Abstract:
The problem of constructing solutions to a system of two coupled nonlinear parabolic equations of the reaction–diffusion type is considered. Solutions in the form of diffusion waves propagating over zero background with a finite speed are investigated. The theorem on the construction of exact solutions by reduction to the Cauchy problem for a system of ordinary differential equations is proved. A time-step numerical technique for solving the reaction–diffusion system using radial basis function expansion is proposed. The same approach is used to solve systems of ordinary differential equations that determine the exact solutions of the reaction–diffusion system. Numerical analysis and estimation of the accuracy of solutions to a system of ordinary differential equations are carried out. These solutions are applied to verify the obtained time-step solutions of the original system.
Citation:
A. L. Kazakov, L. F. Spevak, “Exact and approximate solutions to the degenerated reaction–diffusion system”, Prikl. Mekh. Tekh. Fiz., 62:4 (2021), 169–180; J. Appl. Mech. Tech. Phys., 62:4 (2021), 673–683
\Bibitem{KazSpe21}
\by A.~L.~Kazakov, L.~F.~Spevak
\paper Exact and approximate solutions to the degenerated reaction--diffusion system
\jour Prikl. Mekh. Tekh. Fiz.
\yr 2021
\vol 62
\issue 4
\pages 169--180
\mathnet{http://mi.mathnet.ru/pmtf140}
\crossref{https://doi.org/10.15372/PMTF20210417}
\elib{https://elibrary.ru/item.asp?id=46448003}
\transl
\jour J. Appl. Mech. Tech. Phys.
\yr 2021
\vol 62
\issue 4
\pages 673--683
\crossref{https://doi.org/10.1134/S0021894421040179}
Linking options:
https://www.mathnet.ru/eng/pmtf140
https://www.mathnet.ru/eng/pmtf/v62/i4/p169
This publication is cited in the following 6 articles:
A. L. Kazakov, L. F. Spevak, “Tochnye i priblizhennye resheniya kvazilineinoi parabolicheskoi sistemy «khischnik-zhertva» s nulevymi frontami”, Materialy 6 Mezhdunarodnoi konferentsii «Dinamicheskie sistemy i kompyuternye nauki: teoriya i prilozheniya» (DYSC 2024). Irkutsk, 16–20 sentyabrya 2024 g. Chast 3, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 240, VINITI, M., 2025, 19–28
A. L. Kazakov, P. A. Kuznetsov, L. F. Spevak, “O nekotorykh resheniyakh s nulevym frontom evolyutsionnoi parabolicheskoi sistemy”, Differentsialnye uravneniya i optimalnoe upravlenie, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 224, VINITI RAN, M., 2023, 80–88
V.S. Bobrovskiy, A.L. Kazakov, E.M. Rojas, A.V. Sinitsyn, L.F. Spevak, “Steady-states solutions of the Vlasov–Maxwell–Fokker–Planck system of proton channeling in crystals”, Communications in Nonlinear Science and Numerical Simulation, 118 (2023), 107005
A. L. Kazakov, P. A. Kuznetsov, L. F. Spevak, “The Problem of Diffusion Wave Initiation for a Nonlinear Second-Order Parabolic System”, Proc. Steklov Inst. Math. (Suppl.), 321:1 (2023), S109–S126
A. L. Kazakov, L. F. Spevak, “Solutions to a nonlinear degenerating reaction–diffusion system of the type of diffusion waves with two fronts”, J. Appl. Mech. Tech. Phys., 63:6 (2022), 995–1004
A. L. Kazakov, P. A. Kuznetsov, L. F. Spevak, “Postroenie reshenii vyrozhdayuscheisya sistemy «reaktsiya-diffuziya» v sluchayakh tsilindricheskoi i sfericheskoi simmetrii pri nelineinostyakh obschego vida”, Geometriya, mekhanika i differentsialnye uravneniya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 213, VINITI RAN, M., 2022, 54–62
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