Discrete Modified Projection Method for Urysohn Integral Equations with Smooth Kernels

TL;DR

This paper introduces discrete modified projection methods for solving Urysohn integral equations with smooth kernels, emphasizing the importance of quadrature choices to maintain convergence orders, supported by numerical examples.

Contribution

It develops discrete versions of the modified projection method using interpolatory projections at Gauss points for Urysohn equations with smooth kernels, analyzing convergence.

Findings

Convergence orders depend on quadrature accuracy.

Numerical results confirm theoretical convergence rates.

Abstract

Approximate solutions of linear and nonlinear integral equations using methods related to an interpolatory projection involve many integrals which need to be evaluated using a numerical quadrature formula. In this paper, we consider discrete versions of the modified projection method and of the iterated modified projection methodfor solution of a Urysohn integral equation with a smooth kernel. For $r \geq 1,$ a space of piecewise polynomials of degree less than or equal to r - 1 with respect to an uniform partition is chosen to be the approximating space and the projection is chosen to be the interpolatory projection at r Gauss points. The orders of convergence which we obtain for these discrete versions indicate the choice of numerical quadrature which preserves the orders of convergence. Numerical results are given for a specific example.