Numerical Solution of Volterra Integral Equations with Weakly Singular Kernel Based on the DE-Sinc Method

Abstract

A method for numerical solution of Volterra integral equations of the second kind with a weakly singular kernel based on the double exponential (DE) transformation is proposed. In this method we first express the approximate solution in the form of a Sinc expansion based on the double exponential transformation by Takahasi and Mori in 1974 followed by collocation at the Sinc points. We also apply the DE formula to the kernel integration. In every sample equation a numerical solution with very high accuracy is obtained and a nearly exponential convergence rate $\exp(-cM/{\log M})$, $c>0$ in the error is observed where $M$ is a parameter representing the number of terms in the Sinc expansion. We compared the result with the one based on the single exponential (SE) transformation by Riley in 1992 which made us confirm the high efficiency of the present method.