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Model. Anal. Inform. Sist., 2016, Volume 23, Number 3, Pages 259–282 (Mi mais496)  

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

Numerical scheme for the pseudoparabolic singularly perturbed initial-boundary problem with interior transitional layer

A. A. Bykov

Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991, Leninskiye gory, 1, b. 2, Russian Federation

Abstract: Evolution equations are derived for the contrasting-structure-type solution of the generalized Kolmogorov–Petrovskii–Piskunov (GKPP) equation with the small parameter with high order derivatives. The GKPP equation is a pseudoparabolic equation with third order derivatives. This equation describes numerous processes in physics, chemistry, biology, for example, magnetic field generation in a turbulent medium and the moving front for the carriers in semiconductors. The profile of the moving internal transitional layer (ITL) is found, and an expression for drift speed of the ITL is derived. An adaptive mesh (AM) algorithm for the numerical solution of the initial-boundary value problem for the GKPP equation is developed and rigorously substantiated. AM algorithm for the special point of the first kind is developed, in which drift speed of the ITL in the first order of the asymptotic expansion turns to zero. Sufficient conditions for ITL transitioning through the special point within finite time are formulated. AM algorithm for the special point of the second kind is developed, in which drift speed of the ITL in the first order formally turns to infinity. Substantiation of the AM method is given based on the method of differential inequalities. Upper and lower solutions are derived. The results of the numerical algorithm are presented.

Keywords: singularly perturbed equation, interior transitional layer, finite difference method, asymptotic expansion.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00690_а
This work was supported by RFBR, project 16-01-00690-а.


DOI: https://doi.org/10.18255/1818-1015-2016-3-259-282

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Bibliographic databases:

UDC: 517.228.4
Received: 20.05.2016

Citation: A. A. Bykov, “Numerical scheme for the pseudoparabolic singularly perturbed initial-boundary problem with interior transitional layer”, Model. Anal. Inform. Sist., 23:3 (2016), 259–282

Citation in format AMSBIB
\Bibitem{Byk16}
\by A.~A.~Bykov
\paper Numerical scheme for the pseudoparabolic singularly perturbed initial-boundary problem with interior transitional layer
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 3
\pages 259--282
\mathnet{http://mi.mathnet.ru/mais496}
\crossref{https://doi.org/10.18255/1818-1015-2016-3-259-282}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3520848}
\elib{http://elibrary.ru/item.asp?id=26246292}


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    This publication is cited in the following articles:
    1. A. A. Bykov, K. E. Ermakova, “Exact solutions of the equations of a nonstationary front with equilibrium points of an infinite order of degeneracy”, Mosc. Univ. Phys. Bull., 73:6 (2018), 583–591  crossref  mathscinet  isi  scopus
  • Моделирование и анализ информационных систем
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