Verification of the Invariant Imbedding Method for Certain Fredholm Integral Equations.
Previous RAND studies have shown that the solution of the Fredholm integral equation satisfies an initial-value problem. In the present study, the converse is shown to be true: the solution of the initial-value problem is a solution of the integral equation. It is assumed that the kernel is exponential in form. First, the integral equation is rewritten to show the dependence on the upper limit of integration. Next, an initial-value problem for the solution of the integral equation is derived in which the internal point remains fixed while the interval length is varied. During this procedure, the solution to the auxiliary integral equation and the solution of the Sobolov integral equation are introduced. Then, the validity of the Cauchy problem is established. 25 pp. Refs.