy strength 1 / ɛ can be observed numerically and the condition number does not scale with the anisotropy.
NASA Technical Reports Server (NTRS)
Gottlieb, D.; Turkel, E.
1980-01-01
New methods are introduced for the time integration of the Fourier and Chebyshev methods of solution for dynamic differential equations. These methods are unconditionally stable, even though no matrix inversions are required. Time steps are chosen by accuracy requirements alone. For the Fourier method both leapfrog and Runge-Kutta methods are considered. For the Chebyshev method only Runge-Kutta schemes are tested. Numerical calculations are presented to verify the analytic results. Applications to the shallow water equations are presented.
NASA Astrophysics Data System (ADS)
Feng, Bao-Feng; Maruno, Ken-ichi; Ohta, Yasuhiro
2017-02-01
In the present paper, we propose a two-component generalization of the reduced Ostrovsky (Vakhnenko) equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis-Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that the two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and N-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Bäcklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of the two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. In particular, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their N-soliton solutions in terms of pffafians are also provided.
NASA Technical Reports Server (NTRS)
Banyukevich, A.; Ziolkovski, K.
1975-01-01
A number of hybrid methods for solving Cauchy problems are described on the basis of an evaluation of advantages of single and multiple-point numerical integration methods. The selection criterion is the principle of minimizing computer time. The methods discussed include the Nordsieck method, the Bulirsch-Stoer extrapolation method, and the method of recursive Taylor-Steffensen power series.
Integrated technology wing design study
NASA Technical Reports Server (NTRS)
Hays, A. P.; Beck, W. E.; Morita, W. H.; Penrose, B. J.; Skarshaug, R. E.; Wainfan, B. S.
1984-01-01
The technology development costs and associated benefits in applying advanced technology associated with the design of a new wing for a new or derivative trijet with a capacity for 350 passengers and maximum range of 8519 km, entering service in 1990 were studied. The areas of technology are: (1) airfoil technology; (2) planform parameters; (3) high lift; (4) pitch active control system; (5) all electric systems; (6) E to 3rd power propulsion; (7) airframe/propulsion integration; (8) graphite/epoxy composites; (9) advanced aluminum alloys; (10) titanium alloys; and (11) silicon carbide/aluminum composites. These technologies were applied to the reference aircraft configuration. Payoffs were determined for block fuel reductions and net value of technology. These technologies are ranked for the ratio of net value of technology (NVT) to technology development costs.
A new dispersion-relation preserving method for integrating the classical Boussinesq equation
NASA Astrophysics Data System (ADS)
Jang, T. S.
2017-02-01
In this paper, a dispersion-relation preserving method is proposed for nonlinear dispersive waves, starting from the oldest weakly nonlinear dispersive wave mathematical model in shallow water waves, i.e., the classical Boussinesq equation. It is a semi-analytic procedure, however, which preserves, as a distinctive feature, the dispersion-relation imbedded in the model equation without adding (unwelcome) numerical effects, i.e., the proposed method has the same dispersion-relation as the original classical Boussinesq equation. This remarkable (dispersion-relation) preserving property is proved mathematically for small wave motion in present study. The property is also numerically examined by observing both the local wave number and the local frequency of a slowly varying water-wave group. The dispersion-relation preserving method proposed here is powerful as well for observing nonlinear wave phenomena such as solitary waves and their collision. In fact, the main features of nonlinear wave characteristics are clearly seen through not only a single propagating solitary wave but counter-propagating (head-on) solitary wave collisions. They are compared with known (exact) nonlinear solutions, the results of which represent a major improvement over existing solution formulations in the literature.
Studies on the equation of state of mixed carbide fuel
NASA Astrophysics Data System (ADS)
Joseph, M.; Mathews, C. K.; Rao, P. Bhaskar
1989-12-01
The equation of state of reactor fuels is required up to very high temperatures in order to assess the energy release in hypothetical core disruptive accidents (HCM). Though the mixed carbide of uranium and plutonium is a candidate fuel material for fast breeder reactors, much information is not available on its equation of state. This paper reports the results of our studies to obtain the equilibrium vapour pressures of uranium carbide and uranium-plutonium mixed carbide of varying compositions in the temperature range of 1300-9000 K. An extrapolation method based on the principles of equilibrium thermodynamics has been used as also the principle of corresponding states. The agreement between the different results are discussed and their implications in HCDA calculations brought out.
NASA Technical Reports Server (NTRS)
Kaplan, Carl
1951-01-01
The particular integrals of the second-order and third-order Prandtl-Busemann iteration equations for the flow of a compressible fluid are obtained by means of the method in which the complex conjugate variables are utilized as the independent variables of the analysis. The assumption is made that the Prandtl-Glauert solution of the linearized or first-order iteration equation for the two-dimensional flow of a compressible fluid is known. The forms of the particular integrals, derived for subsonic flow, are readily adapted to supersonic flows with only a change in sign of one of the parameters of the problem.
Monte Carlo solution of the volume-integral equation of electromagnetic scattering
NASA Astrophysics Data System (ADS)
Peltoniemi, J.; Muinonen, K.
2014-07-01
Electromagnetic scattering is often the main physical process to be understood when interpreting the observations of asteroids, comets, and meteors. Modeling the scattering faces still many problems, and one needs to assess several different cases: multiple scattering and shadowing by the rough surface, multiple scattering inside a surface element, and single scattering by a small object. Our specific goal is to extend the electromagnetic techniques to larger and more complicated objects, and derive approximations taking into account the most important effects of waves. Here we experiment with Monte Carlo techniques: can they provide something new to solving the scattering problems? The electromagnetic wave equation in the presence of a scatterer of volume V and refractive index m, with an incident wave EE_0, including boundary conditions and the scattering condition at infinity, can be presented in the form of an integral equation EE(rr)(1+suski(rr) Q(ρ))-int_{V-V_ρ}ddrr' GG(rr-rr')suski(rr')EE(rr') =EE_0, where suski(rr)=m(rr)^2-1, Q(ρ)=-1/3+{cal O}(ρ^2)+{O'}(m^2ρ^2), {O}, and {O'} are some second- and higher-order corrections for the finite-size volume V_ρ of radius ρ around the singularity and GG is the dyadic Green's function of the form GG(RR)={exp(im kR)}/{4π R}[unittensor(1+{im}/{R}-{1}/{R^2})-RRRR(1+{3im}/{R}-{3}/{R^2})]. In general, this is solved by extending the internal field in terms of some simple basis functions, e.g., plane or spherical waves or a cubic grid, approximating the integrals in a clever way, and determining the goodness of the solution somehow, e.g., moments or least square. Whatever the choice, the solution usually converges nicely towards a correct enough solution when the scatterer is small and simple, and diverges when the scatterer becomes too complicated. With certain methods, one can reach larger scatterers faster, but the memory and CPU needs can be huge. Until today, all successful solutions are based on more or less
NASA Astrophysics Data System (ADS)
Ying Huang, Shao; Wu, Bae-Ian; Foong, Shaohui
2013-01-01
Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) surface integral equation method is applied for the first time to accurately estimate the surface-enhanced Raman scattering (SERS) enhancement factor distribution for arbitrary nanoparticles and nano-aggregates. It is the first time in literature that the distributions of SERS enhancement factors of nanoparticles of a large variety are reported. It is shown that not every SERS substrate exhibits a long-tail distribution as a dimer consisting of two spheres in close proximity. Generic methods are proposed to evaluate the performance of nanoparticles on SERS substrates. A cumulative distribution is proposed to examine the contributions of hot and warm spots around the nanoparticles. It is used to identify the importance of warm spots on a SERS substrate. A parameter q is proposed to describe the likelihood of a randomly positioned molecule that can be activated. This study provides guidance and insights for the optimization of SERS substrate fabrication techniques.
NASA Technical Reports Server (NTRS)
Atluri, Satya N.; Shen, Shengping
2002-01-01
In this paper, a very simple method is used to derive the weakly singular traction boundary integral equation based on the integral relationships for displacement gradients. The concept of the MLPG method is employed to solve the integral equations, especially those arising in solid mechanics. A moving Least Squares (MLS) interpolation is selected to approximate the trial functions in this paper. Five boundary integral Solution methods are introduced: direct solution method; displacement boundary-value problem; traction boundary-value problem; mixed boundary-value problem; and boundary variational principle. Based on the local weak form of the BIE, four different nodal-based local test functions are selected, leading to four different MLPG methods for each BIE solution method. These methods combine the advantages of the MLPG method and the boundary element method.
ERKN integrators for systems of oscillatory second-order differential equations
NASA Astrophysics Data System (ADS)
Wu, Xinyuan; You, Xiong; Shi, Wei; Wang, Bin
2010-11-01
For systems of oscillatory second-order differential equations y+My=f with M∈R, a symmetric positive semi-definite matrix, X. Wu et al. have proposed the multidimensional ARKN methods [X. Wu, X. You, J. Xia, Order conditions for ARKN methods solving oscillatory systems, Comput. Phys. Comm. 180 (2009) 2250-2257], which are an essential generalization of J.M. Franco's ARKN methods for one-dimensional problems or for systems with a diagonal matrix M=wI [J.M. Franco, Runge-Kutta-Nyström methods adapted to the numerical integration of perturbed oscillators, Comput. Phys. Comm. 147 (2002) 770-787]. One of the merits of these methods is that they integrate exactly the unperturbed oscillators y+My=0. Regretfully, even for the unperturbed oscillators the internal stages Y of an ARKN method fail to equal the values of the exact solution y(t) at t+ch, respectively. Recently H. Yang et al. proposed the ERKN methods to overcome this drawback [H.L. Yang, X.Y. Wu, Xiong You, Yonglei Fang, Extended RKN-type methods for numerical integration of perturbed oscillators, Comput. Phys. Comm. 180 (2009) 1777-1794]. However, the ERKN methods in that paper are only considered for the special case where M is a diagonal matrix with nonnegative entries. The purpose of this paper is to extend the ERKN methods to the general case with M∈R, and the perturbing function f depends only on y. Numerical experiments accompanied demonstrates that the ERKN methods are more efficient than the existing methods for the computation of oscillatory systems. In particular, if M∈R is a symmetric positive semi-definite matrix, it is highly important for the new ERKN integrators to show the energy conservation in the numerical experiments for problems with Hamiltonian H(p,q)=1/2 >pp+1/2 >qMq+V(q) in comparison with the well-known methods in the scientific literature. Those so called separable Hamiltonians arise in many areas of physical sciences, e.g., macromolecular dynamics, astronomy, and classical
An integral equation approach to smooth 3D Navier-Stokes solution
NASA Astrophysics Data System (ADS)
Costin, O.; Luo, G.; Tanveer, S.
2008-12-01
We summarize a recently developed integral equation (IE) approach to tackling the long-time existence problem for smooth solution v(x, t) to the 3D Navier-Stokes (NS) equation in the context of a periodic box problem with smooth time independent forcing and initial condition v0. Using an inverse-Laplace transform of {\\skew5\\hat v} (k, t) - {\\skew5\\hat v}_0 in 1/t, we arrive at an IE for {\\skew5\\hat U} (k, p) , where p is inverse-Laplace dual to 1/t and k is the Fourier variable dual to x. The advantage of this formulation is that the solution {\\skew5\\hat U} to the IE is known to exist a priori for p \\in \\mathbb{R}^+ and the solution is integrable and exponentially bounded at ∞. Global existence of NS solution in this formulation is reduced to an asymptotics question. If \\parallel\\!{\\skew5\\hat U} (\\cdot, p)\\!\\parallel_{{l^{1} (\\mathbb{Z}^3)}} has subexponential bounds as p→∞, then global existence to NS follows. Moreover, if f=0, then the converse is also true in the following sense: if NS has global solution, then there exists n>=1 for which the inverse-Laplace transform of {\\skew5\\hat v} (k, t) - {\\skew5\\hat v}_0 in 1/tn necessarily decays as q→∞, where q is the inverse-Laplace dual to 1/tn. We also present refined estimates of the exponential growth when the solution {\\skew5\\hat U} is known on a finite interval [0, p0]. We also show that for analytic v[0] and f, with finitely many nonzero Fourier-coefficients, the series for {\\skew5\\hat U} (k, p) in powers of p has a radius of convergence independent of initial condition and forcing; indeed the radius gets bigger for smaller viscosity. We also show that the IE can be solved numerically with controlled errors. Preliminary numerical calculations for Kida (1985 J. Phys. Soc. Japan 54 2132) initial conditions, though far from being optimized, and performed on a modest interval in the accelerated variable q show decay in q.
NASA Astrophysics Data System (ADS)
Zinser, Brian
We present two distinct mathematical models where high-order integral equations are applied to electromagnetic problems. The first problem is to find the electric potential in and around ion channels and Janus particles. The second problem is to find the electromagnetic scattering caused by a set of simple geometric objects. In biology, we consider two types of inhomogeneities: the first one is a simple model of an ion channel which consists of a finite height cylindrical cavity embedded in a layered electrolytes/membrane environment, and the second one is a Janus particle made of two different semi-spherical dielectric materials. A boundary element method (BEM) for the Poisson-Boltzmann equation based on Muller's hyper-singular second kind integral equation formulation is used to accurately compute electrostatic potentials. The proposed BEM gives O(1) condition numbers and we show that the second order basis converges faster and is more accurate than the first order basis. For solar cells, we develop a Nystrom volume integral equation (VIE) method for calculating the electromagnetic scattering according to the Maxwell equations. The Cauchy principal values (CPVs) that arise from the VIE are computed using a finite size exclusion volume with explicit correction integrals. Outside the exclusion, the hyper-singular integrals are computed using an interpolated quadrature formulae with tensor-product quadrature nodes. We considered cubes, rectangles, cylinders, spheres, and ellipsoids. As the new quadrature weights are pre-calculated and tabulated, the integrals are calculated efficiently at runtime. Simulations with many scatterers demonstrate the efficiency of the interpolated quadrature formulae. We also demonstrate that the resulting VIE has high accuracy and p-convergence.
Zhang, Yu-Juan; Zhao, Dun; Luo, Hong-Gang
2014-11-15
We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose–Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton. - Highlights: • We consider a unified model for soliton management by an integrable integro-differential Schrödinger equation. • Using Lax pair, the N-fold Darboux transformation for the equation is presented. • The multi-soliton management is considered. • The synchronized dispersive and nonlinear management is suggested.
NASA Astrophysics Data System (ADS)
Mitchell, William H.; Spagnolie, Saverio E.
2017-03-01
A double-layer integral equation for the surface tractions on a body moving in a viscous fluid is derived which allows for the incorporation of a background flow and/or the presence of a plane wall. The Lorentz reciprocal theorem is used to link the surface tractions on the body to integrals involving the background velocity and stress fields on an imaginary bounding sphere (or hemisphere for wall-bounded flows). The derivation requires the velocity and stress fields associated with numerous fundamental singularity solutions which we provide for free-space and wall-bounded domains. Two sample applications of the method are discussed: we study the tractions on an ellipsoid moving near a plane wall, which provides a more detailed understanding of the well-studied glancing and reversing trajectories in the context of particle sedimentation, and the erosion of bodies by a viscous flow, in which the surface is ablated at a rate proportional to the local viscous shear stress. Simulations and analytical estimates suggest that a spherical body in a uniform flow first reduces nearly but not exactly to the drag minimizing profile and then vanishes in finite time. The shape dynamics of an eroding body in a shear flow and near a wall are also investigated. Stagnation points on the body surface lead generically to the formation of cusps, whose number depends on the flow configuration and/or the presence of nearby boundaries.
Hybrid two-chain simulation and integral equation theory : application to polyethylene liquids.
Huimin Li, David T. Wu; Curro, John G.; McCoy, John Dwane
2006-02-01
We present results from a hybrid simulation and integral equation approach to the calculation of polymer melt properties. The simulation consists of explicit Monte Carlo (MC) sampling of two polymer molecules, where the effect of the surrounding chains is accounted for by an HNC solvation potential. The solvation potential is determined from the Polymer Reference Interaction Site Model (PRISM) as a functional of the pair correlation function from simulation. This hybrid two-chain MC-PRISM approach was carried out on liquids of polyethylene chains of 24 and 66 CH{sub 2} units. The results are compared with MD simulation and self-consistent PRISM-PY theory under the same conditions, revealing that the two-chain calculation is close to MD, and able to overcome the defects of the PRISM-PY closure and predict more accurate structures of the liquid at both short and long range. The direct correlation function, for instance, has a tail at longer range which is consistent with MD simulation and avoids the short-range assumptions in PRISM-PY theory. As a result, the self-consistent two-chain MC-PRISM calculation predicts an isothermal compressibility closer to the MD results.
Neural Network Emulation of the Integral Equation Model with Multiple Scattering
Pulvirenti, Luca; Ticconi, Francesca; Pierdicca, Nazzareno
2009-01-01
The Integral Equation Model with multiple scattering (IEMM) represents a well-established method that provides a theoretical framework for the scattering of electromagnetic waves from rough surfaces. A critical aspect is the long computational time required to run such a complex model. To deal with this problem, a neural network technique is proposed in this work. In particular, we have adopted neural networks to reproduce the backscattering coefficients predicted by IEMM at L- and C-bands, thus making reference to presently operative satellite radar sensors, i.e., that aboard ERS-2, ASAR on board ENVISAT (C-band), and PALSAR aboard ALOS (L-band). The neural network-based model has been designed for radar observations of both flat and tilted surfaces, in order to make it applicable for hilly terrains too. The assessment of the proposed approach has been carried out by comparing neural network-derived backscattering coefficients with IEMM-derived ones. Different databases with respect to those employed to train the networks have been used for this purpose. The outcomes seem to prove the feasibility of relying on a neural network approach to efficiently and reliably approximate an electromagnetic model of surface scattering. PMID:22408496
Transport in simple networks described by an integrable discrete nonlinear Schrödinger equation.
Nakamura, K; Sobirov, Z A; Matrasulov, D U; Sawada, S
2011-08-01
We elucidate the case in which the Ablowitz-Ladik (AL)-type discrete nonlinear Schrödinger equation (NLSE) on simple networks (e.g., star graphs and tree graphs) becomes completely integrable just as in the case of a simple one-dimensional (1D) discrete chain. The strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The present work is a nontrivial extension of our preceding one [Sobirov et al., Phys. Rev. E 81, 066602 (2010)] on the continuum NLSE to the discrete case. We find (1) the solution on each bond is a part of the universal (bond-independent) AL soliton solution on the 1D discrete chain, but it is multiplied by the inverse of the square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule; and (3) under findings 1 and 2, there exist an infinite number of constants of motion. As a practical issue, with the use of an AL soliton injected through the incoming bond, we obtain transmission probabilities inversely proportional to the strength of nonlinearity on the outgoing bonds.
NASA Astrophysics Data System (ADS)
Yuan, Zhengwen; Xiao, Hong; Xie, Hongbiao
2014-02-01
Precise strip-shape control theory is significant to improve rolled strip quality, and roll flattening theory is a primary part of the strip-shape theory. To improve the accuracy of roll flattening calculation based on semi-infinite body model, a new and more accurate roll flattening model is proposed in this paper, which is derived based on boundary integral equation method. The displacement fields of the finite length semi-infinite body on left and right sides are simulated by using finite element method (FEM) and displacement decay functions on left and right sides are established. Based on the new roll flattening model, a new 4Hi mill deformation model is established and verified by FEM. The new model is compared with Foppl formula and semi-infinite body model in different strip width, roll shifting value and bending force. The results show that the pressure and flattening between rolls calculated by the new model are more precise than other two models, especially near the two roll barrel edges.
Coupled integral equations for sound propagation above a hard ground surface with trench cuttings.
Wang, Gong Li; Chew, Weng Cho; White, Michael J
2006-09-01
A set of coupled integral equations is formulated for the investigation of sound propagation from an infinitesimal harmonic line source above a hard ground surface corrugated with cuttings. Two half-space Green's functions are employed in the formulation. The first one defined for the upper half space is used to reduce the problem size and eliminate the edge effect resulting from the boundary truncation; the other one for the lower half space is to simplify the representation of the Neumann-Dirichlet map. As a result, the unknowns are only distributed over the corrugated part of the surface, which leads to substantial reduction in the size of the final linear system. The computational complexity of the Neumann-Dirichlet map is also reduced. The method is used to analyze the behavior of sound propagation above textured surfaces the impedance of which is expectedly altered. The effects of number and opening of trench cuttings, and the effect of source height are investigated. The conclusions drawn can be used for reference in a practical problem of mitigating gun blast noise.
NASA Astrophysics Data System (ADS)
Kwieciński, Jan; Maul, Martin
2003-02-01
In this paper we derive an integral equation for the evolution of unintegrated (longitudinally) polarized quark and gluon parton distributions. The conventional Catani-Ciafaloni-Fiorani-Marchesini (CCFM) framework is modified at small x in order to incorporate the QCD expectations concerning the double ln2(1/x) resummation at low x for the integrated distributions. Complete Altarelli-Parisi splitting functions are included that makes the formalism compatible with the leading order Altarelli-Parisi evolution at large and moderately small values of x. The obtained modified polarized CCFM equation is shown to be partially diagonalized by the Fourier-Bessel transformation. Results of the numerical solution for this modifed polarized CCFM equation for the nonsinglet quark distributions are presented.
Heydari, M.H.; Hooshmandasl, M.R.; Cattani, C.; Maalek Ghaini, F.M.
2015-02-15
Because of the nonlinearity, closed-form solutions of many important stochastic functional equations are virtually impossible to obtain. Thus, numerical solutions are a viable alternative. In this paper, a new computational method based on the generalized hat basis functions together with their stochastic operational matrix of Itô-integration is proposed for solving nonlinear stochastic Itô integral equations in large intervals. In the proposed method, a new technique for computing nonlinear terms in such problems is presented. The main advantage of the proposed method is that it transforms problems under consideration into nonlinear systems of algebraic equations which can be simply solved. Error analysis of the proposed method is investigated and also the efficiency of this method is shown on some concrete examples. The obtained results reveal that the proposed method is very accurate and efficient. As two useful applications, the proposed method is applied to obtain approximate solutions of the stochastic population growth models and stochastic pendulum problem.
NASA Astrophysics Data System (ADS)
Zhang, Yu-Juan; Zhao, Dun; Luo, Hong-Gang
2014-11-01
We consider a wide class of integrable nonautonomous nonlinear integro-differential Schrödinger equation which contains the models for the soliton management in Bose-Einstein condensates, nonlinear optics, and inhomogeneous Heisenberg spin chain. With the help of the nonisospectral AKNS hierarchy, we obtain the N-fold Darboux transformation and the N-fold soliton-like solutions for the equation. The soliton management, especially the synchronized dispersive and nonlinear management in optical fibers is discussed. It is found that in the situation without external potential, the synchronized dispersive and nonlinear management can keep the integrability of the nonlinear Schrödinger equation; this suggests that in optical fibers, the synchronized dispersive and nonlinear management can control and maintain the propagation of a multi-soliton.
NASA Technical Reports Server (NTRS)
Bates, J. R.; Semazzi, F. H. M.; Higgins, R. W.; Barros, Saulo R. M.
1990-01-01
A vector semi-Lagrangian semi-implicit two-time-level finite-difference integration scheme for the shallow water equations on the sphere is presented. A C-grid is used for the spatial differencing. The trajectory-centered discretization of the momentum equation in vector form eliminates pole problems and, at comparable cost, gives greater accuracy than a previous semi-Lagrangian finite-difference scheme which used a rotated spherical coordinate system. In terms of the insensitivity of the results to increasing timestep, the new scheme is as successful as recent spectral semi-Lagrangian schemes. In addition, the use of a multigrid method for solving the elliptic equation for the geopotential allows efficient integration with an operation count which, at high resolution, is of lower order than in the case of the spectral models. The properties of the new scheme should allow finite-difference models to compete with spectral models more effectively than has previously been possible.
NASA Technical Reports Server (NTRS)
Gedney, Stephen D.; Lansing, Faiza
1994-01-01
It has been found that the Discrete Integral Equation (DSI)technique is a highly effective technique for the analysis of microwave circuits and devices [1,2]. The DSI is much more robust than the traditional Finite Difference Time Domain (FDTD) method in a number of ways.
ERIC Educational Resources Information Center
Marsh, Herbert W.; Ludtke, Oliver; Robitzsch, Alexander; Trautwein, Ulrich; Asparouhov, Tihomir; Muthen, Bengt; Nagengast, Benjamin
2009-01-01
This article is a methodological-substantive synergy. Methodologically, we demonstrate latent-variable contextual models that integrate structural equation models (with multiple indicators) and multilevel models. These models simultaneously control for and unconfound measurement error due to sampling of items at the individual (L1) and group (L2)…
Hansson, T; Lisak, M; Anderson, D
2012-02-10
It is shown that the evolution equations describing partially coherent wave propagation in noninstantaneous Kerr media are integrable and have an infinite number of invariants. A recursion relation for generating these invariants is presented, and it is demonstrated how to express them in the coherent density, self-consistent multimode, mutual coherence, and Wigner formalisms.
NASA Astrophysics Data System (ADS)
Luo, Hong-Ying; Wang, Chuan-Jian; Liu, Jun; Dai, Zheng-De
2013-06-01
Painlevé integrability has been tested for (2+1)D Boussinesq equation with disturbance term using the standard WTC approach after introducing the Kruskai's simplification. New breather solitary solutions depending on constant equilibrium solution are obtained by using Extended Homoclinic Test Method. Moreover, the spatiotemporal feature of breather solitary wave is exhibited.
NASA Astrophysics Data System (ADS)
Ashyralyev, Allaberen; Sharifov, Y. A.
2012-08-01
The system of nonlinear impulsive differential equations with two-point and integral boundary conditions are investigated. Theorems on existence and uniqueness of a solution are established under some sufficient conditions on nonlinear terms. A simple example of application of the main result of this paper is presented.
NASA Astrophysics Data System (ADS)
Chen, Jinbing
2010-08-01
Each soliton equation in the Korteweg-de Vries (KdV) hierarchy, the 2+1 dimensional breaking soliton equation, and the 2+1 dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation are reduced to two or three Neumann systems on the tangent bundle TSN -1 of the unit sphere SN -1. The Lax-Moser matrix for the Neumann systems of degree N -1 is deduced in view of the Mckean-Trubowitz identity and a bilinear generating function, whose favorite characteristic accounts for the problem of the genus of Riemann surface matching to the number of elliptic variables. From the Lax-Moser matrix, the constrained Hamiltonians in the sense of Dirac-Poisson bracket for all the Neumann systems are written down in a uniform recursively determined by integrals of motion. The involution of integrals of motion and constrained Hamiltonians is completed on TSN -1 by using a Lax equation and their functional independence is displayed over a dense open subset of TSN -1 by a direct calculation, which contribute to the Liouville integrability of a family of Neumann systems in a new systematical way. We also construct the hyperelliptic curve of Riemann surface and the Abel map straightening out the restricted Neumann flows that naturally leads to the Jacobi inversion problem on the Jacobian with the aid of the holomorphic differentials, from which some finite-gap solutions expressed by Riemann theta functions for the 2+1 dimensional breaking soliton equation, the 2+1 dimensional CDGKS equation, the KdV, and the fifth-order KdV equations are presented by means of the Riemann theorem.
Advanced Studies of Integrable Systems.
1984-06-26
Tokamaks , and that instead of nonlinearities dominating, the diffusion effects would dominate with a subsequent loss in the coherence of any such...the rest frame of the soliton. Since we start cone. Inside the light cone, one must use other techniques. with an equilibrium state, we have u, -sin u...are then in equilibrium along go. From the parallel The frequency wi and the wave number k are related by the momentum equation, one then obtains the
NASA Astrophysics Data System (ADS)
Wang, Ya-Le; Gao, Yi-Tian; Jia, Shu-Liang; Lan, Zhong-Zhou; Deng, Gao-Fu; Su, Jing-Jing
2017-01-01
Under investigation in this paper is a (2+1)-dimensional generalized variable-coefficient shallow water wave equation which can be reduced to several integrable equations, such as the Korteweg-de Vries (KdV) equation and the Calogero-Bogoyavlenskii-Schiff (CBS) equation. Bilinear forms, Bäcklund transformation, Lax pair and infinite conservation laws are derived based on the binary Bell polynomials. N-soliton solutions are constructed via the Hirota method. Propagation and interaction of the solitons are illustrated graphically: (i) variable coefficients affect the shape of the N-soliton interaction in the scaled space and time coordinates; (ii) positions of the solitons depend on the sign of wave numbers after each interaction; (iii) interaction of the solitons is elastic, i.e. the amplitude, velocity and shape of each soliton remain invariant after each interaction except for a phase shift.
NASA Astrophysics Data System (ADS)
Xu, Gui-Qiong; Deng, Shu-Fang
2016-11-01
Under investigation in this paper is a (2 + 1)-dimensional generalized NNV equation, which includes as many important nonlinear models as its particular cases. First, we perform the Painlevé test for the generalized NNV equation with the help of symbolic computation, and it is shown that this generalized equation admits the Painlevé property for one set of parametric choices. For the newly obtained integrable equation, we then employ the binary Bell polynomial method to construct the bilinear form, N-soliton solution, bilinear Bäcklund transformation and Lax pair in a systematic way. In addition, some new doubly periodic wave solutions with two arbitrary functions are obtained by means of truncated Painlevé expansions. Finally, the collisions of multiple solitons and periodic waves are interesting and shown by some graphs.
Eastern Renewable Generation Integration Study
Bloom, Aaron; Townsend, Aaron; Palchak, David; Novacheck, Joshua; King, Jack; Barrows, Clayton; Ibanez, Eduardo; O'Connell, Matthew; Jordan, Gary; Roberts, Billy; Draxl, Caroline; Gruchalla, Kenny
2016-08-01
The Eastern Interconnection (EI) is one of the largest power systems in the world, and its size and complexity have historically made it difficult to study in high levels of detail in a modeling environment. In order to understand how this system might be impacted by high penetrations (30% of total annual generation) of wind and solar photovoltaic (PV) during steady state operations, the National Renewable Energy Laboratory (NREL) and the U.S. Department of Energy (DOE) conducted the Eastern Renewable Generation Integration Study (ERGIS). This study investigates certain aspects of the reliability and economic efficiency problem faced by power system operators and planners. Specifically, the study models the ability to meet electricity demand at a 5-minute time interval by scheduling resources for known ramping events, while maintaining adequate reserves to meet random variation in supply and demand, and contingency events. To measure the ability to meet these requirements, a unit commitment and economic dispatch (UC&ED) model is employed to simulate power system operations. The economic costs of managing this system are presented using production costs, a traditional UC&ED metric that does not include any consideration of long-term fixed costs. ERGIS simulated one year of power system operations to understand regional and sub-hourly impacts of wind and PV by developing a comprehensive UC&ED model of the EI. In the analysis, it is shown that, under the study assumptions, generation from approximately 400 GW of combined wind and PV capacity can be balanced on the transmission system at a 5-minute level. In order to address the significant computational burdens associated with a model of this detail we apply novel computing techniques to dramatically reduce simulation solve time while simultaneously increasing the resolution and fidelity of the analysis. Our results also indicate that high penetrations of wind and PV (collectively variable generation (VG
Ferrighi, Lara; Frediani, Luca; Fossgaard, Eirik; Ruud, Kenneth
2006-10-21
We present a parallel implementation of the integral equation formalism of the polarizable continuum model for Hartree-Fock and density functional theory calculations of energies and linear, quadratic, and cubic response functions. The contributions to the free energy of the solute due to the polarizable continuum have been implemented using a master-slave approach with load balancing to ensure good scalability also on parallel machines with a slow interconnect. We demonstrate the good scaling behavior of the code through calculations of Hartree-Fock energies and linear, quadratic, and cubic response function for a modest-sized sample molecule. We also explore the behavior of the parallelization of the integral equation formulation of the polarizable continuum model code when used in conjunction with a recent scheme for the storage of two-electron integrals in the memory of the different slaves in order to achieve superlinear scaling in the parallel calculations.
NASA Astrophysics Data System (ADS)
Ferrighi, Lara; Frediani, Luca; Fossgaard, Eirik; Ruud, Kenneth
2006-10-01
We present a parallel implementation of the integral equation formalism of the polarizable continuum model for Hartree-Fock and density functional theory calculations of energies and linear, quadratic, and cubic response functions. The contributions to the free energy of the solute due to the polarizable continuum have been implemented using a master-slave approach with load balancing to ensure good scalability also on parallel machines with a slow interconnect. We demonstrate the good scaling behavior of the code through calculations of Hartree-Fock energies and linear, quadratic, and cubic response function for a modest-sized sample molecule. We also explore the behavior of the parallelization of the integral equation formulation of the polarizable continuum model code when used in conjunction with a recent scheme for the storage of two-electron integrals in the memory of the different slaves in order to achieve superlinear scaling in the parallel calculations.
The integrated Michaelis-Menten rate equation: déjà vu or vu jàdé?
Goličnik, Marko
2013-08-01
A recent article of Johnson and Goody (Biochemistry, 2011;50:8264-8269) described the almost-100-years-old paper of Michaelis and Menten. Johnson and Goody translated this classic article and presented the historical perspective to one of incipient enzyme-reaction data analysis, including a pioneering global fit of the integrated rate equation in its implicit form to the experimental time-course data. They reanalyzed these data, although only numerical techniques were used to solve the model equations. However, there is also the still little known algebraic rate-integration equation in a closed form that enables direct fitting of the data. Therefore, in this commentary, I briefly present the integral solution of the Michaelis-Menten rate equation, which has been largely overlooked for three decades. This solution is expressed in terms of the Lambert W function, and I demonstrate here its use for global nonlinear regression curve fitting, as carried out with the original time-course dataset of Michaelis and Menten.
Iterative Algorithms for Integral Equations of the First Kind With Applications to Statistics
1992-10-01
n-.) B2.(m-&)Xm where J11 is a nonsingular matrix of Jordan blocks, and N22 is a nilpotent matrix of index t > 1 of Jordan blocks corresponding to a...statistical methodology, and an application to an inverse problem. ) In the first part, singular matrix equations that result from discretizing ill...application to an inverse problem. In the first part, singular matrix equations that result from discretizing ill-posed inte- gral equations of the first
NASA Astrophysics Data System (ADS)
Sinha, Debdeep; Ghosh, Pijush K.
2017-01-01
A two component nonlocal vector nonlinear Schrödinger equation (VNLSE) is considered with a self-induced parity-time-symmetric potential. It is shown that the system possess a Lax pair and an infinite number of conserved quantities and hence integrable. Some of the conserved quantities like number operator, Hamiltonian etc. are found to be real-valued, in spite of the corresponding charge densities being complex. The soliton solution for the same equation is obtained through the method of inverse scattering transformation and the condition of reduction from nonlocal to local case is mentioned.
NASA Technical Reports Server (NTRS)
Joseph, Rose M.; Hagness, Susan C.; Taflove, Allen
1991-01-01
The initial results for femtosecond pulse propagation and scattering interactions for a Lorentz medium obtained by a direct time integration of Maxwell's equations are reported. The computational approach provides reflection coefficients accurate to better than 6 parts in 10,000 over the frequency range of dc to 3 x 10 to the 16th Hz for a single 0.2-fs Gaussian pulse incident upon a Lorentz-medium half-space. New results for Sommerfeld and Brillouin precursors are shown and compared with previous analyses. The present approach is robust and permits 2D and 3D electromagnetic pulse propagation directly from the full-vector Maxwell's equations.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions.
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.
Study of coupled nonlinear partial differential equations for finding exact analytical solutions
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
NASA Astrophysics Data System (ADS)
Kashirin, A. A.; Smagin, S. I.; Taltykina, M. Yu.
2016-04-01
Interior and exterior three-dimensional Dirichlet problems for the Helmholtz equation are solved numerically. They are formulated as equivalent boundary Fredholm integral equations of the first kind and are approximated by systems of linear algebraic equations, which are then solved numerically by applying an iteration method. The mosaic-skeleton method is used to speed up the solution procedure.
Fukuda, Ikuo; Nakamura, Haruki
2006-02-01
For an arbitrary ordinary differential equation (ODE), a scheme for constructing an extended ODE endowed with a time-invariant function is here proposed. This scheme enables us to examine the accuracy of the numerical integration of an ODE that may itself have had no invariant. These quantities are constructed by referring to the Nosé-Hoover molecular dynamics equation and its related conserved quantity. By applying this procedure to several molecular dynamics equations, the conventional conserved quantity individually defined in each dynamics can be reproduced in a uniform, generalized way; our concept allows a transparent outlook underlying these quantities and ideas. Developing the technique, for a certain class of ODEs we construct a numerical integrator that is not only explicit and symmetric, but preserves a unit Jacobian for a suitably defined extended ODE, which also provides an invariant. Our concept is thus to simply build a divergence-free extended ODE whose solution is just a lift-up of the original ODE, and to constitute an efficient integrator that preserves the phase-space volume on the extended system. We present precise discussions about the general mathematical properties of the integrator and provide specific conditions that should be incorporated for practical applications.
Xie, G.; Li, J.; Majer, E.; Zuo, D.
1998-07-01
This paper describes a new 3D parallel GILD electromagnetic (EM) modeling and nonlinear inversion algorithm. The algorithm consists of: (a) a new magnetic integral equation instead of the electric integral equation to solve the electromagnetic forward modeling and inverse problem; (b) a collocation finite element method for solving the magnetic integral and a Galerkin finite element method for the magnetic differential equations; (c) a nonlinear regularizing optimization method to make the inversion stable and of high resolution; and (d) a new parallel 3D modeling and inversion using a global integral and local differential domain decomposition technique (GILD). The new 3D nonlinear electromagnetic inversion has been tested with synthetic data and field data. The authors obtained very good imaging for the synthetic data and reasonable subsurface EM imaging for the field data. The parallel algorithm has high parallel efficiency over 90% and can be a parallel solver for elliptic, parabolic, and hyperbolic modeling and inversion. The parallel GILD algorithm can be extended to develop a high resolution and large scale seismic and hydrology modeling and inversion in the massively parallel computer.
Differential Equations Compatible with Boundary Rational qKZ Equation
NASA Astrophysics Data System (ADS)
Takeyama, Yoshihiro
2011-10-01
We give diffierential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (Cěen, Cn) which in the case of type GLn was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.
Goličnik, Marko
2011-01-01
The Michaelis-Menten rate equation can be found in most general biochemistry textbooks, where the time derivative of the substrate is a hyperbolic function of two kinetic parameters (the limiting rate V, and the Michaelis constant K(M) ) and the amount of substrate. However, fundamental concepts of enzyme kinetics can be difficult to understand fully, or can even be misunderstood, by students when based only on the differential form of the Michaelis-Menten equation, and the variety of methods available to calculate the kinetic constants from rate versus substrate concentration "textbook data." Consequently, enzyme kinetics can be confusing if an analytical solution of the Michaelis-Menten equation is not available. Therefore, the still rarely known exact solution to the Michaelis-Menten equation is presented here through the explicit closed-form equation in terms of the Lambert W(x) function. Unfortunately, as the W(x) is not available in standard curve-fitting computer programs, the practical use of this direct solution is limited for most life-science students. Thus, the purpose of this article is to provide analytical approximations to the equation for modeling Michaelis-Menten kinetics. The elementary and explicit nature of these approximations can provide students with direct and simple estimations of kinetic parameters from raw experimental time-course data. The Michaelis-Menten kinetics studied in the latter context can provide an ideal alternative to the 100-year-old problems of data transformation, graphical visualization, and data analysis of enzyme-catalyzed reactions. Hence, the content of the course presented here could gradually become an important component of the modern biochemistry curriculum in the 21st century.
Advanced Studies of Integrable Systems.
1986-12-18
Fluctuations in Magnetized Plasmas (Phys. Fluids 27, 1169-75 (1984)] (coauthored with S.N. Antani) The nonlinear interactions of whistler waves with density... Dynamica Problems in Soliton Systems, pp 12-22. ed. S. Takeno, Springer-Verlag, NY (1985)]. S 11. Forced Integrable Systems - An Overview, D. J. Kaup...Kaup, P.J. Hansen, S. Roy Choudhury and Gary E. Thomas (accepted for publication in Phys. Fluids ). A singular perturbation method is used to solve this
Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation
NASA Astrophysics Data System (ADS)
Dou, Nicholas G.; Minnich, Austin J.
2016-01-01
Materials that possess low density, low thermal conductivity, and high stiffness are desirable for engineering applications, but most materials cannot realize these properties simultaneously due to the coupling between them. Nanotrusses, which consist of hollow nanoscale beams architected into a periodic truss structure, can potentially break these couplings due to their lattice architecture and nanoscale features. In this work, we study heat conduction in the exact nanotruss geometry by solving the frequency-dependent Boltzmann transport equation using a variance-reduced Monte Carlo algorithm. We show that their thermal conductivity can be described with only two parameters, solid fraction and wall thickness. Our simulations predict that nanotrusses can realize unique combinations of mechanical and thermal properties that are challenging to achieve in typical materials.
Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation
Dou, Nicholas G.; Minnich, Austin J.
2016-01-04
Materials that possess low density, low thermal conductivity, and high stiffness are desirable for engineering applications, but most materials cannot realize these properties simultaneously due to the coupling between them. Nanotrusses, which consist of hollow nanoscale beams architected into a periodic truss structure, can potentially break these couplings due to their lattice architecture and nanoscale features. In this work, we study heat conduction in the exact nanotruss geometry by solving the frequency-dependent Boltzmann transport equation using a variance-reduced Monte Carlo algorithm. We show that their thermal conductivity can be described with only two parameters, solid fraction and wall thickness. Our simulations predict that nanotrusses can realize unique combinations of mechanical and thermal properties that are challenging to achieve in typical materials.
NASA Astrophysics Data System (ADS)
Kitanine, N.; Maillet, J. M.; Niccoli, G.
2014-05-01
We solve the longstanding problem of defining a functional characterization of the spectrum of the transfer matrix associated with the most general spin-1/2 representations of the six-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variables (SOV) representation, hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial form of the Q-function allows us to show that a finite system of generalized Bethe equations can also be used to describe the complete transfer matrix spectrum.
Sourcing for Parameter Estimation and Study of Logistic Differential Equation
ERIC Educational Resources Information Center
Winkel, Brian J.
2012-01-01
This article offers modelling opportunities in which the phenomena of the spread of disease, perception of changing mass, growth of technology, and dissemination of information can be described by one differential equation--the logistic differential equation. It presents two simulation activities for students to generate real data, as well as…
Sasorov, P. V.; Fomin, I. V.
2015-06-15
The collision integral in the kinetic equation for a rarefied spin-polarized gas of fermions (electrons) is derived. The collisions between these fermions and the collisions with much heavier particles (ions) forming a randomly located stationary background (gas) are taken into account. An important new circumstance is that the particle-particle scattering amplitude is not assumed to be small, which could be obtained, for example, in the first Born approximation. The derived collision integral can be used in the kinetic equation, including that for a relatively cold rarefied spin-polarized plasma with a characteristic electron energy below α{sup 2}m{sub e}c{sup 2}, where α is the fine-structure constant.
NASA Astrophysics Data System (ADS)
Stenroos, Matti
2016-11-01
Boundary element methods (BEM) are used for forward computation of bioelectromagnetic fields in multi-compartment volume conductor models. Most BEM approaches assume that each compartment is in contact with at most one external compartment. In this work, I present a general surface integral equation and BEM discretization that remove this limitation and allow BEM modeling of general piecewise-homogeneous medium. The new integral equation allows positioning of field points at junctioned boundary of more than two compartments, enabling the use of linear collocation BEM in such a complex geometry. A modular BEM implementation is presented for linear collocation and Galerkin approaches, starting from the standard formulation. The approach and resulting solver are verified in four ways, including comparisons of volume and surface potentials to those obtained using the finite element method (FEM), and the effect of a hole in skull on electroencephalographic scalp potentials is demonstrated.
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.
NASA Astrophysics Data System (ADS)
Marx, Egon
2007-12-01
The behavior of the field components near the edge has been shown to be that of the static fields, which is derived here without rigor for an infinite wedge. Fields scattered by a finite dielectric wedge illuminated by an arbitrary plane monochromatic wave are computed using either singular or hypersingular integral equations (SIEs or HIEs), derived by the single integral equation method. Field components are then computed near the edge of a finite wedge. Longitudinal components of the fields behave like constants, other components of the electric field behave like those in the transverse magnetic mode, and other components of the magnetic field behave like those in the transverse electric mode. Exceptions occur when approaching the wedge along the bisector. Boundary functions and transverse field components computed with SIEs rise more sharply than predicted approaching the edge after a range in which the agreement with those computed with HIEs is good.
Stenroos, Matti
2016-11-21
Boundary element methods (BEM) are used for forward computation of bioelectromagnetic fields in multi-compartment volume conductor models. Most BEM approaches assume that each compartment is in contact with at most one external compartment. In this work, I present a general surface integral equation and BEM discretization that remove this limitation and allow BEM modeling of general piecewise-homogeneous medium. The new integral equation allows positioning of field points at junctioned boundary of more than two compartments, enabling the use of linear collocation BEM in such a complex geometry. A modular BEM implementation is presented for linear collocation and Galerkin approaches, starting from the standard formulation. The approach and resulting solver are verified in four ways, including comparisons of volume and surface potentials to those obtained using the finite element method (FEM), and the effect of a hole in skull on electroencephalographic scalp potentials is demonstrated.
NASA Technical Reports Server (NTRS)
Chao, W. C.
1982-01-01
With appropriate modifications, a recently proposed explicit-multiple-time-step scheme (EMTSS) is incorporated into the UCLA model. In this scheme, the linearized terms in the governing equations that generate the gravity waves are split into different vertical modes. Each mode is integrated with an optimal time step, and at periodic intervals these modes are recombined. The other terms are integrated with a time step dictated by the CFL condition for low-frequency waves. This large time step requires a special modification of the advective terms in the polar region to maintain stability. Test runs for 72 h show that EMTSS is a stable, efficient and accurate scheme.
NASA Technical Reports Server (NTRS)
Joseph, Rose M.; Goorjian, Peter M.; Taflove, Allen
1993-01-01
We present what are to our knowledge first-time calculations from vector nonlinear Maxwell's equations of femtosecond soliton propagation and scattering, including carrier waves, in two-dimensional dielectric waveguides. The time integration efficiently implements linear and nonlinear convolutions for the electric polarization, and the nonlinear convolution accounts for two quantum effects, the Kerr and Raman interactions. By retaining the optical carrier, the new method solves for fundamental quantities - optical electric and magnetic fields in space and time - rather than a nonphysical envelope function. It has the potential to provide an unprecedented two- and three-dimensional modeling capability for millimeter-scale integrated-optical circuits with submicrometer engineered inhomogeneities.
NASA Astrophysics Data System (ADS)
Jungemann, C.; Pham, A. T.; Meinerzhagen, B.; Ringhofer, C.; Bollhöfer, M.
2006-07-01
The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real spaces by box integration. This ensures exact current continuity for the discrete equations. Spurious oscillations of the distribution function are suppressed by stabilization based on a maximum entropy dissipation principle avoiding the H transformation. The derived formulation can be used on arbitrary grids as long as box integration is possible. The approach works not only with analytical bands but also with full band structures in the case of holes. Results are presented for holes in bulk silicon based on a full band structure and electrons in a Si NPN bipolar junction transistor. The convergence of the spherical harmonics expansion is shown for a device, and it is found that the quasiballistic transport in nanoscale devices requires an expansion of considerably higher order than the usual first one. The stability of the discretization is demonstrated for a range of grid spacings in the real space and bias points which produce huge gradients in the electron density and electric field. It is shown that the resultant large linear system of equations can be solved in a memory efficient way by the numerically robust package ILUPACK.
1992-01-01
mathematical papers which describe various locking effects and analyze methods (mainly mixed methods ) to overcome it. However, the treatment in these...finite element method in various areas, such as the numerical approximation of three-dimensional PDEs anu integral equations, the investigation of mixed ... methods for these versions and, most importantly, the uniform approximation of parameter-dependent problems by these versions. By the p version, we
Penalized differential pathway analysis of integrative oncogenomics studies.
van Wieringen, Wessel N; van de Wiel, Mark A
2014-04-01
Through integration of genomic data from multiple sources, we may obtain a more accurate and complete picture of the molecular mechanisms underlying tumorigenesis. We discuss the integration of DNA copy number and mRNA gene expression data from an observational integrative genomics study involving cancer patients. The two molecular levels involved are linked through the central dogma of molecular biology. DNA copy number aberrations abound in the cancer cell. Here we investigate how these aberrations affect gene expression levels within a pathway using observational integrative genomics data of cancer patients. In particular, we aim to identify differential edges between regulatory networks of two groups involving these molecular levels. Motivated by the rate equations, the regulatory mechanism between DNA copy number aberrations and gene expression levels within a pathway is modeled by a simultaneous-equations model, for the one- and two-group case. The latter facilitates the identification of differential interactions between the two groups. Model parameters are estimated by penalized least squares using the lasso (L1) penalty to obtain a sparse pathway topology. Simulations show that the inclusion of DNA copy number data benefits the discovery of gene-gene interactions. In addition, the simulations reveal that cis-effects tend to be over-estimated in a univariate (single gene) analysis. In the application to real data from integrative oncogenomic studies we show that inclusion of prior information on the regulatory network architecture benefits the reproducibility of all edges. Furthermore, analyses of the TP53 and TGFb signaling pathways between ER+ and ER- samples from an integrative genomics breast cancer study identify reproducible differential regulatory patterns that corroborate with existing literature.
On testing a subroutine for the numerical integration of ordinary differential equations
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1973-01-01
This paper discusses how to numerically test a subroutine for the solution of ordinary differential equations. Results obtained with a variable order Adams method are given for eleven simple test cases.-
Some results on the integral transforms and applications to differential equations
Eltayeb, Hassan; Kilicman, Adem
2010-11-11
In this paper we give some remark about the relationship between Sumudu and Laplace transforms, further; for the comparison purpose, we apply both transforms to solve partial differential equations to see the differences and similarities.
NASA Astrophysics Data System (ADS)
Zieniuk, Eugeniusz; Kapturczak, Marta; Sawicki, Dominik
2016-06-01
In solving of boundary value problems the shapes of the boundary can be modelled by the curves widely used in computer graphics. In parametric integral equations system (PIES) such curves are directly included into the mathematical formalism. Its simplify the way of definition and modification of the shape of the boundary. Until now in PIES the B-spline, Bézier and Hermite curves were used. Recent developments in the computer graphics paid our attention, therefore we implemented in PIES possibility of defining the shape of boundary using the NURBS curves. The curves will allow us to modeling different shapes more precisely. In this paper we will compare PIES solutions (with applied NURBS) with the solutions existing in the literature.
Energy preserving integration of the strongly coupled nonlinear Schrödinger equation
NASA Astrophysics Data System (ADS)
Akkoyunlu, C.
2015-03-01
In this paper, average vector field method (AVF) is derived for strongly coupled Schrödinger equation (SCNLS). The SCNLS equation is discretized in space by finite differences and is solved in time by structure preserving AVF method. Numerical results for different paremeter compare with the Lobatto IIIA-IIIB method. The results indicate that AVF method are effective to preserve global energy and momentum.
Numerical integration of the master equation in some models of stochastic epidemiology.
Jenkinson, Garrett; Goutsias, John
2012-01-01
The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation--up to a desired precision--in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1.
Surface effects in anti-plane deformations of a micropolar elastic solid: integral equation methods
NASA Astrophysics Data System (ADS)
Sigaeva, Taisiya; Schiavone, Peter
2016-03-01
The theory of linear micropolar elasticity is used in conjunction with a new representation of micropolar surface mechanics to develop a comprehensive model for the deformations of a linearly micropolar elastic solid subjected to anti-plane shear loading. The proposed model represents the surface effect as a thin micropolar film of separate elasticity, perfectly bonded to the bulk. This model captures not only the micro-mechanical behavior of the bulk which is known to be considerable in many real materials but also the contribution of the surface effect which has been experimentally well observed for bodies with significant size-dependency and large surface area to volume ratios. The contribution of the surface mechanics to the ensuing boundary-value problem gives rise to a highly nonstandard boundary condition not accommodated by classical studies in this area. Nevertheless, the corresponding interior and exterior mixed boundary-value problems are formulated and reduced to systems of singular integro-differential equations using a representation of solutions in the form of modified single-layer potentials. Analysis of these systems demonstrates that the classical Noether theorems reduce to Fredholms theorems leading to results on well-posedness of the corresponding mathematical model.
Xue, Chuan
2015-01-01
Chemotaxis of single cells has been extensively studied and a great deal on intracellular signaling and cell movement is known. However, systematic methods to embed such information into continuum PDE models for cell population dynamics are still in their infancy. In this paper, we consider chemotaxis of run-and-tumble bacteria and derive continuum models that take into account of the detailed biochemistry of intracellular signaling. We analytically show that the macroscopic bacterial density can be approximated by the Patlak-Keller-Segel equation in response to signals that change slowly in space and time. We derive, for the first time, general formulas that represent the chemotactic sensitivity in terms of detailed descriptions of single-cell signaling dynamics in arbitrary space dimensions. These general formulas are useful in explaining relations of single cell behavior and population dynamics. As an example, we apply the theory to chemotaxis of bacterium Escherichia coli and show how the structure and kinetics of the intracellular signaling network determine the sensing properties of E. coli populations. Numerical comparison of the derived PDEs and the underlying cell-based models show quantitative agreements for signals that change slowly, and qualitative agreements for signals that change extremely fast. The general theory we develop here is readily applicable to chemotaxis of other run-and-tumble bacteria, or collective behavior of other individuals that move using a similar strategy.
Neural network training by integration of adjoint systems of equations forward in time
NASA Technical Reports Server (NTRS)
Toomarian, Nikzad (Inventor); Barhen, Jacob (Inventor)
1992-01-01
A method and apparatus for supervised neural learning of time dependent trajectories exploits the concepts of adjoint operators to enable computation of the gradient of an objective functional with respect to the various parameters of the network architecture in a highly efficient manner. Specifically, it combines the advantage of dramatic reductions in computational complexity inherent in adjoint methods with the ability to solve two adjoint systems of equations together forward in time. Not only is a large amount of computation and storage saved, but the handling of real-time applications becomes also possible. The invention has been applied it to two examples of representative complexity which have recently been analyzed in the open literature and demonstrated that a circular trajectory can be learned in approximately 200 iterations compared to the 12000 reported in the literature. A figure eight trajectory was achieved in under 500 iterations compared to 20000 previously required. The trajectories computed using our new method are much closer to the target trajectories than was reported in previous studies.
Neural Network Training by Integration of Adjoint Systems of Equations Forward in Time
NASA Technical Reports Server (NTRS)
Toomarian, Nikzad (Inventor); Barhen, Jacob (Inventor)
1999-01-01
A method and apparatus for supervised neural learning of time dependent trajectories exploits the concepts of adjoint operators to enable computation of the gradient of an objective functional with respect to the various parameters of the network architecture in a highly efficient manner. Specifically. it combines the advantage of dramatic reductions in computational complexity inherent in adjoint methods with the ability to solve two adjoint systems of equations together forward in time. Not only is a large amount of computation and storage saved. but the handling of real-time applications becomes also possible. The invention has been applied it to two examples of representative complexity which have recently been analyzed in the open literature and demonstrated that a circular trajectory can be learned in approximately 200 iterations compared to the 12000 reported in the literature. A figure eight trajectory was achieved in under 500 iterations compared to 20000 previously required. Tbc trajectories computed using our new method are much closer to the target trajectories than was reported in previous studies.
A comparative study of different ferrofluid constitutive equations.
NASA Astrophysics Data System (ADS)
Kaloni, Purna
2011-11-01
Ferrofluids are stable colloidal suspensions of fine ferromagnetic monodomain nanoparticles in a non-conducting carrier fluid. The particles are coated with a surfacant to avoid agglomeration and coagulation.Brownian motion keeps the nanoparticles from settling under gravity. In recent years these fluids have found several applications including in liquid seals in rotary shafts for vacuum system and in hard disk drives of personal computers, in cooling and damping of loud speakers, in shock absorbers and in biomedical applications. A continuum description of ferrofluids was initiated by Neuringer and Rosensweig but the theory had some limitations. In subsequent years,several authors have proposed generalization of the above theory.Some of these are based upon the internal particle rotation concept, some are phemonological, some are based upon a thermodynamic framework, some employ statistical approach and some have used the dynamic mean field approach. The results based upon these theories ane in early stages and inconclusive. Our purpose is, first, to critically examine the basic foundations of these equations and then study the pedictions obtained in all the theories related to an experimental as well as a theoretical study.
Comparative study of homotopy continuation methods for nonlinear algebraic equations
NASA Astrophysics Data System (ADS)
Nor, Hafizudin Mohamad; Ismail, Ahmad Izani Md.; Majid, Ahmad Abd.
2014-07-01
We compare some recent homotopy continuation methods to see which method has greater applicability and greater accuracy. We test the methods on systems of nonlinear algebraic equations. The results obtained indicate the superior accuracy of Newton Homotopy Continuation Method (NHCM).