Mathematical Methods for Physics and Engineering : A Comprehensive Guide

INTEGRAL EQUATIONS

inhomogeneous Fredholm equation of the first kind may be written as

0=f+λKy,

which has the unique solutiony=−K−^1 f/λ, provided thatf= 0 and the inverse

operatorK−^1 exists.

Similarly, we may write the corresponding Fredholm equation of the second

kind as

y=f+λKy. (23.6)

In the homogeneous case, wheref= 0, this reduces toy=λKy,whichis

reminiscent of an eigenvalue problem in linear algebra (except thatλappears on

the other side of the equation) and, similarly, only has solutions for at most a

countably infinite set ofeigenvaluesλi. The corresponding solutionsyiare called

the eigenfunctions.

In the inhomogeneous case (f= 0), the solution to (23.6) can be written

symbolically as

y=(1−λK)−^1 f,

again provided that the inverse operator exists. It may be shown that, in general,

(23.6) does possess a unique solution ifλ=λi,i.e.whenλdoes not equal one of

the eigenvalues of the corresponding homogeneous equation.

Whenλdoes equal one of these eigenvalues, (23.6) may have either many

solutions or no solution at all, depending on the form off. If the functionfis

orthogonal toeveryeigenfunction of the equation

g=λ∗K†g (23.7)

that belongs to the eigenvalueλ∗,i.e.

〈g|f〉=

∫b

a

g∗(x)f(x)dx=0

for every functiongobeying (23.7), then it can be shown that (23.6) has many

solutions. Otherwise the equation has no solution. These statements are discussed

further in section 23.7, for the special case of integral equations with Hermitian

kernels, i.e. those for whichK=K†.

23.4 Closed-form solutions

In certain very special cases, it may be possible to obtain a closed-form solution

of an integral equation. The reader should realise, however, when faced with an

integral equation, that in general it will not be soluble by the simple methods

presented in this section but must instead be solved using (numerical) iterative

methods, such as those outlined in section 23.5.