Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL EQUATIONS

We shall illustrate the principles involved by considering the differential equa-

tion

y′′(x)=f(x, y), (23.1)

wheref(x, y) can be any function ofxandybut not ofy′(x). Equation (23.1)

thus represents a large class of linear and non-linear second-order differential

equations.

We can convert (23.1) into the corresponding integral equation by first inte-

grating with respect toxto obtain

y′(x)=

∫x

0

f(z, y(z))dz+c 1.

Integrating once more, we find

y(x)=

∫x

0

du

∫u

0

f(z, y(z))dz +c 1 x+c 2.

Provided we do not change the region in theuz-plane over which the double

integral is taken, we can reverse the order of the two integrations. Changing the

integration limits appropriately, we find

y(x)=

∫x

0

f(z, y(z))dz

∫x

z

du+c 1 x+c 2 (23.2)

=

∫x

0

(x−z)f(z, y(z))dz+c 1 x+c 2 ; (23.3)

this is a non-linear (for generalf(x, y))Volterraintegral equation.

It is straightforward to incorporate any boundary conditions on the solution

y(x) by fixing the constantsc 1 andc 2 in (23.3). For example, we might have the

one-point boundary conditiony(0) =aandy′(0) =b, for which it is clear that

we must setc 1 =bandc 2 =a.

23.2 Types of integral equation

From (23.3), we can see that even a relatively simple differential equation such

as (23.1) can lead to a corresponding integral equation that is non-linear. In this

chapter, however, we will restrict our attention tolinearintegral equations, which

have the general form

g(x)y(x)=f(x)+λ

∫b

a

K(x, z)y(z)dz. (23.4)

In (23.4),y(x) is the unknown function, while the functionsf(x),g(x)andK(x, z)

are assumed known.K(x, z) is called thekernelof the integral equation. The

integration limitsaandbare also assumed known, and may be constants or

functions ofx,andλis a known constant or parameter.