Acceleration of convergence in approximate solutions of Urysohn integral equations with Green's kernels

TL;DR

This paper introduces an improved approximation method for solving Urysohn integral equations with Green's kernels, achieving faster convergence and higher accuracy than traditional collocation techniques.

Contribution

The paper develops a new approximation approach using interpolatory projections onto piecewise polynomial spaces, enhancing convergence speed for Urysohn integral equations with Green's kernels.

Findings

Enhanced accuracy over classical collocation methods

Numerical examples confirm theoretical convergence improvements

Applicable to integral equations with Green's function kernels

Abstract

Consider a non-linear operator equation $x - K(x) = f$, where $f$ is a given function and $K$ is a Urysohn integral operator with Green's function type kernel defined on $L^\infty [0, 1]$. We apply approximation methods based on interpolatory projections onto the approximating space $\mathcal{X}_n$, which is the space of piecewise polynomials of even degree with respect to a uniform partition of $[0, 1]$. The approximate solutions obtained from these methods demonstrate enhanced accuracy compared to the classical collocation solution for the same equation. Numerical examples are given to support our theoretical results.