Showing papers on "Integro-differential equation published in 1984"
01 Jan 1984
TL;DR: In this paper, an equation for the distribution function describing Brownian motion was first derived by Fokker [11] and Planck [12] and it is shown that expectation values for nonlinear Langevin equations (367, 110) are much more difficult to obtain.
Abstract: As shown in Sects 31, 2 we can immediately obtain expectation values for processes described by the linear Langevin equations (31, 31) For nonlinear Langevin equations (367, 110) expectation values are much more difficult to obtain, so here we first try to derive an equation for the distribution function As mentioned already in the introduction, a differential equation for the distribution function describing Brownian motion was first derived by Fokker [11] and Planck [12]: many review articles and books on the Fokker-Planck equation now exist [15 – 15]
1,412 citations
TL;DR: The main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region to solve Laplace's and the biharmonic equations on irregular regions with smooth boundaries.
Abstract: We present fast methods for solving Laplace’s and the biharmonic equations on irregular regions with smooth boundaries. The methods used for solving both equations make use of fast Poisson solvers on a rectangular region in which the irregular region is embedded. They also both use an integral equation formulation of the problem where the integral equations are Fredholm integral equations of the second kind. The main idea is to use the integral equation formulation to define a discontinuous extension of the solution to the rest of the rectangular region. Fast solvers are then used to compute the extended solution. Aside from solving the equations we have also been able to compute derivatives of the solutions with little loss of accuracy when the data was sufficiently smooth.
283 citations
01 Jan 1984
196 citations
TL;DR: In this paper, a non-local linear integral equation is studied, which allows for Backlund transformations in the measure, and the compatibility of three of these transformations leads to an integrable nonlinear three-dimensional lattice equation.
130 citations
TL;DR: In this paper, a 6th order method was developed for approximate numerical integration of the Schrodinger equation, which was shown to be generally superior to the widely used Numerov method.
125 citations
68 citations
Book•
01 Aug 1984
TL;DR: In this paper, an Integral Equation Method for the Solution of Singular Slow Flow Problems was proposed for the study of Slow Flow in Bearings with Arbitrary Geometrics.
Abstract: Content.- 1 - General Introduction.- 2 - An Integral Equation Method for the Solution of Singular Slow Flow Problems.- 3 - Modified Integral Equation Solution of Viscous Flows Near Sharp Corners.- 4 - Solution of Nonlinear Elliptic Equations with Boundary Singularities by an Integral Equation Method.- 5 - Boundary Integral Equation Solution of Viscous Flows with Free Surfaces.- 6 - A Boundary Integral Equation Method for the Study of Slow Flow in Bearings with Arbitrary Geometrics.- 7 - General Conclusions.
66 citations
TL;DR: In this article, it was shown that the Percus-Yevick integral equation for the pair distribution function of a fluid interacting with a hard core potential can be solved not only in one and three dimensions, where the solution is well known, but more generally in all odd dimensions.
Abstract: It is shown that the Percus-Yevick integral equation for the pair distribution function of a fluid interacting with a hard-core potential can be solved not only in one and three dimensions, where the solution is well known, but more generally in all odd dimensions. The nonlinear integral equation is reduced to an algebraic equation of order d−3 for odd dimensions d greater than three. As an example the direct correlation function in five dimensions is derived explicitly.
63 citations
TL;DR: An exact integral equation for the pair-connectedness function gDouble Dagger (12) is derived in this paper, which gives rise to a sequence of approximations, starting with the Percus-Yevick approximation.
Abstract: An exact integral equation for the pair-connectedness function gDouble Dagger (12) is derived on the basis of the author's earlier exact equation for the pair-distribution function g(12). The new equation gives rise to a sequence of approximations for gDouble Dagger (12), starting with the Percus-Yevick approximation. Significant aspects of the first two approximations are noted, as is the relevance of the blocking functions g-gDouble Dagger for ionic and chemically associating particles.
48 citations
TL;DR: An exact analytical solution for travelling waves of the Fisher equation with a general nonlinearity is found in this article, where boundary values, the boundedness and the stability of the solution are discussed.
TL;DR: In this article, it was shown that the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the Benjamin-Ono equation for any δ > 0.
Abstract: The intermediate long wave equation is a physically important singular integrodifferential equation containing a parameter, referred to here as δ. For δ → ∞ it reduces to the Benjamin–Ono equation. It has been recently shown that the inverse scattering transform schemes of the above equations have certain significant differences. Here it is shown that for δ → ∞, the inverse scattering transform scheme of the intermediate long wave equation reduces to that of the Benjamin–Ono equation.
TL;DR: In this paper, global and local boundedness results for the solutions of a certain class of A2-degenerate parabolic equations are proved for a class of parabolic functions.
Abstract: Global and local boundedness results for the solutions of a certain class of A2-degenerate parabolic equations are proved.
TL;DR: The system of age dependent master equations (ADME) as mentioned in this paper is a generalization of the phenomenological master equation (PME), which reduces to a Volterra integral equation.
TL;DR: In this article, a recursion operator for the classical Boussinesq equation is given, which yields infinitely many symmetries and conservation laws, each of which is a hamiltonian system.
TL;DR: In this article, a hierarchy of nonlinear evolution equations associated with a generalization of the Schrodinger spectral problem is derived, and it is shown that each equation is Hamiltonian and that their flows commute.
Abstract: A hierarchy of nonlinear evolution equations associated with a generalization of the Schrodinger spectral problem is derived. It is shown that each equation is Hamiltonian and that their flows commute. The spectral equation is examined and certain difficulties in the inverse problem are pointed out.
TL;DR: In this article, it was shown that the diffusivity has a power-law dependence on the water content and the conductivity is proportional to water content, which can be used to check and improve the accuracy of the iterative schemes.
Abstract: Solutions of the infiltration equation have to be obtained, in general, by numerical or analytical, but approximate iterative schemes. We show here that solutions can be obtained without iterations if the diffusivity has a power law dependence on the water content and the conductivity is proportional to the water content. Even though this last condition may be too restrictive to adequately describe most soils, these solutions can be of great use to check and improve the accuracy of the iterative schemes. For a particular dependence of the surface flux on time, a solution is obtained that appears to be the only existing fully analytical solution for a realistic diffusivity. We use this particular solution to assess the accuracy of a very general optimization technique.
TL;DR: In this article, the spherical Raman-Nath differential equation is solved using a generalisation of an operatorial approach successfully applied to differential-difference equations which are particular cases of the one discussed here.
Abstract: The authors discuss the solution of the spherical Raman-Nath differential equation. This type of equation appears in diverse physical problems, one of which is stimulated Compton scattering. The solution technique exploits the generalisation of an operatorial approach successfully applied to differential-difference equations which are particular cases of the one discussed here.
TL;DR: In this article, the Harmonic Raman-Nath equation is analyzed and a non-trivial perturbative solution is presented, and the connection with the conventional Raman Nith equation is discussed.
Abstract: Analyses the Harmonic Raman-Nath equation and present a non-trivial perturbative solution. The connection with the conventional Raman-Nath equation is also discussed.
TL;DR: In this paper, analytic expressions are presented for some of the integrals involving the product of the Legendre polynomials with the exponential integral function and the exponentials.
Abstract: The integrals involving the product of the Legendre polynomials with the exponential integral function and the exponentials arise in the solution of equation of radiative transfer by a suitable expansion in the space variable in terms of the Legendre polynomials. The success of such methods of solution depends on the availability of rapidly converging analytic expressions for such integrals. In this work, analytic expressions are presented for some of such integrals.
TL;DR: In this article, a new integral formula is obtained for solving the equation of steady-state thermoelasticity in a three-dimensional region, differing from the well-known formula /1/ in containing no volume integral.
TL;DR: In this paper, numerical and analytic solutions to the perturbed sine-Gordon equation were presented for long Josephson tunnel junctions, where the current through the junction is larger than the critical current.
TL;DR: The A. J. Burton and G. F. Miller integral equation formulation for the exterior Neumann problem for the Helmholtz equation was studied in this paper, where appropriate space settings were found and a rigorous existence and uniqueness proof was given.
TL;DR: An integral equation of boundary value problems for the Laplace equation is investigated in this paper, where an integral integral equation for the boundary value problem of boundary problems with Laplace equations is investigated.
Abstract: An integral equation of boundary value problems for the Laplace equation is investigated. Bibliography: 14 titles.
TL;DR: In this article, the authors consider the use of the somme de la serie d'harmoniques spheriques for resoudre l'equation de Laplace for diverses situations.
TL;DR: In this article, a Volterra integral equation, which relates the Fourier coefficients of the projection (in polar coordinates) with the corresponding coefficients of an unknown density function, is deduced, which holds for both divergent and parallel beam projections.
Abstract: A Volterra integral equation, which relates the Fourier coefficients of the projection (in polar coordinates) with the corresponding coefficients of the unknown density function, is deduced. The same equation holds for both divergent and parallel beam projections.The problem is shown to be “ill-conditioned.” A numerical solution, based on the recursive evaluation of certain integrals, is proposed and a Tikhonov regularization procedure is applied to the discrete problem. Numerical examples are also reported.
TL;DR: The dynamic stability of a thin plate in supersonic flow based on 2-dimensional linear theory leads to the study of a new problem in mathematical physics: complex eigenvalue problem for a non-self-adjoint fourth-order integro-differential equation of Volterra's type as discussed by the authors.
Abstract: The dynamic stability of a thin plate in supersonic flow based on 2-dimensional linear theory leads to the study of a new problem in mathematical physics: complex eigenvalue problem for a non-self-adjoint fourth-order integro-differential equation of Volterra's type.
TL;DR: In this article, the Falkner-Skan equation is solved through the use of pseudo-spectral methods for different values of the similarity parameter β and a convergence criteria is established for each method.