Construction of a Self-Similar Solution to Mutual Diffusion Problems
DOI:
https://doi.org/10.71310/pcam.4_68.2025.12Keywords:
mutual diffusion, self-similar solutions, nonlinear differential equations, approximate solution, asymptotic analysis, multicomponent media, scale invariance, stationary equation, mathematical modeling, diffusion processesAbstract
The article considers the problem of constructing self-similar solutions of a system of nonlinear differential equations modeling mutual diffusion processes in multicomponent media. An analysis of a mathematical model taking into account complex interactions of components and the nonlinear nature of transfer processes is carried out. Approximate solutions of the system are found, allowing one to describe the behavior of concentration profiles in various modes. Asymptotic representations of solutions for regular, unlimited and limited cases are obtained, and the behavior of a two-sided linear stationary equation arising at intermediate stages of analysis is studied. The results are of interest for diffusion theory and applied problems of mathematical modeling of complex physical processes.
References
Samarskiy A.A Teoriya raznostnykh skhem // Moskva: Nauka, – 1977. – 656 s.
Samarskiy A.A i dr. Rezhimy s obostreniyem v zadachakh dlya kvazilineynykh parabolicheskikh uravneniy // Moskva: Nauka, – 1987. – 480 s.
Samarsky A.A., Mikhailov A.P Mathematical modeling // Moscow: Pedagogika, – 1998. –420 p.
Aripov M.M. Metody etalonnykh uravneniy dlya resheniya nelineynykh krayevykh zadach // Tashkent: Fan, – 1988. – 136 s.
Aripov M.M., Cadullayeva SH.A. Komp'yuternoye modelirovaniye nelineynykh diffuzionnykh protsessov // Tashkent: Universitet, – 2020. – 656 s.
Aripov M. The Fujita and Secondary Type Critical Exponents in Nonlinear Parabolic Equations and Systems // Differential Equations and Dynamical Systems. Springer Proceedings in Mathematics & Statistics, – 2017. – Vol. 268. doi: http://dx.doi.org/10.1007/978-3-030-01476-6_2.
Aripov M.M., Mukhamediyeva D.K. Kross-diffuzionnyye modeli konvektivnogo perenosa s dvoynoy nelineynost'yu // Problemy vychislitel'noy i prikladnoy matematiki, – 2015. – No 1. – C. 5–9.
Aripov M.M.,Matyakubov A.S., Imomnazarov B.K. The cauchy problem for a nonlinear degenerate parabolic system in non–divergence form // Mathematical Notes of NEFU, –2020. – No 27(3). – P. 27–38.
Aripov М.М., Matyakubov A.S., Khasanov J.O., Bobokandov M.M., On Some Properties of the Blow-Up Solutions of a Nonlinear Parabolic System Non-divergent Form with Cross-Diffusion // Journal of Physics: Conference Series, – 2021. – No 2131(3).
Matyakubov A., Raupov D. TOn Some Properties of the Blow-Up Solutions of a Nonlinear Parabolic System Non-divergent Form with Cross-Diffusion // Lecture Notes in Civil Engineering, – 2022. – No 180. – P. 289–301.
Aripov M.M., Raimbekov J.R. The Critical Curves of a Doubly Nonlinear Parabolic Equation in Non-divergence form with a Source and a Nonlinear Boundary Flux // Journal of Siberian Federal University. Mathematics & Physics 2019, – 2019. – No 12(1). – P. 112–124. doi: http://dx.doi.org/10.17516/1997-1397-2019-12-1-112-124.
Aripov M., Mukimov A., Sayfullayeva M. To asymptotic of the solution of the heat conduction problem with double nonlinearity, variable density, absorption at a critical parameter // International journal of innovative technology and exploring engineering, – 2019. – Vol. 9. – Issue 1. – P. 3407–3412.
Aripov M., Mukimov A., Mirzayev B. To Asymptotic of the Solution of the Heat Conduction Problem with Double Nonlinearity with Absorption at a Critical Parameter // Mathematics and Statistics, – 2019. – 7(5). – P. 205–217. doi: http://dx.doi.org/10.13189/ms.2019.070507.
Muhamediyeva D.K., Nurumova А.Y., Muminov S.Y. Cauchy Problem and Boundary-Value Problems for Multicomponent Cross-Diffusion Systems // Tashkent: International Conference on Information Science and Communications Technologies. – 2021. doi: http://dx.doi.org/10.1109/ICISCT52966.2021.9670380.
Muhamediyeva D.K., Nurumova А.Y., Muminov S.Y. Numerical modeling of cross-diffusion processes // Tashkent: V International Scientific Conference “Construction Mechanics, Hydraulics and Water Resources Engineering” – 2023. doi: http://dx.doi.org/110.1051/e3sconf/202340105060.
Muhamediyeva D.K., Muminov S.Y, Shaazizova M.E., Khidirova Ch.,Bahromova Yu. Limited different schemes for mutual diffusion problems // Tashkent: V International Scientific Conference “Construction Mechanics, Hydraulics and Water Resources Engineering”. – 2023. doi: http://dx.doi.org/110.1051/e3sconf/202340105057.
Muminov S.YU. Avtomodel'nyye resheniye sistemy nelineynykh differentsial'nykh uravneniy kross-diffuzii // Problemy vychislitel'noy i prikladnoy matematiki, – 2023. – No 4(51). – C. 53–58.
Muminov S., Agarwal P., Muhamediyeva D. Qualitative properties of the mathematical model of nonlinear cross-diffusion processes // Nanosystems: Phys. Chem. Math. – 2024. –15(6). – P. 742–748. doi: http://dx.doi.org/10.17586/2220-8054-2024-15-6-742-748.
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