Numerical solution of two-dimensional weakly singular Volterra integral equations with non-smooth solutions

Abstract

Various numerical methods have been proposed for the solution of weakly singular Volterra integral equations but, for the most part, authors have dealt with linear or one-dimensional weakly singular Volterra integral equations, or have assumed that these equations have smooth solutions. The main purpose of this paper is to propose and analyse a numerical method for the solution of two-dimensional nonlinear weakly singular Volterra integral equations of the second kind. In general the solutions of these equations exhibit singularities in their derivatives at t=0 even if the forcing functions are smooth. To overcome these difficulties a simple smoothing change of variables is proposed. By applying this transformation an equation is obtained which, while still being weakly singular, can have a solution as smooth as is required. We then solve this transformed integral equation using Navot’s quadrature rule for computing integrals with an end point singularity. A new extension of a discrete Gronwall inequality allows us to prove convergence and obtain an error estimate. The theoretical results are then verified by numerical examples.