A Cauchy problem for Milne's integral equation with a pincherle-goursat kernel

In a series of our preceding papers (cf. Bellman and Ueno, 1971a, 1971b, 1972a, 1972b, 1972c; Ueno 1972), with the aid of invariant imbedding technique, we transformed the two point boundary value problem for integral equations into the initial-value problem. In this paper, extending the procedure to the Milne's integral equation, whose kernel is expanded into a double Fourier cosine series, we show how to find a Cauchy system for the required solution of Milne's integral equation and the Fredholm resolvent. The obtained integro-differential equations are reduced to large system of ordinary differential equations with known initial conditions. Then, they are suitable for the numerical computation of the solution by the modern high-speed computer.