The Numerical Solution of Second-Order Weakly Singular Volterra Integro-Differential Equations

Abstract

In this paper we investigate the attainable order of convergence of collocation approximations in certain polynomial spline spaces for solutions of a class of second-order volterra integro-differential equations with weakly singular kernels. While the use of quasi-uniform meshes leads, due to the nonsmooth nature of these solutions, to convergence of order less than one, regardless of the degree of the approximating spling function, collocation on suitably graded meshes will be shown to yield optimal convergence rates.