Solutions of some Fredholm integral equations using fractional integration, with an application to a forced convection problem

Abstract

It is shown that certain Fredhom integral equations of the first kind can be written in terms of fractional integrals. The inversion of these operators, and thus the solutions of the integral equations, can than be deduced from well known properties of fractional integrals. The particular cases studied involve equations arising in heat transfer, viscous flow, and elastic half-space problems. The equations may be solved in terms of classically integrable functions or in a generalised function space. In the latter case the solutions are not unique but the most general solution of the equations is constructed. In the course of this analysis a uniqueness theorem for the classical cases is derived as a by-product. A relationship between certain special functions which enables series solutions to be obtained is shown to be valid in the generalised function space after suitably modifying certain definitions and identities.