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 secondkind 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 writtensymbolically 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 manysolutions 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〉=∫bag∗(x)f(x)dx=0for 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.