Solution of Fredholm integral equations by conversion to known transforms

By manipulation of the kernel, it is possible to convert some types of Fredholm integral equations of the first kind to a well known transform or to a more tractable integral equation. In this paper manipulation of the kernel is used to replace the unknown function by a differential equation involving it, with the result that the integral equation has a simplified form. The technique is illustrated by a number of examples of conversion to Laplace and Mellin transform. Although the method requires only elementary manipulations, it highlights the interesting interplay between the boundary values of the unknown function and the solution of the integral equation.