Superconvergence of system of Volterra integral equations by spectral approximation method

Abstract

In this article, we apply Jacobi spectral Galerkin, multi-Galerkin methods and their iterated versions to approximate the system of Volterra integral equations for smooth as well as weakly singular kernels and obtain the superconvergence results. Before doing this, first, we develop the regularity properties of the system of linear Volterra integral equations. We show that the Jacobi spectral iterated Galerkin approximation yields better converge rates over Jacobi spectral Galerkin method. We make the improvement of the superconvergence rates further for both smooth and weakly singular kernels in Jacobi spectral iterated multi-Galerkin method. Numerical results are provided for the illustration of the theoretical results.