The conversion of Fredholm integral equations to equivalent cauchy problems

摘要

A new method is developed for converting various classes of Fredholm integral equations into equivalent initial value problems. In contrast with previous methods, which accomplished this by imbedding the equation, with respect to some parameter, in a family of similar ones, our approach is parameter free. To effect the conversion the integral equation is first shown to be equivalent to a two point boundary value problem. The application of various invariant imbedding algorithms completes the task. An extensive examination of linear equations is made, and it is shown that our procedure leads to a substantial reduction of dimensionality over previous methods. New techniques for solving critical length and continuation problems are another important consequence of our approach.