چکیده (انگلیسی):

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 n! 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.