Topic
Integro-differential equation
About: Integro-differential equation is a research topic. Over the lifetime, 4264 publications have been published within this topic receiving 80207 citations.
Papers published on a yearly basis
Papers
More filters
Book•
08 Dec 1993
TL;DR: Fractional integrals and derivatives on an interval fractional integral integrals on the real axis and half-axis further properties of fractional integral and derivatives, and derivatives of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations with special function kernels applications to differential equations as discussed by the authors.
Abstract: Fractional integrals and derivatives on an interval fractional integrals and derivatives on the real axis and half-axis further properties of fractional integrals and derivatives other forms of fractional integrals and derivatives fractional integrodifferentiation of functions of many variables applications to integral equations of the first kind with power and power-logarithmic kernels integral equations fo the first kind with special function kernels applications to differential equations.
7,096 citations
Book•
01 Jan 1968
TL;DR: In this article, a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation can be found is presented, where the main tool used is the first remarkable series of integrals discovered by Kruskal and Zabusky.
Abstract: In Section 1 we present a general principle for associating nonlinear equations evolutions with linear operators so that the eigenvalues of the linear operator integrals of the nonlinear equation. A striking instance of such a procedure discovery by Gardner, Miura and Kruskal that the eigenvalues of the Schrodinger operator are integrals of the Korteweg-de Vries equation.
In Section 2 we prove the simplest case of a conjecture of Kruskal and Zabusky concerning the existence of double wave solutions of the Korteweg-de Vries equation, i.e., of solutions which for |I| large behave as the superposition of two solitary waves travelling at different speeds. The main tool used is the first of remarkable series of integrals discovered by Kruskal and Zabusky.
2,124 citations
TL;DR: In this article, the equation of state and pair distribution for the Percus- Yevick integral equation for the radiai distribution function of a classical fluid are obtained in closed form for the prototype of interacting hard spheres.
Abstract: ABS>The equation of state and the pair distribution for the Percus- Yevick integral equation for the radiai distribution function of a classical fluid are obtained in closed form for the prototype of interacting hard spheres. (D.C.W.)
1,420 citations
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 application of integral equation methods to exterior boundary-value problems for Laplace's equation and for the Helmholtz (or reduced wave) equation is discussed in this article, where it is shown that uniqueness can be restored by deriving a second integral equation and suitably combining it with the first.
Abstract: The application of integral equation methods to exterior boundary-value problems for Laplace’s equation and for the Helmholtz (or reduced wave) equation is discussed. In the latter case the straightforward formulation in terms of a single integral equation may give rise to difficulties of non-uniqueness; it is shown that uniqueness can be restored by deriving a second integral equation and suitably combining it with the first. Finally, an outline is given of methods for transforming the integral operators with strongly singular kernels which occur in the second equation.
1,127 citations