Mathematical Methods for Physics and Engineering : A Comprehensive Guide

23.6 Fredholm theory

common ratioλ/3. Thus,provided|λ|<3, this infinite series converges to the value
λ/(3−λ), and the solution to (23.39) is

y(x)=x+

λx

3 −λ

=

3 x

3 −λ

. (23.40)

Finally, we note that the requirement that|λ|<3 may also be derived very easily from
the condition (23.38).

23.6 Fredholm theory

In the previous section, we found that a solution to the integral equation (23.34)

can be obtained as a Neumann series of the form (23.36), where the resolvent

kernelR(x, z;λ) is written as an infinite power series inλ. This solution is valid

provided the infinite series converges.

A related, but more elegant, approach to the solution of integral equations

using infinite series was found by Fredholm. We will not reproduce Fredholm’s

analysis here, but merely state the results we need. Essentially,Fredholm theory

provides a formula for the resolvent kernelR(x, z;λ) in (23.36) in terms of the

ratio of two infinite series:

R(x, z;λ)=

D(x, z;λ)

d(λ)

. (23.41)

The numerator and denominator in (23.41) are given by

D(x, z;λ)=

∑∞

n=0

(−1)n

n!

Dn(x, z)λn, (23.42)

d(λ)=

∑∞

n=0

(−1)n

n!

dnλn, (23.43)

where the functionsDn(x, z) and the constantsdnare found from recurrence

relations as follows. We start with

D 0 (x, z)=K(x, z)andd 0 =1, (23.44)

whereK(x, z) is the kernel of the original integral equation (23.34). The higher-

order coefficients ofλin (23.43) and (23.42) are then obtained from the two

recurrence relations

dn=

∫b

a

Dn− 1 (x, x)dx, (23.45)

Dn(x, z)=K(x, z)dn−n

∫b

a

K(x, z 1 )Dn− 1 (z 1 ,z)dz 1. (23.46)

Although the formulae for the resolvent kernel appear complicated, they are

often simple to apply. Moreover, for the Fredholm solution the power series

(23.42) and (23.43) are both guaranteed to converge for all values ofλ, unlike