On the asymptotic solutions of a differential equation with multiple singular points

Abstract

A singularly perturbed model problem with multiple distinct regular singular points is studied. The uniform approximation of Wazwaz and Hanson (1986) is effectively extended to many singular points, thus establishing a generalized version of that theorem where the classic inner and outer expansions are not employed. A leading order general asymptotic solution is correctly represented by a set of matched exponential asymptotic expansions, where each approximation contains dominant and recessive terms. The resonance criteria due to the influence of multiple singular points are discussed. For an even number of singular points, the eigenvalues were found to be positive with a minimal eigenvalue. However, negative eigenvalues with a maximal eigenvalue arise for an odd number of singular points. The maximal and the minimal eigenvalue may not be unique and each is 0(1).