A note on improvement by iteration for the approximate solutions of second kind Fredholm integral equations with Green's kernels

TL;DR

This paper investigates how iterative modifications to collocation methods can enhance the convergence rate when solving second kind Fredholm integral equations with Green's function kernels.

Contribution

It demonstrates that a modified collocation method improves convergence order specifically for Green's function type kernels, extending previous results to this class.

Findings

Iteration improves convergence order for Green's kernels

Modified collocation method outperforms standard approaches

Results applicable to piecewise polynomial collocation schemes

Abstract

Consider a linear operator equation $x - Kx = f$, where $f$ is given and $K$ is a Fredholm integral operator with a Green's function type kernel defined on $C[0, 1]$. For $r \geq 0$, we employ the interpolatory projection at $2r + 1$ collocation points (not necessarily Gauss points) onto a space of piecewise polynomials of degree $\leq 2r$ with respect to a uniform partition of $[0, 1]$. Previous researchers have established that, in the case of smooth kernels with piecewise polynomials of even degree, iteration in the collocation method and its variants improves the order of convergence by projection methods. In this article, we demonstrate the improvement in order of convergence by modified collocation method when the kernel is of Green's function type.