Jacobi spectral methods for system of nonlinear Volterra Urysohn integral equations

Abstract

In this article, Galerkin, multi-Galerkin methods and their iterated versions based on Jacobi polynomials are exerted to approximate and obtain superconvergence rates for the system of nonlinear Volterra integral equations of the Urysohn type with both smooth and weakly singular kernels. Firstly, we establish the regularity behaviors of the solutions of the system of the nonlinear second kind Volterra integral equations. We determine convergence results for Jacobi spectral Galerkin method and its iterated version in both weighted-$L^2$ as well as infinity norms and show that the iterated version provides better approximation. Furthermore, we improve the superconvergence rates for both the smooth as well as the weakly singular kernels in Jacobi spectral iterated multi-Galerkin method. The reliability and efficiency of the theoretical results are verified with numerical experiments.