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Existence and Stability of Contrast Structures in Multidimensional Singularly Perturbed Reaction-Diffusion-Advection Problems

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Numerical Analysis and Its Applications (NAA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10187))

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Abstract

We consider stationary solutions with boundary and internal transition layers (contrast structures) for a nonlinear singularly perturbed equation that is referred to in applications as the stationary reaction-diffusion-advection equation. We construct an asymptotic approximation of an arbitrary-order accuracy to such solutions and prove the existence theorem. We suggest an afficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed asymptotics, we use and develop, to this class of problems, an asymptotic method of differential inequalities, which also permits one to prove the Lyapunov stability of such stationary solutions. The results can be used to create the numerical method which uses the asymptotic analyses to create space non uniform meshes to describe internal layer behavior of the solution.

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References

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Authors and Affiliations

  1. Faculty of Physics, Department of Mathematics, Lomonosov Moscow State University, Moscow, 119991, Russia

    M. A. Davydova & N. N. Nefedov

Authors
  1. M. A. Davydova
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  2. N. N. Nefedov
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Correspondence to M. A. Davydova .

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Editors and Affiliations

  1. Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Ivan Dimov

  2. Department of Applied Analysis, Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. Department of Applied Mathematics and Statistics, University of Rousse, Ruse, Bulgaria

    Lubin Vulkov

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Davydova, M.A., Nefedov, N.N. (2017). Existence and Stability of Contrast Structures in Multidimensional Singularly Perturbed Reaction-Diffusion-Advection Problems. In: Dimov, I., Faragó, I., Vulkov, L. (eds) Numerical Analysis and Its Applications. NAA 2016. Lecture Notes in Computer Science(), vol 10187. Springer, Cham. https://doi.org/10.1007/978-3-319-57099-0_29

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  • DOI: https://doi.org/10.1007/978-3-319-57099-0_29

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57098-3

  • Online ISBN: 978-3-319-57099-0

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