Turning point problems and resonance

Document Type

Article

Publication Date

3-1-1982

Abstract

Consider the boundary value problem: ε2yn + (xp(x) + ε2f(x, ε))y' + g(x, ε)y = 0, y(a) = A, y(b) = B, where a < 0 < b, p(x) < p(x) < 0, and p, f, and g are analytic. We investigate the solution of this problem for small positive values of the parameter ε. If-g(0, 0)/p(0) c where c ε N = {0, 1, 2, 3,...}, then so-called resonance does not occur, and y = o(εn) on closed subintervals of (a, b), for any n ε N, with expected boundary layer behaviour at the end-points. If -g(0, 0)/p(0) = c, c ε N, then further transformations of dependent and independent variables may still expose resonance or non-resonance. The set of necessary conditions that is developed is compared to other authors' criteria, most notably, Olver's sufficiency condition, and the necessary conditions of Cook & Eckhaus, Lakin, and Matkowsky. Finally, it is proved that these conditions are necessary for resonance. © 1982, by Academic Press Inc. (London) Ltd.

Publication Title

IMA Journal of Applied Mathematics (Institute of Mathematics and Its Applications)

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