Rigorous convergence analysis of Jacobi spectral Galerkin methods for Volterra integral equations with noncompact kernels

Abstract

This article deals with the implementation of Jacobi spectral and pseudospectral Galerkin approaches for solving weakly singular Volterra integral equations with noncompact kernels. The computational domain of the main problem and intervals of the involved integrals are transformed into [-1,1] by some appropriate change of variables. Therefore, one can apply the Jacobi Galerkin method and impose the inner product conditions on the residual of the considered equation. Also, Jacobi Gauss quadrature rules are implemented to approximate the integral operator and the inner products associated to the Jacobi weight function in the pseudospectral Galerkin method. For both cases of spectral and pseudospectral Galerkin schemes, rigorous convergence analysis is provided under some mild conditions on the smoothness of kernel and source functions in L∞ and weighted L2 norms. Several numerical examples are also given to confirm the theoretical prediction.