phone +7 (3412) 91 60 92
mail imi@udsu.ru
ru-RU Russian
Home » Issues » Archive of Issues » 2023 - 2 issue » pp. 225-239

Archive of Issues

Russia Irkutsk
Year
2023
Volume
33
Issue
2
Pages
225-239
<<
>>
Section Mathematics
Title On multidimensional exact solutions of a nonlinear reaction-diffusion system
Author(-s) Kosov A.A.a, Semenov E.I.a, Tirskikh V.V.b
Affiliations Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciencesa, Irkutsk State Transport Universityb
Abstract We study a multidimensional case of a nonlinear reaction-diffusion system modeled by a system of two parabolic equations with power nonlinearities. Such systems can be used to simulate the process of propagation in space of interacting distributed formations of robots of two types. Such equations also describe the processes of nonlinear diffusion in reacting two-component continuous media. An original version of the reduction method is proposed, which reduces the construction of the dependence of the exact solution on spatial variables to the solution of the Helmholtz equation, and the dependence on time to the solution of a linear system of ordinary differential equations. A number of examples of multiparameter families of exact solutions given by elementary functions are constructed.
Keywords reaction-diffusion system, reduction, exact solutions
UDC 35K57, 35K55, 35Q35
MSC 517.957
DOI 10.35634/vm230203
Received 3 February 2023
Language Russian
Citation Kosov A.A., Semenov E.I., Tirskikh V.V. On multidimensional exact solutions of a nonlinear reaction-diffusion system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2023, vol. 33, issue 2, pp. 225-239.
References
  1. Polyanin A.D., Kutepov A.M., Kazenin D.A., Vyazmin A.V. Hydrodynamics, mass and heat transfer in chemical engineering, London: CRC Press, 2001. https://doi.org/10.1201/9781420024517
  2. Kaptsov O.V. Metody integrirovaniya uravnenii s chastnymi proizvodnymi (Integration methods for partial differential equations), Moscow: Fizmatlit, 2009.
  3. Zhuravlev V.M. On a class of models of autowaves in active media with diffusion that admit exact solutions, Pis'ma v Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki, 1997, vol. 65, no. 3, pp. 285-290 (in Russian).
  4. Shmidt A.V. Exact solutions of systems of equations of the reaction-diffusion type, Vychislitel'nye Tekhnologii, 1998, vol. 3, no. 4, pp. 87-94 (in Russian).
  5. Ma Luyi, Niu Hong-Tao, Wang Zhi-Cheng. Pyramidal traveling fronts in the Belousov-Zhabotinskii reaction-diffusion system in $\mathbb{R}$${3}$ , Electronic Journal of Differential Equations, 2020, vol. 2020, no. 112, pp. 1-29.
  6. Wei Jieqiang, Fridman E., Johansson K.H. A PDE approach to deployment of mobile agents under leader relative position measurements, Automatica, 2019, vol. 106, pp. 47-53. https://doi.org/10.1016/j.automatica.2019.04.040
  7. Kosov A.A., Semenov E.I. Distributed model of space exploration by two types of interacting robots and its exact solutions, Journal of Physics: Conference Series, 2021, vol. 1847, no. 1, 012007. https://doi.org/10.1088/1742-6596/1847/1/012007
  8. Elamvazhuthi K., Berman S. Mean-field models in swarm robotics: a survey, Bioinspiration and Biomimetics, 2019, vol. 15, no. 1, 015001. https://doi.org/10.1088/1748-3190/ab49a4
  9. Álvarez-Caudevilla P., Du Yihong, Peng Rui. Qualitative analysis of a cooperative reaction-diffusion system in a spatiotemporally degenerate environment, SIAM Journal on Mathematical Analysis, 2014, vol. 46, issue 1, pp. 499-531. https://doi.org/10.1137/13091628x
  10. Rao Feng, Kang Yun. The complex dynamics of a diffusive prey-predator model with an Allee effect in prey, Ecological Complexity, 2016, vol. 28, pp. 123-144. https://doi.org/10.1016/j.ecocom.2016.07.001
  11. Amorim P., Telch B., Villada L.M. A reaction-diffusion predator-prey model with pursuit, evasion, and nonlocal sensing, Mathematical Biosciences and Engineering, 2019, vol. 16, issue 5, pp. 5114-5145. https://doi.org/10.3934/mbe.2019257
  12. Chen Zhenglong, Li Shunjie, Zhang Xuebing. Analysis of a delayed reaction-diffusion predator-prey system with fear effect and anti-predator behaviour, Mathematics, 2022, vol. 10, issue 18, 3270. https://doi.org/10.3390/math10183270
  13. Song Yingwei, Zhang Tie, Li Jinpeng. Dynamical behavior in a reaction-diffusion system with prey-taxis, Electronic Journal of Differential Equations, 2022, vol. 2022, no. 37, pp. 1-15.
  14. Kuang Yang, Wang Kaifa. Coexistence and extinction in a data-based ratio-dependent model of an insect community, Mathematical Biosciences and Engineering, 2020, vol. 17, issue 4, pp. 3274-3293. https://doi.org/10.3934/mbe.2020187
  15. Cherniha R., King J.R. Non-linear reaction–diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions, Journal of Mathematical Analysis and Applications, 2005, vol. 308, issue 1, pp. 11-35. https://doi.org/10.1016/j.jmaa.2004.10.034
  16. Kosov A.A., Semenov E.I. On exact multidimensional solutions of a nonlinear system of reaction–diffusion equations, Differential Equations, 2018, vol. 54, issue 1, pp. 106-120. https://doi.org/10.1134/S0012266118010093
  17. Polyanin A.D. Zaitsev V.F., Zhurov A.I. Metody resheniya nelineinykh uravnenii matematicheskoi fiziki i mekhaniki (Methods for solving nonlinear equations of mathematical physics and mechanics), Moscow: Fizmatlit, 2005.
  18. Polyanin A.D., Zhurov A.I. Metody razdeleniya peremennykh i tochnye resheniya nelineinykh uravnenii matematicheskoi fiziki (Methods of separation of variables and exact solutions of nonlinear equations of mathematical physics), Moscow: Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, 2020.
  19. Polyanin A.D., Zhurov A.I. Separation of variables and exact solutions to nonlinear PDEs, Chapman and Hall/CRC, 2021. https://doi.org/10.1201/9781003042297
  20. Galaktionov V.A., Svirshchevskii S.R. Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics, Chapman and Hall/CRC, 2006. https://doi.org/10.1201/9781420011623
  21. Awasthi P., Kumar M., Nec Yana. Effects of anisotropy in tridimensional diffusion: Flow patterns and transport efficiency, SIAM Journal on Applied Mathematics, 2023, vol. 83, issue 2, pp. 460-483. https://doi.org/10.1137/22M1476642
  22. Kosov A.A., Semenov E.I. Anisotropic solutions of a nonlinear kinetic model of elliptic type, Vestnik Yuzhno-Ural'skogo Universiteta. Seriya Matematicheskoe Modelirovanie i Programmirovanie, 2020, vol. 13, issue 4, pp. 48-57 (in Russian). https://doi.org/10.14529/mmp200404
  23. Kosov A.A., Semenov E.I. New exact solutions of the diffusion equation with power nonlinearity, Siberian Mathematical Journal, 2022, vol. 63, no. 6, pp. 1102-1116. https://doi.org/10.1134/S0037446622060106
  24. Kamke E. Spravochnik po obyknovennym differentsial'nym uravneniyam (Handbook of ordinary differential equations), Moscow: Nauka, 1971.
Full text
/doc/issues-2023/23-02-03.pdf
<< Previous article
Next article >>