Spectral approximated superconvergent method for nonlinear Volterra Hammerstein integral equations with weakly singular kernels

Abstract

In this paper, we apply Jacobi spectral Galerkin and multi-Galerkin methods using Kumar–Sloan technique for obtaining approximations of weakly singular Volterra integral equation of Hammerstein type and obtain superconvergence results. We derive the enhanced superconvergence results for the Kumar–Sloan approximation based on Galerkin and multi-Galerkin methods in both cases: when the exact solution is smooth and when the exact solution is non-smooth, in both infinity and weighted-$L^2$ norms. We conclude that without the need for the iterated versions, we achieve superconvergence rates as high as the superconvergence rates of iterated Galerkin and iterated multi-Galerkin methods. The numerical results are presented to demonstrate the theoretical ones.