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Model. Anal. Inform. Sist., 2016, Volume 23, Number 5, Pages 559–567 (Mi mais522)  

This article is cited in 1 scientific paper (total in 1 paper)

Asymptotic approximation of the stationary solution with internal layer for FitzHugh–Nagumo system

A. A. Mel'nikova, R. L. Argun

Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow 119991, Russian Federation

Abstract: Creating adequate mathematical models of processes in living nature is an important task of modern biophysics. Blood clotting, nerve impulse propagation, reduction of the heart muscle, the pattern-formation in nature are auto-wave processes. FitzHugh–Nagumo system of equations is used to describe the auto-wave processes in active media. Such math problems are usually solved by numerical methods. The use of resource-intensive algorithms is required in the case of auto-wave solutions with sharp gradients. Therefore, it is appropriate to use the analytical methods for this type of problems. In this paper, the asymptotic method of contrast structures theory is used to obtain an approximate solution of a singularly perturbed system of FitzHugh–Nagumo type. The method allows to reduce the non-linear system of equations to a number of problems that can be solved analytically or with a stable numerical algorithm. This study presents the asymptotic approximation of a stationary auto-wave solution of the considered system. Additionally, this paper provides a formula that specifies the location of internal transition layers. The results were compared with the numerical solution. The application of contrast structures theory to the study of active media models can be used for analytical studies of other such systems, improving existing models and increasing the efficiency of the numerical calculations.

Keywords: asymptotic approximation, small parameter, singular perturbation, inner transition layer, activator-inhibitor system.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00437_а
15-01-04619_а
This study was supported by grants of the Russian Foundation for Basic Research projects No. 16-01-00437, 15-01-04619.


DOI: https://doi.org/10.18255/1818-1015-2016-5-559-567

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Bibliographic databases:

UDC: 517.9
Received: 15.06.2016

Citation: A. A. Mel'nikova, R. L. Argun, “Asymptotic approximation of the stationary solution with internal layer for FitzHugh–Nagumo system”, Model. Anal. Inform. Sist., 23:5 (2016), 559–567

Citation in format AMSBIB
\Bibitem{MelArg16}
\by A.~A.~Mel'nikova, R.~L.~Argun
\paper Asymptotic approximation of the stationary solution with internal layer for FitzHugh--Nagumo system
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 5
\pages 559--567
\mathnet{http://mi.mathnet.ru/mais522}
\crossref{https://doi.org/10.18255/1818-1015-2016-5-559-567}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3569852}
\elib{http://elibrary.ru/item.asp?id=27202305}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Melnikova, M. Chen, “Existence and asymptotic representation of the autowave solution of a system of equations”, Comput. Math. Math. Phys., 58:5 (2018), 680–690  mathnet  crossref  crossref  isi  elib
  • Моделирование и анализ информационных систем
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