Abstract:
This work presents a review and analysis of modern asymptotic methods for analysis of singularly perturbed problems with interior and boundary layers. The central part of the work is a review of the work of the author and his colleagues and disciples. It highlights boundary and initial-boundary value problems for nonlinear elliptic and parabolic partial differential equations, as well as periodic parabolic problems, which are widely used in applications and are called reaction–diffusion and reaction–diffusion–advection equations. These problems can be interpreted as models in chemical kinetics, synergetics, astrophysics, biology, and other fields. The solutions of these problems often have both narrow boundary regions of rapid change and inner layers of various types (contrasting structures, moving interior layers: fronts), which leads to the need to develop new asymptotic methods in order to study them both formally and rigorously. A general scheme for a rigorous study of contrast structures in singularly perturbed problems for partial differential equations, based on the use of the asymptotic method of differential inequalities, is presented and illustrated on relevant problems. The main achievements of this line of studies of partial differential equations are reflected, and the key directions of its development are indicated.
Citation:
N. N. Nefedov, “Development of methods of asymptotic analysis of transition layers in reaction–diffusion–advection equations: theory and applications”, Zh. Vychisl. Mat. Mat. Fiz., 61:12 (2021), 2074–2094; Comput. Math. Math. Phys., 61:12 (2021), 2068–2087
\Bibitem{Nef21}
\by N.~N.~Nefedov
\paper Development of methods of asymptotic analysis of transition layers in reaction–diffusion–advection equations: theory and applications
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2021
\vol 61
\issue 12
\pages 2074--2094
\mathnet{http://mi.mathnet.ru/zvmmf11333}
\crossref{https://doi.org/10.31857/S0044466921120103}
\elib{https://elibrary.ru/item.asp?id=46713031}
\transl
\jour Comput. Math. Math. Phys.
\yr 2021
\vol 61
\issue 12
\pages 2068--2087
\crossref{https://doi.org/10.1134/S0965542521120095}
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Linking options:
https://www.mathnet.ru/eng/zvmmf11333
https://www.mathnet.ru/eng/zvmmf/v61/i12/p2074
This publication is cited in the following 47 articles:
E. I. Nikulin, B. T. Volkov, D. A. Karmanov, “Periodic Inner Transition Layers in the Reaction–Diffusion Problem in the Case of Weak Reaction Discontinuity”, VMU, 80:№1, 2025 (2025)
N. N. Nefedov, “Existence, Asymptotics, and Lyapunov Stability of Solutions of Periodic Parabolic Problems for Tikhonov-Type Reaction–Diffusion Systems”, Math. Notes, 115:2 (2024), 232–239
S. A. Kaschenko, “Chains with Diffusion-Type Couplings Containing a Large Delay”, Math. Notes, 115:3 (2024), 323–335
E. I. Nikulin, V. T. Volkov, D. A. Karmanov, “Internal Transition Layer Structure
in the Reaction–Diffusion Problem for the Case
of a Balanced Reaction with a Weak Discontinuity”, Diff Equat, 60:1 (2024), 65
Sergey Kashchenko, “Chains with Connections of Diffusion and Advective Types”, Mathematics, 12:6 (2024), 790
V. I. Uskov, “Uravnenie vetvleniya dlya differentsialnogo uravneniya pervogo poryadka v banakhovom prostranstve c kvadratichnymi vozmuscheniyami malogo parametra”, Materialy Voronezhskoi mezhdunarodnoi vesennei matematicheskoi shkoly «Sovremennye metody kraevykh zadach.
Pontryaginskie chteniya—XXXIV», Voronezh, 3-9 maya 2023 g. Chast 4, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 233, VINITI RAN, M., 2024, 99–106
E. I. Nikulin, V. T. Volkov, A. G. Nikitin, “On contrast structures in a problem of the baretting effect
theory”, Theoret. and Math. Phys., 220:1 (2024), 1193–1200
N. N. Nefedov, A. O. Orlov, “Existence and stability of stationary solutions with boundary layers in a system of fast and slow reaction–diffusion–advection equations with KPZ nonlinearities”, Theoret. and Math. Phys., 220:1 (2024), 1178–1192
P. E. Bulatov, Han Cheng, Yuxuan Wei, V. T. Volkov, N. T. Levashova, “Boundary control problem for the reaction–advection–diffusion equation with a modulus discontinuity of advection”, Theoret. and Math. Phys., 220:1 (2024), 1097–1109
N. T. Levashova, E. A. Chunzhuk, A. O. Orlov, “Stabilization of the front in a medium with discontinuous characteristics”, Theoret. and Math. Phys., 220:1 (2024), 1139–1156
E. I Nikulin, V. T Volkov, D. A Karmanov, “STRUKTURA VNUTRENNEGO PEREKhODNOGO SLOYa V ZADAChE REAKTsIYa–DIFFUZIYa V SLUChAE SBALANSIROVANNOY REAKTsII SO SLABYM RAZRYVOM”, Differencialʹnye uravneniâ, 60:1 (2024), 64
A. A. Bykov, “Two-dimensional transient contrasting structure evolution in an inhomogeneous media with the advection”, VMU, 2024, no. №2_2024, 2420101–1
A. Liubavin, Mingkang Ni, “Application of Asymptotic Methods to the Question of Stability in Stationary Solution with Discontinuity on a Curve”, Comput. Math. and Math. Phys., 64:6 (2024), 1286
S. A. Kashchenko, “Logistic Equation with Long Delay Feedback”, Diff Equat, 60:2 (2024), 145
D. A. Maslov, “About One Method for Numerical Solution of the Cauchy Problem for Singularly Perturbed Differential Equations”, Comput. Math. and Math. Phys., 64:5 (2024), 1029
A. A. Bykov, “Evolution of a Two-Dimensional Moving Contrast Structure in an Inhomogeneous Medium with Advection”, Moscow Univ. Phys., 79:2 (2024), 140
N. N. Nefedov, K. A. Kotsubinsky, “Existence and Stability of a Stationary Solution in a Two-Dimensional
Reaction-Diffusion System with Slow and Fast Components”, VMU, 2024, no. №3_2024, 2430101–1
S. A. Kashchenko, “Logistic equation with long delay feedback”, Differencialʹnye uravneniâ, 60:2 (2024)
N. N. Nefedov, K. A. Kotsubinsky, “Existence and Stability of a Stationary Solution in a Two-Dimensional Reaction-Diffusion System with Slow and Fast Components”, Moscow Univ. Phys., 79:3 (2024), 301
E.I. Nikulin, N.N. Nefedov, A.O. Orlov, “Existence and Asymptotic Stability of Solutions for Periodic Parabolic Problems in Tikhonov-Type Reaction–Diffusion–Advection Systems with KPZ Nonlinearities”, Russ. J. Math. Phys., 31:3 (2024), 504
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