Volterra integral equations solved in Fredholm form using Walsh functions

Abstract

Recently Walsh function methods have been developed for the numerical solution of several classes of problems, mainly linear and nonlinear integral equations of both Volterra and Fredholm types. In addition, modifications of the basic approach have led to the solution of functional differential equations, variational problems and parameter estimation problems. Linear Volterra integral equations are re-written as Fredholm integral equations with appropriately modified kernels. In this Fredholm equation form, the Walsh function solution method is more efficient than directly solving the Volterra equation. Walsh function methods are spectral methods but they have a natural grid interpretation. Multigrid methods and a variation on the use of Richardson extrapolation are used on six well known Volterra test problems, re-written in Fredholm form, to illustrate that these methods provide effective and efficient solution methods.