ABSTRACT
In this introductory chapter we describe briefl y basic defi nitions concerning integral equations in general, and singular integral equations in particular. Integral equations arise in a natural way in various branches of mathematics and mathematical physics. Many initial and boundary value problems associated with linear ordinary and partial differential equations can be cast into problems of solving integral equations. Here we present some basic defi nitions and concepts involving singular integral equations and their occurrences in problems of mathematical physics such as mechanics, elasticity and linearised theory of water waves
1.1 BASIC DEFINITIONS
Example 1.1.1
Example 1.1.2
Example 1.1.3
[ ]23 3 3 3( ) ( , ) ( ) ( ), b
x K x t t dt f x a x bϕ ϕ+ = ≤ ≤∫
The integral equations in the Examples 1.1.1 and 1.1.2 above are examples of linear integral equations, since the unknown functions 1 2,ϕ ϕ there, appear linearly, whereas the integral equation in the Example 1.1.3, in which the unknown function appears nonlinearly, is an example of nonlinear integral equation. In the present book we will consider, only linear integral equations.
