Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 50. no. 1. P. 92 – 110. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2025-50-1-92-110
Research Article
Full text in Russian
MSC 78-10

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A Method for the Analytical Study of Autowave Processes in Weakly Conductive and Conductive Liquids

S. A. Kovalenko¹, V. S. Chekanov²^{\ast} , N. V. Kandaurova², M. H. Urtenov¹

¹Kuban State University, 350040, Krasnodar, Stavropol str., 149, Russia
²Branch of RTU MIREA in Stavropol, 355038, Stavropol, Kulakov ave., 8, Russia

Abstract. The article presents mathematical modeling of autowaves in a weakly conductive liquid (ferrocolloid) and in a conductive liquid (salt solution), taking into account ion recharge in the regions of spatial charge in the form of a boundary value problem for a nonstationary system of Nernst-Planck-Poisson equations. Using the proposed mathematical model, a theoretical study of the occurrence of the self-organization process – autowaves in an electromembrane system and in a ferrocolloid (magnetic liquid) was carried out. The main mechanisms in the occurrence of autowaves are the recharging of magnetic particles (in the case of a ferrocolloid) and the recharging of ions (cations and anions) in the case of an electromembrane system. Impurity ions are involved only in the charge transfer process. A numerical analysis of the boundary value problem of the mathematical model is carried out and the main regularities of the phenomenon of recharging on the transfer of magnetic particles are established. The existence of soliton-like autowave solutions for magnetic particles and salt ions is shown. The total concentration of basic ions practically does not change, in contrast, the total concentration of impurity ions decreases. The article proposes a new mathematical method for the approximate analytical solution of a boundary value problem based on the derivation of a nonlinear partial differential equation for a potential in the region of a single wave. It is shown that this equation can be reduced using a number of transformations, including the Hopf–Cole transformation, to a canonical parabolic equation, and in this sense an exact analytical solution is found. In addition, a simple analytical approximation for a single wave was found, a comparison with the numerical solution was carried out and their qualitative and quantitative coincidence was shown (with an accuracy of ∼ 3%).

Key words: autowaves, electromembrane system, ion exchange membrane, spatial charge, Nernst-Planck-Poisson equations, asymptotic solution, singularly perturbed boundary value problems, galvanodynamic regime

Received: 20.01.2025; Revised: 21.03.2025; Accepted: 24.03.2025; First online: 25.03.2025

For citation. Kovalenko S. A., Chekanov V. S., Kandaurova N. V., Urtenov M. H. A method for the analytical study of autowave processes in weakly conductive and conductive liquids. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 50: 1, 92-110. EDN: AVPSUS. https://doi.org/10.26117/2079-6641-2025-50-1-92-110.

Funding. The research was supported by the Russian Science Foundation, research project No. 24-19-00648, https://rscf.ru/project/24-19-00648 .

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: oranjejam@mail.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Kovalenko S. A., Chekanov V. S., Kandaurova N. V., Urtenov M. H., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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Information about the authors

Kovalenko Savva Andreevich – 1st year Master of the Faculty of Computer Technology and Applied Mathematics, Kuban State University, Krasnodar, Russia, ORCID 0009-0009-4849-0487.

Chekanov Vladimir Sergeevich – D. Sci. (Physics and Mathematics), Docent, Head of the Department of Information Technology, Branch of RTU MIREA in Stavropol, Stavropol, Russia, ORCID 0000-0002-2680-2883.

Kandaurova Natalia Vladimirovna – D. Sci. (Engineering), Professor, Deputy Director of Science and Development, Branch of RTU MIREA in Stavropol, Stavropol, Russia, ORCID 0000-0003-0928-4022.

Urtenov Mahamet Ali Huseevich – D. Sci. (Physics and Mathematics), Professor, Professor of the Department of Applied Mathematics, Kuban State University, Krasnodar, Russia, ORCID 0000-0002-0252-6247.

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