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Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales

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Abstract

We consider a coupled system of two singularly perturbed reaction-diffusion equations, with two small parameters 0 < ε ≤ μ ≤ 1, each multiplying the highest derivative in the equations. The presence of these parameters causes the solution(s) to have boundary layers which overlap and interact, based on the relative size of ε and μ. We show how one can construct full asymptotic expansions together with error bounds that cover the complete range 0 < ε ≤ μ ≤ 1. For the present case of analytic input data, we present derivative growth estimates for the terms of the asymptotic expansion that are explicit in the perturbation parameters and the expansion order.

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Authors and Affiliations

  1. Institut für Analysis und Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040, Wien, Austria

    J. M. Melenk

  2. Department of Mathematics and Statistics, University of Cyprus, P.O. BOX 20537, Nicosia, 1678, Cyprus

    C. Xenophontos

  3. Department of Mathematics and Statistics, Loyola University Maryland, 4501 N. Charles Street, Baltimore, MD, 21210, USA

    L. Oberbroeckling

Authors
  1. J. M. Melenk
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  2. C. Xenophontos
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  3. L. Oberbroeckling
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Correspondence to J. M. Melenk.

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Communicated by: M. Stynes.

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Melenk, J.M., Xenophontos, C. & Oberbroeckling, L. Analytic regularity for a singularly perturbed system of reaction-diffusion equations with multiple scales. Adv Comput Math 39, 367–394 (2013). https://doi.org/10.1007/s10444-012-9284-x

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  • DOI: https://doi.org/10.1007/s10444-012-9284-x

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