Mathematical Methods for Physics and Engineering - Matematica.NET

INTEGRAL EQUATIONSinhomogeneous Fredholm equation of the first kind may be written as0=f + λKy,which has the unique solution y = −K −1 f/λ, provided that f ≠ 0 and the inverseoperator K −1 exists.Similarly, we may write the corresponding Fredholm equation of the secondkind asy = f + λKy. (23.6)In the homogeneous case, where f = 0, this reduces to y = λKy, which isreminiscent of an eigenvalue problem in linear algebra (except that λ appears onthe other side of the equation) and, similarly, only has solutions for at most acountably infinite set of eigenvalues λ i . The corresponding solutions y i are calledthe eigenfunctions.In the inhomogeneous case (f ≠ 0), the solution to (23.6) can be writtensymbolically asy =(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 if λ ≠ λ i ,i.e.whenλ does not equal one ofthe eigenvalues of the corresponding homogeneous equation.When λ does equal one of these eigenvalues, (23.6) may have either manysolutions or no solution at all, depending on the form of f. If the function f isorthogonal to every eigenfunction of the equationthat belongs to the eigenvalue λ ∗ ,i.e.〈g|f〉 =g = λ ∗ K † g (23.7)∫ bag ∗ (x)f(x) dx =0for every function g obeying (23.7), then it can be shown that (23.6) has manysolutions. Otherwise the equation has no solution. These statements are discussedfurther in section 23.7, for the special case of integral equations with Hermitiankernels, i.e. those for which K = K † .23.4 Closed-form solutionsIn certain very special cases, it may be possible to obtain a closed-form solutionof an integral equation. The reader should realise, however, when faced with anintegral equation, that in general it will not be soluble by the simple methodspresented in this section but must instead be solved using (numerical) iterativemethods, such as those outlined in section 23.5.806