The Singularly Perturbed Turning-Point Problem: A Geometric Approach

Abstract

This chapter discusses a geometric approach of the singularly perturbed turning-point problem. In the boundary layer, f(x,ε) is almost constant so the nonconstant solutions are growing or dying exponentials, depending on the sign of f. For f > 0, the exponential solutions decay so the boundary layer must be placed at the left; for f < 0, the boundary layer is on the right. The point x = 0, where f changes sign, is called a turning point. As f > 0 for x < 0 and f < 0 for x > 0, boundary layers are possible at both the left and right endpoints. Indeed for almost all equations with a turning point, there is a unique solution y(x, ɛ) to the boundary value problem, and this solution has both boundary layers; in the interior, y(x, ɛ) converge s to zero as ɛ ε 0. There are, however, some special cases in which the solution does not decay to zero as ɛ ε 0. On the turning point problem, resonance is treated in the context of a boundary value problem: The solution to some boundary value problem does not decay to zero as ɛ ε 0. Resonance as a phenomenon is regarded depending only on the equation, independent of boundary conditions. In general, it is impossible to check for resonance using asymptotic expansions in powers of ɛ: The Matkowsky criteria are really nonresonance criteria, useful for ruling out resonance.