A simplification to Fredholm’s solution to the Fredholm integral equation of the second kind

摘要

Fredholm’s classic solution to his linear integral equation of the second kind has been generally considered to be too complex to provide a practical solution method. It gives the resolvent kernel as a ratio of two infinite series, with terms in each series containing multiple integrals of determinants such that the evaluation of the nth term would seem to require different n-fold multi-integrations, a formidable task even for modest values of n. This paper shows, however, that by using the methods of the combinatorics of permutations and by solving a well-known recursion relation, Fredholm’s solution can be simplified considerably, In this simplified form, the nth term of each series requires the evaluation of just one new multi-integral. Thus presented is a new expression for the Fredholm’s solution, and the practical use of this expression is demonstrated on some example problems.