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Zh. Vychisl. Mat. Mat. Fiz., 2013, Volume 53, Number 3, Pages 365–376 (Mi zvmmf9884)  

This article is cited in 10 scientific papers (total in 10 papers)

Contrast structures in the reaction-diffusion-advection equations in the case of balanced advection

N. T. Levashova, N. N. Nefedov, A. V. Yagremtsev

M. V. Lomonosov Moscow State University, Faculty of Physics

Abstract: For a singularly perturbed parabolic equation termed in applications as the reaction-diffusion-advection equation, stationary solutions with internal transition layers (contrast structures) are studied. An arbitrary-order asymptotic approximation of such solutions is constructed, and an existence theorem is proved. An efficient algorithm for constructing an asymptotic approximation of the transition point is proposed. The constructed asymptotic approximation is justified by applying the asymptotic method of differential inequalities, which is extended to the class of problems under study. This method is also used to establish the Lyapunov stability of such stationary solutions.

Key words: singularly perturbed parabolic problems, reaction-diffusion equations, internal layers, asymptotic methods, method of differential inequalities, Lyapunov stability.

DOI: https://doi.org/10.7868/S0044466913030095

Full text: PDF file (214 kB)
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English version:
Computational Mathematics and Mathematical Physics, 2013, 53:3, 273–283

Bibliographic databases:

UDC: 519.63
MSC: 35K57
Received: 12.01.2012
Revised: 09.10.2012

Citation: N. T. Levashova, N. N. Nefedov, A. V. Yagremtsev, “Contrast structures in the reaction-diffusion-advection equations in the case of balanced advection”, Zh. Vychisl. Mat. Mat. Fiz., 53:3 (2013), 365–376; Comput. Math. Math. Phys., 53:3 (2013), 273–283

Citation in format AMSBIB
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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. E. A. Antipov, N. T. Levashova, N. N. Nefedov, “Asymptotics of the front motion in the reaction-diffusion-advection problem”, Comput. Math. Math. Phys., 54:10 (2014), 1536–1549  mathnet  crossref  crossref  elib
    2. M. A. Davydova, “Existence and Stability of Solutions with Boundary Layers in Multidimensional Singularly Perturbed Reaction-Diffusion-Advection Problems”, Math. Notes, 98:6 (2015), 909–919  mathnet  crossref  crossref  mathscinet  isi  elib
    3. N. T. Levashova, O. A. Nikolaeva, A. D. Pashkin, “Simulation of the temperature distribution at the water-air interface using the theory of contrast structures”, Mosc. Univ. Phys. Bull., 70:5 (2015), 341–345  crossref  mathscinet  isi  elib  scopus
    4. N. T. Levashova, J. V. Muhartova, M. A. Davydova, N. E. Shapkina, A. V. Oltchev, “The application of the theory of contrast structures for describing wind field in spatially heterogeneous vegetation cover”, Mosc. Univ. Phys. Bull., 70:3 (2015), 167–174  crossref  mathscinet  isi  elib  scopus
    5. A. A. Melnikova, R. L. Argun, “Asimptotika statsionarnogo resheniya s vnutrennim perekhodnym sloem dlya sistemy tipa FittsKhyu–Nagumo”, Model. i analiz inform. sistem, 23:5 (2016), 559–567  mathnet  crossref  mathscinet  elib
    6. N. T. Levashova, O. A. Nikolaeva, “Asimptoticheskoe issledovanie resheniya uravneniya teploprovodnosti vblizi granitsy razdela dvukh sred”, Model. i analiz inform. sistem, 24:3 (2017), 339–352  mathnet  crossref  elib
    7. M. A. Davydova, N. N. Nefedov, “Existence and stability of contrast structures in multidimensional singularly perturbed reaction-diffusion-advection problems”, Numerical Analysis and Its Applications, NAA 2016, Lecture Notes in Computer Science, 10187, eds. I. Dimov, I. Farago, L. Vulkov, Springler, 2017, 277–285  crossref  mathscinet  zmath  isi  scopus
    8. M. A. Davydova, S. A. Zakharova, N. T. Levashova, “On one model problem for the reaction-diffusion-advection equation”, Comput. Math. Math. Phys., 57:9 (2017), 1528–1539  mathnet  crossref  crossref  isi  elib  elib
    9. N. T. Levashova, N. N. Nefedov, A. V. Yagremtsev, “Existence of a solution in the form of a moving front of a reaction-diffusion-advection problem in the case of balanced advection”, Izv. Math., 82:5 (2018), 984–1005  mathnet  crossref  crossref  adsnasa  isi  elib
    10. 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
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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