A Variational Approach to Singularly Perturbed Boundary Value Problems for Ordinary and Partial Differential Equations with Turning Points

Abstract

In studying singularity perturbed boundary value problems for second order linear differential equations with a simple turning point, R. C. Ackerberg and R. E. O’Malley [2] pointed out a number of interesting anomalies. In particular they observed that standard application of the method of matched asymptotic expansions did not suffice to uniquely determine the asymptotic expansion of the solution. They further noted that the standard construction in that method led to boundary layers at both ends of the interval, even for problems where in fact there is only one boundary layer located at one or other of the endpoints. In this paper we employ a variational formulation of the problem to resolve the question of the number and location of the boundary layers as well as to uniquely determine the asymptotic expansion of the solution. The results are then extended to analogous problems for partial differential equations, and new results are obtained for a class of singularly perturbed elliptic boundary value problems with turning points.