Numerical Solution of Linear Volterra Integral Equation Systems of Second Kind by Radial Basis Functions

Pedro González-Rodelas,Miguel Pasadas,Basim Mustafa,Abdelouahed Kouibia

doi:10.3390/math10020223

Abstract

In this paper we propose an approximation method for solving second kind Volterra integral equation systems by radial basis functions. It is based on the minimization of a suitable functional in a discrete space generated by compactly supported radial basis functions of Wendland type. We prove two convergence results, and we highlight this because most recent published papers in the literature do not include any. We present some numerical examples in order to show and justify the validity of the proposed method. Our proposed technique gives an acceptable accuracy with small use of the data, resulting also in a low computational cost.

Highlights

A considerable large amount of research literature and books on the theory and applications of Volterra’s integral equations have emerged over many decades since the apparition of Volterra’s book “Leçons sur les équations intégrales et intégro-différentielles” [1] in 1913.The applications include elasticity, plasticity, semi-conductors, scattering theory, seismology, heat and mass conduction or transfer, metallurgy, fluid flow dynamics, chemical reactions, population dynamics, and oscillation theory, among many others
Volterra integral equations (VIEs) appear naturally when we try to transform an initial value problem into integral form, so that the solution of this integral equation is usually much easier to obtain than the original initial value problem
Some nonlinear Volterra integral equations are equivalent to an initial-value problem for a system of ordinary differential equations (ODEs)

Summary

Introduction

A considerable large amount of research literature and books on the theory and applications of Volterra’s integral equations have emerged over many decades since the apparition of Volterra’s book “Leçons sur les équations intégrales et intégro-différentielles” [1] in 1913. Several numerical methods based on different triangular type and delta orthogonal functions were designed for approximating the solution of integral and/or integro-differential Volterra equations (see for example [15,16,17], and the references therein). All these publications have demonstrated and revealed that these techniques based on PCBF and wavelets are effective to obtain the solution of such integral equations.

Notations and Preliminaries

Discretization Space

Convergence Result

Numerical Examples

Conclusions