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Qualitative properties of the mathematical model of nonlinear cross-diffusion processes

https://doi.org/10.17586/2220-8054-2024-15-6-742-748

Abstract

The work is devoted to developing a self-similar solution for a system of nonlinear differential equations that describe diffusion processes. Various techniques are used to examine the capacity for generating self-similar solutions that can estimate and predict system behavior under diffusion conditions. The focus is on developing and applying numerical algorithms, as well as using theoretical tools such as asymptotic analysis, to obtain more accurate and reliable results. The study’s results can be applied to various scientific and technical fields, such as physics, chemistry, biology, and engineering, where diffusion processes play an essential role. The development of self-similar solutions for systems of nonlinear differential equations related to diffusion opens novel opportunities for modeling and analyzing complex systems and enhancing diffusion processes in various fields.

About the Authors

S. Muminov
Mamun university; Urgench state university
Uzbekistan

Sokhibjan Muminov

Bolkhovuz street 2, Khiva 220900

Khamid Alimjan street 14, Urgench 220100



P. Agarwal
Nonlinear Dynamics Research Center (NDRC); Department of Mathematics, Anand International College of Engineering; International Center for Basic and Applied Sciences
India

Praveen Agarwal

Ajman, UAE

Jaipur 303012

Jaipur-302029



D. Muhamediyeva
Tashkent University of information technologies named after Muhammad Al-Khwarizmi
Uzbekistan

Dildora Muhamediyeva – Department of Software of Information Technologies

Amir Temur Avenue 108, Tashkent 100084



References

1. Samarsky A.A., Mikhailov A.P. Mathematical Modeling, Fizmatlib, Moscow, 2001, 320 p.

2. Aripov M.M. Methods of Reference Equations for Solving Nonlinear Boundary Value Problems., Fan, Tashkent, 1988, 136 p.

3. Aripov M.M., Sadullaeva Sh.A. Computer Modeling of Nonlinear Diffusion Processes. University, Tashkent, 2020, 656 p.

4. Aripov M.M., Matyakubov A.S., Imomnazarov B.Kh. The cauchy problem for a nonlinear degenerate parabolic system in non-divergence form. Mathematical Notes of NEFU, 2020, 27(3), P. 27–38.

5. Matyakubov A.S., Raupov D. On some properties of the blow-up solutions of a nonlinear parabolic system non-divergent form with cross-diffusion. Lec. Notes in Civil Engineering, 2022, 180, P. 289–301.

6. Aripov M.M., Matyakubov A.S., Xasanov J.O. To the qualitative properties of self-similar solutions of a cross-diffusion parabolic system not in divergence form with a source. AIP Conf. Proc., 2023, 2781.

7. Aripov M.M., Matyakubov A.S., Khasanov J.O., Bobokandov M.M. Mathematical modeling of double nonlinear problem of reaction diffusion in not divergent form with a source and variable density. J. Phys. : Conf. Ser., 2021, 2131(3).

8. Muhamediyeva D.K. Properties of self similar solutions of reaction-diffusion systems of quasilinear equations. Int. J. of Mech. and Prod. Eng. Research and Development, 2018, 36(8), P. 555–566.

9. Muhamediyeva D.K. Qualitative properties of wave solutions of the equation of reaction-diffusion of a biological population. Conference proceedings of “2020 International Conference on Information Science and Communications Technologies (ICISCT)”, Tashkent, Uzbekistan, 04-06 November 2020.

10. Muhamediyeva D.K., Nurumova A.Y, Muminov S.Y. Fuzzy evaluation of cotton varieties in the natural climatic. In IOP Conf. Ser.: Earth and Environmental Science, 2022, 1076, P. 012043.

11. Muhamediyeva D.K., Madrakhimov A.Kh., Kodirov Z.Z. Construction of a system of differential equations taking into account convective transfer. Proceedings of “Computer applications for management and sustainable development of production and industry (CMSD2022)”. Dushanbe, Tajikistan, 21-23 December 2022.

12. Muhamediyeva D.K., Muminov S.Y., Shaazizova M.E., Hidirova Ch., and Bahromova Yu. Limited different schemes for mutual diffusion problems. E3S Web of Conf, 2023, 401, P. 05057.

13. Muhamediyeva D.K., Nurumova A.Y, Muminov S.Y. Numerical modeling of cross-diffusion processes. E3S Web of Conf, 2023, 401, P. 05060.

14. Muhamediyeva D.K., Nurumova A.Y., Muminov S.Y. Cauchy problem and boundary-value problems for multicomponent cross-diffusion systems. Proceedings of “Int. Conf. on Inf. Sci. and Comm. Tech.”. Tashkent 2021. 01-05.

15. Muminov S.Y. Construction of self-similar solutions of the system of nonlinear differential equations of cross-diffusion. Proceedings of “The int. Sci. and Prac. Conf.”, China 2023. 57–60.

16. Rakhmonov Z.R., Alimov A.A. Properties of solutions for a nonlinear diffusion problem with a gradient nonlinearity. Int. J. App. Math., 2023, 36(3), P. 1–20.

17. Yang Liu, Yanwei Du, Hong Li, Jichun Li, and Siriguleng He. A two-grid mixed finite element method for a nonlinear fourth-order reaction-diffusion problem with time-fractional derivative. Comp. and Math. with Appl., 2015, 70(10), P. 2474–2492.

18. Farina A., Gianni R. Self-similar solutions for the heat equation with a positive non-Lipschitz continuous, semilinear source term. Nonl. Anal.: Real World Appl., 2024, 79.

19. Aripov M.M., Raimbekov J.R. The critical curves of a doubly nonlinear parabolic equation in non-divergent form with a source and nonlinear boundary flux. J. Sib. Fed. Univ. - Math. and Phys., 2019, 12(1), P. 112–124.

20. Topaev T.N., Popov A.I., Popov I.Yu. On Keller-Rubinow model for Liesegang structure formation. Nanosystems: physics, chemistry, mathematics, 2022, 13(4), P. 365–371.

21. Fedorov E.G. Properties of an oriented ring of neurons with the Fitzhugh-Nagumo model. Nanosystems: physics, chemistry, mathematics, 2021, 12(5), P. 553–562.


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For citations:


Muminov S., Agarwal P., Muhamediyeva D. Qualitative properties of the mathematical model of nonlinear cross-diffusion processes. Nanosystems: Physics, Chemistry, Mathematics. 2024;15(6):742-748. https://doi.org/10.17586/2220-8054-2024-15-6-742-748

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ISSN 2220-8054 (Print)
ISSN 2305-7971 (Online)