On Some Basic Concepts in the Analysis of Singular Perturbations

Abstract

This chapter discusses some basic concepts in the analysis of singular perturbations. It describes the theory of singular layer problems. The frequent occurrence of singular perturbations in applications can be explained by the observation that whenever some basic mathematic model of some phenomena is improved by incorporating some of the effects that were first neglected; the improved model is most likely to be a problem of singular perturbations. Analysis of singular perturbations is not a straight-forward generalization and extension of classical asymptotic analysis and perturbation theory but rather an entirely new discipline. Every practitioner of singular perturbations uses, implicitly or explicitly, certain concepts that are commonly accepted as the basis for the method of analysis. The elegant classical perturbation analysis combines the construction of approximations and the proof of their validity into one line of thinking. In singular perturbations, the complete analysis requires various different ingredients. The extension theorem, which is the rigorous ingredient in the derivation of the matching rule, only asserts the existence of the extended domain but gives no information on the functions.