Showing papers on "Integro-differential equation published in 1992"
TL;DR: In this article, an asymptotics of the Boltzmann equation leading to the Fokker-Planck-Landau equation was shown to be valid in the context of linearized equations and an extension to the Ka[cbreve] equation.
Abstract: We deal in this work with an asymptotics of the Boltzmann equation leading to the Fokker-Planck-Landau equation. We prove its mathematical validity in the context of linearized equations and give an extension to the Ka[cbreve] equation.
185 citations
TL;DR: In this paper, the Galerkin finite element method of an integro-differential equation of parabolic type with a memory term containing a weakly singular kernel is used to give error estimates for the numerical solution.
Abstract: We give error estimates for the numerical solution by means of the Galerkin finite element method of an integro-differential equation of parabolic type with a memory term containing a weakly singular kernel. Optimal-order estimates are shown for spatially semidiscrete and completely discrete methods. Special attention is paid to the regularity of the exact solution.
102 citations
TL;DR: In this paper, a variational formulation of the electrostatic potential and standard multi-dimensional maximization methods are used to solve the full non-linear Poisson-Boltzmann equation.
Abstract: The Poisson–Boltzmann equation can be used to calculate the electrostatic potential field of a molecule surrounded by a solvent containing mobile ions. The Poisson–Boltzmann equation is a non-linear partial differential equation. Finite-difference methods of solving this equation have been restricted to the linearized form of the equation or a finite number of non-linear terms. Here we introduce a method based on a variational formulation of the electrostatic potential and standard multi-dimensional maximization methods that can be used to solve the full non-linear equation. © 1992 by John Wiley & Sons, Inc.
100 citations
IBM1
TL;DR: In this paper, the authors present a method for the rapid, high order accurate evaluation of certain volume integrals in potential theory on general irregular regions, where the kernels of the integrals are either a fundamental solution, or a linear combination of the derivatives of the fundamental solution of a second-order linear elliptic differential equation.
66 citations
TL;DR: In this paper, the authors make use of the Sherman-Lauricella integral equation which is a Fredholm equation with bounded kernel and describe a fast algorithm for the evaluation of the integral operators appearing in that equation.
63 citations
59 citations
TL;DR: In this paper, the stability properties of a numerical scheme applied to a Volterra integro-differential equation with a finite memory were discussed, in which the solution is determined by an initial function.
44 citations
TL;DR: In this article, the 2D sine-Gordon equation was reduced to an algebraic system consisting of three equations, and three classes of solutions for this equation were found, which are a generalization of the solutions of the solution of the 1D SING equation, phi xx+y=sin phi (x,y).
Abstract: After the Lamb substitution the 2D sine-Gordon equation was solved. Three classes of solutions for this equation were found. Ad hoc the 2D sine-Gordon equation was reduced to an algebraic system consisting of three equations. The solutions given of the 2D sine-Gordon equation are a generalization of the solutions of the 1D sine-Gordon equation. They are also a generalization of the solutions of the equation phi xx+yy=sin phi (x,y).
40 citations
TL;DR: Differential equations with singular coefficients such as Bessel's equation, Legendre's equation and Chebychev's equation are solved by the decomposition method as mentioned in this paper, and Laguerre's solution is solved by decomposition.
29 citations
01 Jan 1992
TL;DR: Laplace's equation for steady state heat flow in a two-dimensional region with fixed temperatures on the boundaries is discussed, using successive overrelaxation with parity ordering of the grid elements to computed equilibrium temperatures.
Abstract: This tutorial discusses Laplace's equation for steady state heat flow in a two-dimensional region with fixed temperatures on the boundaries. The equilibrium temperatures are computed for a square grid using successive overrelaxation with parity ordering of the grid elements. The numerical method is illustrated by a Pascal algorithm. We assume that the reader is familiar with elementary calculus.
29 citations
TL;DR: This paper shows that this linear differential equation is naturally introduced through this Dirac equation after it is constructed on an elastic rod embedded into (2+1)-dimensional space-time.
Abstract: The 2×2-matrix-valued first-order linear differential equation appears when we solve the modified Korteweg-de Vries (MKdV) equation. This linear differential equation is regarded as a fictitious quantum equation. On the other hand, it is known that the dynamics of an elastic rod is governed by the MKdV equation. In this paper, after we construct a Dirac equation on an elastic rod embedded into (2+1)-dimensional space-time, we show that this linear differential equation is naturally introduced through this Dirac equation
Journal Article•
TL;DR: In this paper, a simple and accurate procedure for solving the infiltration equation for nonlinear diffusivity and conductivity is given, which overcomes some of the numerical difficulties associated with an earlier procedure given by Hogarth et al. but confirms the accuracy of their results.
Abstract: A simple and accurate procedure is given for solving the infiltration equation for nonlinear diffusivity and conductivity. It overcomes some of the numerical difficulties associated with an earlier procedure given by Hogarth et al. (1989), but confirms the accuracy of their results.
TL;DR: In this article, the integro-differential equation approach (IDEA) was extended from few nucleon to closed-shell and closed-subshell nuclei and the analytical methods required for the calculation of the density functions, which entered into the IDEA, were presented.
Abstract: The authors extend the integro-differential equation approach (IDEA) from few nucleon to closed-shell and closed-subshell nuclei and outline the analytical methods required for the calculation of the density functions, which enter into the integro-differential equations. These contain all the physics for a system of fermions associated with the Pauli principle. In order to test the accuracy of the IDEA comparisons are made of the binding energies of 4He, 12C and 16O obtained with effective potentials using the hypercentral approximation (HCA) providing a variational solution without correlations, the IDEA which fully includes the two-body correlations, the S-state integro-differential equation (SIDE) valid for potentials operating only on pairs in the S-state and those calculated by several variational or perturbative methods in the literature.
TL;DR: In this paper, a method for solving the Schrodinger equation with spherically symmetric potentials is presented, which is based on the use of a certain integral transform for the radial wavefunction.
Abstract: The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.
TL;DR: In this paper, the identification of damping parameters in a Volterra integro-differential system with a weakly singular kernel was studied. But the method was adapted to weakly-singular systems whose kernels are not continuous at the origin.
TL;DR: In this article, a survey is given of new results of the Painleve test and nonlinear evolution equations where ordinary-and partial-differential equations are considered, including the self-dual Yang-Mills equation.
Abstract: A survey is given of new results of the Painleve test and nonlinear evolution equations where ordinary- and partial-differential equations are considered. We study the semiclassical Jaynes-Cumming model, the energy-eigenvalue-level-motion equation, the Kadomtsev-Petviashvili equation, the nonlinear Klein-Gordon equation and the self-dual Yang-Mills equation.
TL;DR: In this paper, the reduction of an SISO system is formulated as the minimisation of the l 2 criterion, and an efficient method for evaluation of the required scalar products of repeated integrals and derivatives is proposed.
Abstract: The reduction of an SISO system is formulated as the minimisation of the l2 criterion. Expansions of a time-response function in terms of its derivatives and integrals are proposed. Approximation of the quadratic criterion reduces the nonlinear determination of the denominator to a simple linear system resolution. An efficient method for evaluation of the required scalar products of repeated integrals and derivatives is proposed.
TL;DR: A reconstruction procedure for electrical conductance tomography developed by solving a linear Fredholm integral equation of the first kind obtained from a linearized Poisson's equations is discussed.
Abstract: A reconstruction procedure for electrical conductance tomography developed by solving a linear Fredholm integral equation of the first kind is discussed. The integral equation is obtained from a linearized Poisson's equations. Properties of the integral equation are discussed, and problems associated with numerical solution of the equation are treated. The reconstruction requires only one matrix multiplication and therefore can be computed in a short time. Test results of the algorithm using both simulated and measured data are presented. >
TL;DR: In this paper, energy inequalities for the integro-partial differential equations with the Riemann-Liouville integrals are presented, and the proofs depend on the Fourier analysis and the probability methods.
Abstract: This paper presents energy inequalities for the integro-partial differential equations with the Riemann–Liouville integrals. These equations interpolate between the heat equation and the wave equation. This fact is reflected in the energy inequalities so that they correspond to the energy equality for the wave equation. The proofs depend on the Fourier analysis and the probability methods.
01 Jan 1992
TL;DR: In this article, a Fredholm integral equation of the second kind for the Green's function associated to a heterogenous medium is derived based on the idea to interpret the space dependent propagation speed of the wave equation as a perturbation of a constant reference velocity.
Abstract: In this paper a Fredholm integral equation of the second kind for the Green’s function associated to a heterogenous medium is derived. This approach is based on the idea to interpret the space dependent propagation speed of the wave equation as a perturbation of a constant reference velocity. The integral equation can be solved by standard quadrature methods. Once the Green’s function is calculated, the seismic response to an arbitrary source function can be calculated by a simple convolution. Also, since the calculations are performed in the frequency domain, an incorporation of attenuation mechanisms can be applied easily. Some numerical examples for one-dimensional acoustic and viscoacoustic media are presented. The derived integral equation can also be interpreted as first kind integral equation for the perturbation potential. Thus it can be applied to the inverse problem as well. In order to treat the ill-posedness of this problem it is necessary to apply certain regularization methods such as truncated singular value decomposition or Tikhonov regularization. Some numerical results for Born inversion applied to synthetic data obtained by the integral equation modeling are presented.
TL;DR: The spinmomentum equation as mentioned in this paper is the eigenvalue equation for the inner product of spin and kinetic momentum operators in the absence of a scalar potential, and it can be solved directly for a constant magnetic field, and the solution is compared with the well known Pauli equation solutions.
Abstract: The gyromagnetic ratio or g factor of the electron is correctly predicted by the Dirac equation to be 2. However, it is also possible to obtain this value from nonrelativistic quantum mechanics, as shown by Sakurai, who credits Feynman. It is noted here that the same trick used by Sakurai allows a factorization of the time independent Pauli equation. The result is an equivalent first‐order equation which is referred to as the spin‐momentum equation. In the absence of a scalar potential, the spin‐momentum equation is the eigenvalue equation for the inner product of spin and kinetic momentum operators. The spin‐momentum equation is solved directly for a constant magnetic field, and the solution is compared with the well‐known Pauli equation solutions.
TL;DR: Considering a generalized form of the Emden-Fowler equation and using a direct and simple method, this paper found sufficient conditions of integrability which enable us to obtain its first integrals as well as the solutions.
Abstract: Considering a generalized form of the Emden-Fowler equation and using a direct and simple method, we find sufficient conditions of integrability which enable us to obtain its first integrals as well as the solutions. As special cases, we discuss the integrals of the Emden-Fowler equation ẍ + ( k 1 / t ) x = Kt k 2 x R . Illustrative examples are given.
29 Dec 1992
TL;DR: A review of the various integral equation formuiations that have been employed for the inverse source problem for the inhomogeneous scalar Heimhoitz equation can be found in this paper.
Abstract: This paper presents a brief review of the various integral equation formuiations that have been employedfor the inverse source problem for the inhomogeneous scalar Heimhoitz equation. It is shown that theseformulations apply only in cases where either the data are prescribed on a closed surface surrounding theunknown source or where the unknown source lies entirely on one side of an open measurement surface.A generalized integral equation is derived that applies to the more general case where unknown sourcescan exist on both sides of an open measurement surface. This latter problem arises in geophysical remote sensing and the derived integral equation offers an approach to this class of problems not offered by currently employed techniques. 1. Introduction R.P. Porter [1,2,3} and N. Bojarski [4] independently derived an integral equation that relates an unknownsource p to the inhomogeneous Helmholtz equation to an image of this source generated from field dataspecified over a closed surface surrounding the source. This integral equation, and certain generalizationsknown collectively as the "Porter-Bojarski integral equations", have formed the basis for a number of appli-cations in various inverse problems related to the Helmholtz equation [5,6,7] .
TL;DR: In this paper, the oscillatory properties of the solutions of the equation [(Lx)(t)](n) have been studied in terms of oscillations of the solution of the problem.
Abstract: In the present paper the oscillatory properties of the solutions of the equation[(Lx)(t)](n)
TL;DR: In this paper, the Bethe-Salpeter equation for two-fermion scattering states is reduced to an equivalent Pauli-Schrodinger equation, which provides a new approach to the relativistic scattering problem.
Abstract: The Bethe-Salpeter equation for two-fermion scattering states is reduced to an equivalent Pauli-Schrodinger equation. The latter equation provides a new approach to the relativistic scattering problem. Since this equation may avoid the problem of solving coupled equations, it appears to be more convenient than the Bethe-Salpeter equation in practical applications.