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Zh. Vychisl. Mat. Mat. Fiz., 2010, Volume 50, Number 2, Pages 276–285 (Mi zvmmf4827)  

This article is cited in 27 scientific papers (total in 27 papers)

Front motion in a parabolic reaction-diffusion problem

Yu. V. Bozhevol'nov, N. N. Nefëdov

Faculty of Physics, Moscow State University, Moscow, 119992

Abstract: A singularly perturbed initial-boundary value problem is considered for a parabolic equation known in applications as the reaction-diffusion equation. An asymptotic expansion of solutions with a moving front is constructed, and an existence theorem for such solutions is proved. The asymptotic expansion is substantiated using the asymptotic method of differential inequalities, which is extended to the class of problems under study. The method is based on well-known comparison theorems and is a development of the idea of using formal asymptotics for the construction of upper and lower solutions in singularly perturbed problems with internal and boundary layers.

Key words: singularly perturbed parabolic problems, reaction-diffusion equation, internal layers, fronts, asymptotic methods, differential inequalities.

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English version:
Computational Mathematics and Mathematical Physics, 2010, 50:2, 264–273

Bibliographic databases:

UDC: 519.633
Received: 27.03.2009

Citation: Yu. V. Bozhevol'nov, N. N. Nefëdov, “Front motion in a parabolic reaction-diffusion problem”, Zh. Vychisl. Mat. Mat. Fiz., 50:2 (2010), 276–285; Comput. Math. Math. Phys., 50:2 (2010), 264–273

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. V. T. Volkov, N. E. Grachev, A. V. Dmitriev, N. N. Nefedov, “Front formation and dynamics in the reaction-diffusion-advection model”, Math. Models Comput. Simul., 3:2 (2011), 158–164  mathnet  crossref  mathscinet
    2. Grachev N.E., Dmitriev A.V., Senin D.S., Volkov V.T., Nefedov N.N., “Modelirovanie dinamiki fronta vnutriplastovogo goreniya”, Vychislitelnye metody i programmirovanie: novye vychislitelnye tekhnologii, 11:1 (2010), 306–312  mathnet  elib
    3. Bykov A.A., Sharlo A.S., “Nonstationary contrasting structures for the generalized Kolmogorov-Petrovskiy-Piskunov equation”, Mosc. Univ. Phys. Bull., 67:2 (2012), 147–153  crossref  mathscinet  adsnasa  isi  elib  elib  scopus
    4. Dmitriev M.G., Pavlov A.A., Petrov A.P., “Nonstationary Fronts in the Singularly Perturbed Power-Society Model”, Abstract Appl. Anal., 2013, 172654  crossref  mathscinet  zmath  isi  elib  scopus
    5. A. A. Bykov, A. S. Sharlo, “Nonstationary contrast structures in adjacency of the special point”, Math. Models Comput. Simul., 7:2 (2015), 165–178  mathnet  crossref
    6. A. A. Bykov, N. N. Nefedov, A. S. Sharlo, “Contrast structures for a quasilinear Sobolev-type equation with unbalanced nonlinearity”, Comput. Math. Math. Phys., 54:8 (2014), 1234–1243  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. 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
    8. Levashova N.T., Nikolaeva O.A., Pashkin A.D., “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
    9. Nefedov N.N., Nikulin E.I., “Existence and Stability of Periodic Contrast Structures in the Reaction-Advection-Diffusion Problem”, Russ. J. Math. Phys., 22:2 (2015), 215–226  crossref  mathscinet  zmath  isi  elib  scopus
    10. Mikhailov E.A., “Problems With a Small Parameter and Propagation of Fronts in the Galactic Dynamo Theory”, Mosc. Univ. Phys. Bull., 70:2 (2015), 101–106  crossref  mathscinet  isi  elib  scopus
    11. Levashova N.T. Mel'nikova A.A., “Step-Like Contrast Structure in a Singularly Perturbed System of Parabolic Equations”, Differ. Equ., 51:3 (2015), 342–361  crossref  mathscinet  zmath  isi  elib  scopus
    12. Volkov V., Nefedov N., Antipov E., “Asymptotic-Numerical Method For Moving Fronts in Two-Dimensional R-D-a Problems”, Finite Difference Methods, Theory and Applications, Lecture Notes in Computer Science, 9045, eds. Dimov I., Farago I., Vulkov L., Springer-Verlag Berlin, 2015, 408–416  crossref  mathscinet  zmath  isi  scopus
    13. A. A. Bykov, “Chislennoe reshenie nachalno-kraevoi zadachi dlya psevdoparabolicheskogo uravneniya s vnutrennim perekhodnym sloem”, Model. i analiz inform. sistem, 23:3 (2016), 259–282  mathnet  crossref  mathscinet  elib
    14. N. T. Levashova, A. A. Melnikova, S. V. Bytsyura, “Primenenie metoda differentsialnykh neravenstv dlya obosnovaniya resheniya sistemy parabolicheskikh uravnenii v vide dvizhuschegosya fronta”, Model. i analiz inform. sistem, 23:3 (2016), 317–325  mathnet  crossref  mathscinet  elib
    15. Nefedov N., “Multiple Scale Reaction-Diffusion-Advection Problems with Moving Fronts”, Murphys-Hsfs-2014: 7Th International Workshop on Multi-Rate Processes & Hysteresis (Murphys) & the 2Nd International Workshop on Hysteresis and Slow-Fast Systems (Hsfs), Journal of Physics Conference Series, 727, eds. Klein O., Dimian M., Gurevich P., Knees D., Rachinskii D., Tikhomirov S., IOP Publishing Ltd, 2016, UNSP 012011  crossref  mathscinet  isi  scopus
    16. Mikhailov E.A., “Wavefronts of the Magnetic Field in Galaxies: Asymptotic and Numerical Approaches”, Magnetohydrodynamics, 52:1-2 (2016), 117–124  isi  elib
    17. E. A. Antipov, V. T. Volkov, N. T. Levashova, N. N. Nefedov, “Reshenie vida dvizhuschegosya fronta dvumernoi zadachi reaktsiya-diffuziya”, Model. i analiz inform. sistem, 24:3 (2017), 259–279  mathnet  crossref  elib
    18. A. A. Bykov, K. E. Ermakova, “Resheniya uravnenii nestatsionarnogo fronta reaktsii s vyrozhdennymi tochkami ravnovesiya”, Model. i analiz inform. sistem, 24:3 (2017), 309–321  mathnet  crossref  elib
    19. Nefedov N., “Asymptotic Analysis of Reaction-Diffusion-Advection Problems: Fronts With Periodic Motion and Blow-Up”, 8Th Workshop on Multi-Rate Processes and Hysteresis and the Hysteresis and Slow-Fast Systems (Hsfs) Workshop, Journal of Physics Conference Series, 811, eds. Gurevich P., Korobeinikov A., Rachinskii D., Sobolev V., IOP Publishing Ltd, 2017, UNSP 012008  crossref  mathscinet  isi  scopus
    20. Mikhailov E.A., “Galactic Magnetic Field Reversals and Vorticity of Transition Layers”, Magnetohydrodynamics, 53:2, SI (2017), 357–363  isi
    21. 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
    22. N. N. Nefedov, O. V. Rudenko, “On front motion in a Burgers-type equation with quadratic and modular nonlinearity and nonlinear amplification”, Dokl. Math., 97:1 (2018), 99–103  mathnet  crossref  crossref  mathscinet  zmath  isi  scopus
    23. Melnikova A.A., Derugina N.N., “The Dynamics of the Autowave Front in a Model of Urban Ecosystems”, Mosc. Univ. Phys. Bull., 73:3 (2018), 284–292  crossref  isi  scopus
    24. 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
    25. 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
    26. A. A. Bykov, K. E. Ermakova, “Exact solutions of equations of a nonstationary front with equilibrium points of a fractional order”, Comput. Math. Math. Phys., 58:12 (2018), 1977–1988  mathnet  crossref  crossref  isi  elib
    27. Bykov A.A. Ermakova K.E., “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  isi
  • Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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