Automatically identifying the asymptotic behavior of nonlinear singularly perturbed boundary value problems

Abstract

We describe a computational procedure designed to automatically analyze the behavior of certain general classes of nonlinear singular perturbation problems by applying the combined results of a body of theory that proves the existence of solutions for these problems. We have also created a computer program that implements the computational procedure. The core mathematical knowledge contained in our program is composed of rules that embody the results of mathematical theorems from nonlinear singular perturbation theory. The principle method of proof used in the mathematical theory yields an estimate of a solution by constructing sharp bounding functions that define a region in which the solution exists uniquely. As a result, a successful application of our program produces an approximation of a solution as a side effect. In addition, the mathematical theory can be used to show the existence of multiple solutions for a nonlinear singularly perturbed boundary value problem. This feature is also reflected in the results obtained from our program. The ability to construct such a program depends critically on the successful coupling of a non-deterministic programming technique called path-finding with the capabilities of a computer algebra system.