Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel

In this study, a new collocation method based on the Bernstein polynomials is introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel. If the exact solution is polynomial, then the exact solution can be obtained. If the exact solution is not a polynomial, then an accurate solution can be obtained with a combination of choice in the number of nodes and the number of digits in the solver. In addition, the method is presented with error and stability analysis.     

On the numerical solution of nonlinear Volterra integro-differential equations

Piecewise polynomialss(x) of degreem2 and of continuity classC ¹ are used to obtain approximating functions to the exact solution of a given (ordinary) integro-differential equation of Volterra type. The unknown coefficients ofs(x) are computed recursively, by requiring thats(x) satisfy the integro-differential equation on a finite set of suitably chosen points. Results on the order of convergence of this method are given, together with a numerical illustration.This research was supported by the National Research of Canada (Grant No. A-4805). Received March 13, 1973. Revised July 2, 1973.     

Explicit solutions of Cauchy singular integral equations with weighted Carleman shift

Existence and uniqueness of solutions, as well as their explicit representations, are obtained for singular integral equations with weighted Carleman shift which cannot be reduced to binomial boundary value problems.     

On a class of integro-differential equations modeling complex systems with nonlinear interactions

This work deals with the qualitative analysis of the initial value problem for a class of large systems of interacting entities in the framework of the mathematical kinetic theory for active particles. The contents are specifically focused on the case where the system interacts with the outer environment and the entities are subject to nonlinearly additive interactions.     

On a family of singular integro-differential equations

Integro-differential systems in which the matrix multiplying the derivative of the unknown vector function is identically singular are analyzed. This analysis is based on the special properties of the matrix polynomials associated with the original system. An existence theorem is proved, and a numerical method for finding the solution is proposed and justified.     

Numerical methods for singular nonlinear integro-differential equations

For the numerical integration of singular nonlinear integro-differential equations we consider fractional linear multistep methods. We prove convergence of these methods and discuss their stability (as an extension of A-stability for stiff differential equations). Numerical experiments with the Basset equation are included.     

A note on a class of singular integro-differential equations

A simplified analysis is employed to handle a class of singular integro-differential equations for their solutions     

On the approximate solution of weakly singular integro-differential equations of Volterra type

Recently, the convergence rate of the collocation method for integral and integro-differential equations with weakly singular kernels has been studied in a series of papers [1–7]. The present paper belongs to the same series. We analyze the possibility of constructing approximate solutions of high-order accuracy on a uniform or almost uniform grid for weakly singular integro-differential equations of Volterra type.Translated from Differentsialnye Uravneniya, Vol. 40, No. 9, 2004, pp. 1271–1279.Original Russian Text Copyright © 2004 by Pedas.     

On a certain class of a nonlinear evolution partial differential equations

An existence result and a priori bound for the solution of a second-order nonlinear parabolic equation are established. Also a generalized tanh-function method is used for constructing exact travelling wave solutions for the nonlinear diffusion equation of Fisher type originated from the considered partial differential equation. And new multiple soliton solutions are obtained.     

On the analytical separation of variables solution for a class of partial integro-differential equations

The present work deals with the derivation of analytical solutions for a particular class of partial intregro-differential equations, by employing the separation of variables technique. The equation under consideration consists of parabolic and hyperbolic terms, so that the character of the equation is determined by their relative weight. The key feature for the analysis presented here is that the eigenfunctions of the problem, which can be found in a closed form, do not comprise an orthogonal set of functions. The convergence of the new series solution is getting faster as the parabolic term of the integro-differential equation dominates over the hyperbolic (integral) one.     

Numerical solution of separable nonlinear equations with a singular matrix at the solution

Numerical Algorithms - We present a numerical method for solving the separable nonlinear equation A(y)z + b(y) =?0, where A(y) is an m × N matrix and b(y) is a vector, with y ∈Rn...     

On the uniqueness of the solution of a boundary-value problem for weakly nonlinear integro-differential equations with parameters

The conditions of existence of the unique solution of a boundary-value problem for weakly nonlinear integro-differential equations with parameters and the equivalence of such problem and the appropriate integral equation are established.     

Quadrature-difference methods for solving linear and nonlinear singular integro-differential equations

Here we propose and justify quadrature-difference methods for solving different kinds (linear, nonlinear and multidimensional) of periodic singular integro-differential equations.     

On the numerical solution of linear and nonlinear volterra integral and integro-differential equations

Sinc bases are developed to approximate the solutions of linear and nonlinear Volterra integral and integro-differential equations. Properties of these sinc bases and some operational matrices are first presented. These properties are then used to reduce the integral and integro-differential equations to systems of linear or nonlinear algebraic equations. Numerical examples illustrate the pertinent features of the method and its applicability to a large variety of problems. The examples include convolution type, singular as well as singularly-perturbed problems.     

Asymptotic behavior of the solution of a system of nonlinear integro-differential equations

We study the asymptotic behavior as time tends to infinity of the solution of an initial-boundary value problem for a system of nonlinear integro-differential equations that arises in the mathematical modeling of penetration of electromagnetic field into a medium whose electric conductivity substantially depends on temperature. Both homogeneous and inhomogeneous boundary conditions are considered. The exponential stabilization of the solution is established.