Approximation methods for second kind weakly singular Volterra integral equations

Abstract

Projection methods are applied to obtain the convergence rates for Volterra integral equations with weakly singular kernels. We consider Galerkin and multi Galerkin methods and their iterated versions to solve Volterra integral equations with weakly singular kernels, in the space of piecewise polynomials subspaces based on graded mesh. We will show that the iterated multi-Galerkin method improves over iterated Galerkin method. In fact, we show that iterated multi-Galerkin solution converges with the convergence rates O(n−3m) and O(n−3m(logn)2), for algebraic and logarithmic type kernels, respectively. We prove that iterated Galerkin method, for algebraic kernel, converges with the convergence rate O(n−2m) and for logarithmic type kernel converges with the convergence rate O(n−2mlogn), where n denotes the number of partition points and m is the highest order of the polynomials employed in the approximations. Theoretical results are justified by the numerical results.