On Certain Fredholm Integral Equations Reducible to Initial Value Problems

Nonlinear initial value problems (of a rather unusual nature) will be obtained which have as their solutions so(x) and so(y). From these values Sp(z) may be found by solving another related initial value problem. The equations derived are amenable to numerical treatment and may very well provide advantages over the more customary direct methods of computing $o(z) from (1.1). The representation (1.2) actually encompasses a very large class of kernels. If, for example, a(s) = s, we have the usual Laplace transform. Kernels expressed in terms of such Laplace integrals arise often in transport theory. A problem in gas dynamics analyzed in [10] involves a kernel represenited in this way. Again, if a(s) = -log s, then K is a Mellin transform. We shall also find that a(s) = i log s is admissible in our theory, and this leads to a Fourier integral: