Approximated superconvergent methods for Volterra Hammerstein integral equations

Abstract

In this article, we apply Kumar–Sloan based Galerkin and multi-Galerkin methods for solving the nonlinear Volterra–Hammerstein integral equations with both smooth and weakly singular kernels. For smooth kernels, we use piecewise polynomials basis functions based on uniform mesh, whereas for weakly singular kernels, we use piecewise polynomials basis functions based on graded mesh, to derive the superconvergence results. We obtain the improved superconvergence rates using Kumar–Sloan based Galerkin and multi-Galerkin methods without going to the iterated Galerkin and iterated multi-Galerkin methods. Infact without going to the iterated versions, we obtain the superconvergence rates in this method as in the case of iterated Galerkin and iterated multi-Galerkin methods. Numerical examples are given to verify the theoretical results.