Averaged Nystr\"om interpolants for the solution of Fredholm integral equations of the second kind

TL;DR

This paper investigates the use of averaged Nyström methods with Gauss quadrature rules to solve Fredholm integral equations of the second kind, focusing on error estimation and stability improvements.

Contribution

It introduces new stability properties for averaged and weighted averaged Gauss quadrature rules in Nyström methods for integral equations.

Findings

Averaged quadrature rules enhance error estimation accuracy.

New stability properties improve solution reliability.

Applicable to equations on finite and infinite intervals.

Abstract

Fredholm integral equations of the second kind that are defined on a finite or infinite interval arise in many applications. This paper discusses Nystr\"om methods based on Gauss quadrature rules for the solution of such integral equations. It is important to be able to estimate the error in the computed solution, because this allows the choice of an appropriate number of nodes in the Gauss quadrature rule used. This paper explores the application of averaged and weighted averaged Gauss quadrature rules for this purpose, and introduces new stability properties for them.