Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL EQUATIONS

Neumann series, which converge only if the condition (23.38) is satisfied. Thus the

Fredholm method leads to a unique, non-singular solution, provided thatd(λ)=0.

In fact, as we might suspect, the solutions ofd(λ) = 0 give the eigenvalues of the

homogeneous equation corresponding to (23.34), i.e. withf(x)≡0.

Use Fredholm theory to solve the integral equation (23.39).

Using (23.36) and (23.41), the solution to (23.39) can be written in the form

y(x)=x+λ

∫ 1

0

R(x, z;λ)zdz=x+λ

∫ 1

0

D(x, z;λ)

d(λ)

zdz. (23.47)

In order to find the form of the resolvent kernelR(x, z;λ), we begin by setting

D 0 (x, z)=K(x, z)=xz and d 0 =1

and use the recurrence relations (23.45) and (23.46) to obtain

d 1 =

∫ 1

0

D 0 (x, x)dx=

∫ 1

0

x^2 dx=

1

3

,

D 1 (x, z)=

xz

3

−

∫ 1

0

xz 12 zdz 1 =

xz

3

−xz

[

z 13

3

] 1

0

=0.

Applying the recurrence relations again we find thatdn=0andDn(x, z)=0forn>1.
Thus, from (23.42) and (23.43), the numerator and denominator of the resolvent respectively
are given by

D(x, z;λ)=xz and d(λ)=1−

λ
3
Substituting these expressions into (23.47), we find that the solution to (23.39) is given
by

y(x)=x+λ

∫ 1

0

xz^2

1 −λ/ 3

dz

=x+λ

[

x

1 −λ/ 3

z^3

3

] 1

0

=x+

λx

3 −λ

=

3 x

3 −λ

,

which, as expected, is the same as the solution (23.40) found by constructing a Neumann
series.

23.7 Schmidt–Hilbert theory

The Schmidt–Hilbert (SH) theory of integral equations may be considered as

analogous to the Sturm–Liouville (SL) theory of differential equations, discussed

in chapter 17, and is concerned with the properties of integral equations with

Hermitiankernels. An Hermitian kernel enjoys the property

K(x, z)=K∗(z, x), (23.48)

and it is clear that a special case of (23.48) occurs for a real kernel that is also

symmetric with respect to its two arguments.