Singular Perturbations of Boundary Value Problems Involving Ordinary Differential Equations

Abstract

In this lecture we shall consider boundary value problems Pε in which the order of the differential equation drops, or its type changes, as ε → 0 so that the boundary conditions prescribed in Pε are not appropriate when ε = 0, and it is not at all obvious how P0 should be defined. It is usually clear in such cases that limyε, if it exists, cannot satisfy all the limiting boundary conditions. In many cases limyε will not be attained uniformly, indeed limyε may be a discontinuous function; and the derivatives of yε may fail to approach a limit or may be unbounded functions of ε. A further characteristic feature of such "singular perturbation problems" is the nonanalytic dependence of yε on ε even in cases in which Pε depends on ε in a very simple manner. This nonanalytic dependence expresses itself frequently in a markedly different behaviour as ε approaches zero through positive or negative values.