The proposed research involves the application of asymptotic methods of singular perturbation type to boundary value problems for ordinary and partial differential equations, integral equations and difference equations. In particular, the problems to be studied center on two types that are relevant to applications: the effects of small random forces on the overall behavior of solutions of deterministic dynamical systems and the evolution of time-dependent solutions of boundary value problems to steady states that possess interior layers. A host of phenomena in nature are described by sets of equations that contain one or more formally small terms. If one were to try solving such equations numerically, for example, one usually runs into difficulty because the small terms can complicate the problems dramatically. An alternative method of solution, the one espoused in this proposal, is to take advantage of the small terms by first simplifying the equations that contain the small terms and then solving the simplified equations. The PI's will use this idea to solve various problems in probability theory and chemical physics.
