An efficient numerical method for Volterra integral equation of the second kind with a weakly singular kernel

Abstract

In this paper, we propose an efficient numerical method combining with a series solution and conformal maps for solving Volterra integral equation (VIE) of the second kind with a weakly singular kernel. A convergent series solution is applied to deal with the initial singularity within arbitrary accuracy. While for the solution away from the singular point, with the aid of conformal maps, Method 1 and Method 2 proposed in Hale et al. (2008) are utilized to derive efficient numerical schemes. Rigorously theoretical analysis is carried out to obtain the convergence rate of the proposed methods. Numerical experiments are designed to show the performance with the theoretical results.