NAIVE SINGULAR PERTURBATION THEORY

Abstract

The paper demonstrates, via extremely simple examples, the shocks, spikes, and initial layers that arise in solving certain singularly perturbed initial value problems for first-order ordinary differential equations. As examples from stability theory, they are basic to many asymptotic techniques. First, we note that limiting solutions of linear homogeneous equations [Formula: see text] on t≥0 are specified by the zeros of [Formula: see text], rather than by the turning points where a(t) becomes zero. Furthermore, solutions to the solvable equations [Formula: see text] for k=1, 2 or 3 can feature canards, where the trivial limit continues to apply after it becomes repulsive. Limiting solutions of the separable equation [Formula: see text] may likewise involve switchings between the zeros of c(x) located immediately above and below x(0), if they exist, at zeros of A(t). Finally, limiting solutions of many other problems follow by using asymptotic expansions for appropriate special functions. For example, solutions of [Formula: see text] can be given in terms of the Bessel functions Kj(t4/4ε) and Ij(t4/4ε) for j=3/8 and -5/8.