An Initial-Value Theory for Fredholm Integral Equations With Semidegenerate Kernels

Abstract

The Fredholm integral equation where the kernel is semidegenerate has many applications. The solution of this integral equation may be studied as a function of the upper limit of integration x , while t remains fixed. It is shown that the solution satisfies an initial-value problem. This reformulation is well suited to numerical solution by analog and digital computers. The present paper is one of a series on initial-value methods for Fredholm integral equations. Its considerations are of practical significance since an arbitrary kernel may be approximated by a degenerate kernel to a desired degree of accuracy using standard techniques. Furthermore, the important cases in which the kernel is a Green's function and in which the integral equation is a Volterra equation are both covered by this treatment.