Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 48. no. 3. P. 83 – 94. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2024-48-3-83-94
Research Article
Full text in Russian
MSC 34A08, 34A34

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Mathematical Fractional Zeeman Model for Describing Cardiac Contractions

G. S. Israyiljanova^{\ast}¹, Sh. T. Karimov¹, R. I. Parovik²³

¹Fergana State University, 150100, Fergana, Murabbiylar str., 19, Uzbekistan
²National University of Uzbekistan named after Mirzo Ulugbek, 100174, Tashkent, Universitetskaya str., 4, Uzbekistan
³Kamchatka State University named after Vitus Bering, 683009, Petropavlovsk-Kamchatsky, Pogranichnaya str., 4. Russia

Abstract. The article proposes a fundamentally new generalization of the previously known mathematical Zeeman model of heart contractions due to electrochemical action. This generalization is due to the presence of heredity effects in the oscillatory system, which indicate that it can store information about its previous states. From the mathematical point of view, the property of heredity can be described using integrodifferential equations of the Volterra type with power difference kernels or using fractional derivatives. In the article, fractional differentiation operators in the sense of Gerasimov-Caputo were introduced into the Zeeman model equations, as well as the characteristic time for matching dimensions in the model equations. The resulting mathematical fractional Zeeman model was studied due to its nonlinearity using numerical methods – a nonlocal finite-difference scheme. The numerical algorithm was implemented in Python in the PyCharm 2024.1 environment, which implemented the ability to visualize calculations using oscillograms and phase trajectories. The interpretation of the modeling results was carried out.

Key words: heart contractions, fractional mathematical Zeeman model, fractional derivative of Gerasimov Caputo, numerical algorithm, oscillogram, phase trajectory.

Received: 30.09.2024; Revised: 01.11.2024; Accepted: 14.11.2024; First online: 20.11.2024

For citation. Israyiljanova G. S., Karimov Sh. T., Parovik R. I. Mathematical fractional Zeeman model for describing cardiac contractions. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 48: 3, 83-94. EDN: SGWSTH. https://doi.org/10.26117/2079-6641-2024-48-3-83-94.

Funding. The research was not supported by any foundation.

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: gulbaxor19802020@gmail.com

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

© Israyiljanova G. S., Karimov Sh. T., Parovik R. I., 2024

© Institute of Cosmophysical Research and Radio Wave Propagation, 2024 (original layout, design, compilation)

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Information about the authors

Israyiljanova Gulbahor Saminjanovna – Applicant of the Department
of Applied Mathematics and Informatics, Fergana State University, Fergana, Uzbekistan, ORCID 0009-0007-9356-3476.

Karimov Shakhobiddin Tuychiboyevich – D. Sc. (Phys. & Math.), Professor, Professor of the Department of Applied Mathematics and Computer Science, Fergana State University, Fergana, Uzbekistan, ORCID 0000-0001-8325-998X.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor,
Associate Professor, Professor of the Department of Computer Science and Mathematics, Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky, Russia, ORCID 0000-0002-1576-1860.

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