A new method for eliminating the indeterminacy in the singularly perturbed problems with Ackerberg-O’Malley resonance

• Jacques Laforgue Departamento de Matemáticas, Núcleo de Sucre, Universidad de Oriente, Cumaná, Estado Sucre, Venezuela https://orcid.org/0000-0002-8823-9694

Keywords: Singular perturbations, Boundary layer resonance, Matched asymptotic expansions

Abstract

In the singularly perturbed problems with resonant character in the sense of Ackerberg and O’Malley, the traditional method of matched asymptotic expansions fails to determine the resonance’s amplitude. A new method is presented, based on established procedures from the theory of ordinary differential equations, for eliminating such indeterminacy taking advantage of the incomplete result of the matched asymptotic expansions and eliminating in a natural fashion the superfluous degree of freedom, through the derivation and imposition of an additional exact boundary condition that relates the slopes at both extremities of the domain. The new method is effective for the variety of problems recognized as resonant, including those exhibiting supersensitivity and also for those of a different structure but with analogous indeterminacy, for example involving a partial differential equation.

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