Numerical Analysis of Fredholm Integral Equations

Mathematics Subject Classification

65R20

Short Definition

Numerical methods are described for solving Fredholm integral equations of the second kind. Related equations are also described briefly.

Introduction

A Fredholm linear integral equation of the second kind has the form

$$\displaystyle{ \lambda x\left (s\right ) -\int _{\varOmega }K\left (s,t\right )x\left (t\right )\,dt = y\left (s\right ),\quad \quad s \in \varOmega }$$

(1)

with $\lambda \neq 0$. The region Ω is assumed to be a closed set and to be contained in the d-dimensional space $\mathbb{R}^{d}$ for some d ≥ 1. Ω can be a d-dimensional region or something of smaller dimension such as a curve or surface, and usually Ω is bounded. The function x is unknown and the remaining functions and parameters are given. For notational convenience, the Eq. (1) is written symbolically as $\left (\lambda -\mathcal{K}\right )x = y$, and $\mathcal{K}$ denotes the integral operator. Throughout this article, assume that (1) has a...