APPROXIMATION OF WEAKLY SINGULAR NON-LINEAR VOLTERRA-URYSOHN INTEGRAL EQUATIONS BY PIECEWISE POLYNOMIAL PROJECTION METHODS BASED ON GRADED MESH

In this article, we address the approximation solution of Volterra-Urysohn integral equations which involves weakly singular kernels. In order to get better convergence rates, projection methods namely Galerkin and multi Galerkin methods, along with their iterated versions are used in the space of piecewise polynomials subspaces based on the graded mesh. In addition, we compute the superconvergence results for the proposed integral equation and show that iterated Galerkin method outperforms Galerkin method in terms of order of convergence. Further, we demonstrate numerical examples to verify the proposed theoretical framework.