Legendre–Kantorovich method for Fredholm integral equations of the second kind

Abstract

In the present paper, we consider polynomially based Kantorovich method for the numerical solution of Fredholm integral equation of the second kind with a smooth kernel. The used projection is either the orthogonal projection or an interpolatory projection using Legendre polynomial bases. The order of convergence of the proposed method and those of superconvergence of the iterated versions are established. We show that these orders of convergence are valid in the corresponding discrete methods obtained by replacing the integration by a quadrature rule. Numerical examples are given to illustrate the theoretical estimates.