Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 45. no. 4. P. 36-51. ISSN 2079-6641

Research Article
Full text in Russian
MSC 34A08, 34A34

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Solution of the Inverse Problem of Identifying the Order of the Fractional Derivative in a Mathematical Model of the Dynamics of Solar Activitythe at Rising Phase

D. A. Tverdyi^\ast, R. I. Parovik

Institute for Cosmophysical Research and Radio Propagation FEB RAS, 684034, v. Paratunka, Mirnaya st., 7, Russia

Abstract. The article refines the mathematical model of solar activity dynamics by solving the inverse problem. Experimental data on the observation of Wolf number values are used as additional information. This parameter of solar activity reflects the number of spots on the surface of the sun, and is considered an indicator of its activity. This process is characterized by observable cyclicality, periods of growth and decline. The analysis and processing of the initial data is carried out in order to isolate from the time series areas corresponding to an increase in solar activity. To describe this dynamic process, a previously proposed mathematical model for describing cycles 23 and 24 is used. The model is a Cauchy problem for a fractional analogue of the nonlinear Riccati equation, where the first-order derivative is replaced by the Gerasimov-Caputo fractional differentiation operator with an order from 0 to 1. The order of the fractional derivative is associated with the intensity of the process. This model equation is solved numerically using a nonlocal implicit finite-difference scheme. To clarify the values of the order of the fractional derivative, the one-dimensional optimization problem was solved using the second-order Levenberg-Marquardt iterative method, based on processed experimental data. It is shown that it is possible to refine the order of the fractional derivative in the solar activity model by solving the corresponding inverse problem, and the results obtained are in better agreement with the data.

Key words: mathematical modeling, reverse problem, solar activity, Wolf number, sunspots, dynamic processes, nonlinear equations, Riccati equation, saturation effect, fractional derivatives, ereditarity, MATLAB, C, parallel algorithms

Received: 02.11.2023; Revised: 16.11.2023; Accepted: 23.11.2023; First online: 11.12.2023

For citation. Tverdyi D. A., Parovik R. I. Solution of the inverse problem of identifying the order of the fractional derivative in a mathematical model of the dynamics of solar activitythe at rising phase. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 45: 4, 24-39. EDN: VBZQIO.

Funding. The research was carried out within the framework of the Russian Science Foundation grant No. 22-11-00064 on the topic «Modeling of dynamic processes in the geospheres taking into account heredity»

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.

^\astCorrespondence: E-mail:

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

© Tverdyi D. A., Parovik R. I., 2023

© Institute of Cosmophysical Research and Radio Wave Propagation, 2023 (original layout, design, compilation)


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Information about authors

Tverdyi Dmitrii Alexsandrovich – Ph. D. (Phys. & Math.), Researcher laboratory of electromagnetic propogation Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0001-6983-5258.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor, Leading researcher laboratory of modeling physical processes Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, Paratunka, Russia, ORCID 0000-0002-1576-1860.