High accurate convergent spectral Galerkin methods for nonlinear weakly singular Volterra integro-differential equations

Abstract

This paper contributes to investigate the Jacobi spectral and pseudo-spectral Galerkin techniques to solve a general form of nonlinear weakly singular Volterra integro-differential equations of the first order. By applying some suitable change of variables, we have made the solution of the mentioned equations to be smooth. Then, by applying the spectral and pseudo-spectral Jacobi Galerkin schemes, accurate solutions are computed efficiently. Rigorous convergence analysis associated with both the spectral and pseudo-spectral Jacobi Galerkin approaches are discussed in detail. Some numerical test problems are given to depict the accuracy of the presented numerical schemes with respect to some recent approximate methods in the literature.