Approximation methods for system of linear Fredholm integral equations of second kind

Abstract

In this paper, Galerkin, multi-Galerkin methods and their iterated versions are developed for solving the system of linear Fredholm integral equations of the second kind for both smooth and weakly singular algebraic and logarithmic type kernels. Here first we develop the theoretical framework for Galerkin and iterated Galerkin methods to solve the system of linear second kind Fredholm integral equations using piecewise polynomials as basis functions and then obtain the superconvergence results similar to that of single linear Fredholm integral equation of the second kind. We show that iterated Galerkin approximation yields better convergence rates than Galerkin approximate solution. Further we enhance these results by using multi-Galerkin and iterated-multi-Galerkin methods and show that the iterated multi-Galerkin approximation yields improved superconvergence rates over iterated Galerkin and multi-Galerkin approximations. The theoretical results are justified by the numerical results.