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Model. Anal. Inform. Sist., 2017, Volume 24, Number 3, Pages 259–279 (Mi mais563)  

This article is cited in 5 scientific papers (total in 5 papers)

Moving front solution of the reaction-diffusion problem

E. A. Antipov, V. T. Volkov, N. T. Levashova, N. N. Nefedov

Lomonosov Moscow State University, Faculty of Physics, 1, bld. 2 Leninskiye Gory, Moscow 119991, Russia

Abstract: In this paper, we study the moving front solution of the reaction-diffusion initial-boundary value problem with a small diffusion coefficient. Problems in such statements can be used to model physical processes associated with the propagation of autowave fronts, in particular, in biophysics or in combustion. The moving front solution is a function the distinctive feature of which is the presence in the domain of its definition of a subdomain where the function has a large gradient. This subdomain is called an internal transition layer. In the nonstationary case, the position of the transition layer varies with time which, as it is well known, complicates the numerical solution of the problem as well as the justification of the correctness of numerical calculations. In this case the analytical method is an essential component of the study. In the paper, asymptotic methods are applied for analytical investigation of the solution of the problem posed. In particular, an asymptotic approximation of the solution as an expansion in powers of a small parameter is constructed by the use of the Vasil'eva algorithm and the existence theorem is carried out using the asymptotic method of differential inequalities. The methods used also make it possible to obtain an equation describing the motion of the front. For this purpose a transition to local coordinates takes place in the region of the front localization. In the present paper, in comparison with earlier publications dealing with two-dimensional problems with internal transition layers the transition to local coordinates in the vicinity of the front has been modified, that led to the simplification of the algorithm of determining the equation of the curve motion.

Keywords: reaction-diffusion problem, two-dimensional moving front, asymptotic representation, small parameter, asymptotic method of differential inequalities.

Funding Agency Grant Number
Russian Foundation for Basic Research 16-01-00437_a
This work was supported by RFBR, project No 16-01-00437.


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UDC: 517.957
Received: 15.12.2016

Citation: E. A. Antipov, V. T. Volkov, N. T. Levashova, N. N. Nefedov, “Moving front solution of the reaction-diffusion problem”, Model. Anal. Inform. Sist., 24:3 (2017), 259–279

Citation in format AMSBIB
\by E.~A.~Antipov, V.~T.~Volkov, N.~T.~Levashova, N.~N.~Nefedov
\paper Moving front solution of the reaction-diffusion problem
\jour Model. Anal. Inform. Sist.
\yr 2017
\vol 24
\issue 3
\pages 259--279

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    This publication is cited in the following articles:
    1. E. A. Antipov, N. T. Levashova, N. N. Nefedov, “Asimptoticheskoe priblizhenie resheniya uravneniya reaktsiya-diffuziya-advektsiya s nelineinym advektivnym slagaemym”, Model. i analiz inform. sistem, 25:1 (2018), 18–32  mathnet  crossref  elib
    2. A. A. Melnikova, N. N. Deryugina, “Periodicheskie izmeneniya avtovolnovogo fronta v dvumernoi sisteme parabolicheskikh uravnenii”, Model. i analiz inform. sistem, 25:1 (2018), 112–124  mathnet  crossref  elib
    3. D. V. Lukyanenko, V. T. Volkov, N. N. Nefedov, A. G. Yagola, “Application of asymptotic analysis for solving the inverse problem of determining the coefficient of linear amplification in Burgers' equation”, Mosc. Univ. Phys. Bull., 74:2 (2019), 131–136  crossref  mathscinet  isi  scopus
    4. D. V. Lukyanenko, V. B. Grigorev, V. T. Volkov, M. A. Shishlenin, “Solving of the coefficient inverse problem for a nonlinear singularly perturbed two-dimensional reaction-diffusion equation with the location of moving front data”, Comput. Math. Appl., 77:5 (2019), 1245–1254  crossref  mathscinet  isi  scopus
    5. N. N. Nefedov, V. T. Volkov, “Asymptotic solution of the inverse problem for restoring the modular type source in Burgers' equation with modular advection”, J. Inverse Ill-Posed Probl., 28:5 (2020), 633–639  crossref  mathscinet  zmath  isi  scopus
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