Approximate Solutions to Fredholm Integral Equations by Iterated Petrov-Galerkin Method with Regular Pairs

Abstract

In this work we use Petrov-Galerkin method for solving classical Fredholm integral equations of the second kind. The notion of regular pair of finite dimensional subspaces – simply characterized by the positive definiteness of a correlation matrix - makes it easier to guarantee solvability and numerical stability of the approximation scheme. By an iteration of the method the approximate solution can be improved. We show how the error is reduced by means of this procedure and build a better approximation.