Modified projection method for Urysohn integral equations with non-smooth kernels

Abstract

Consider a nonlinear operator equation x−K(x)=f, where K is a Urysohn integral operator with a Green’s function type kernel. Using the orthogonal projection onto a space of discontinuous piecewise polynomials, previous authors have investigated approximate solution of this equation using the Galerkin and the iterated Galerkin methods. They have shown that the iterated Galerkin solution is superconvergent. In this paper, a solution obtained using the iterated modified projection method is shown to converge faster than the iterated Galerkin solution. The improvement in the order of convergence is achieved by retaining the size of the system of equations same as for the Galerkin method. Numerical results are given to illustrate the improvement in the order of convergence.