Inverse problem on determining variable-order fractional derivative in mathematical model of anomalous variations in radon volumetric activity

D.A. Tverdyi, R.I. Parovik

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Abstract: The seismicity issue in the Kamchatka region highlights the importance of fundamental research to understand the processes occurring in the Earth’s crust. Anomalous changes in the concentration of radioactive radon gas (222Rn) have been observed to be precursor to earthquakes. Monitoring involves collecting data on the volume activity of 222Rn in the recording chamber over time and detecting any unusual patterns or anomalies. However, the mechanisms behind these irregularities are still not well understood. Therefore, the authors have previously proposed new mathematical hereditary VAR models to describe the unusual migration capacity of 222Rn, taking into account the time-dependent transport
process in a heterogeneous fractal geoenvironment. The key parameter of the models is the variable order of the Gerasimov-Caputo fractional derivative, related to the intensity of the 222Rn transport process with changes in the permeability of the geoenvironment.
Аim. The study is to solve the coefficient inverse problem of identifying values in the mathematical hereditary model of anomalous variations of the OAR.
Research methods. Methods of mathematical modeling of processes occurring in the geological environment are used, as well as solving coefficient inverse problems for these models using an algorithm based on the Levenberg-Marquardt method (IP-LM).
Results. A series of results are obtained for solving the inverse problem with various parameters controlling the IP-LM motion. The results are divided into two types: implausible – due to going beyond the range of acceptable values and the initial approximation of identified values close to the reference point, similar to manually selected values; plausible – due to the initial approximation close to 0, good agreement between the results and the OAR data, where the increase in values from 0 to 1 is maintained despite the loss of the reference point’s apparent periodicity.
Conclusions. The results suggest that it is possible to solve the formulated two-parameter inverse problem based on the experimental data of the OAR. The obtained results are plausible; however, the outcome of the inverse problem solution depends on the initial approximation of the identified values.

Keywords: mathematical modeling, radon volumetric activity, fractional derivative of variable order, inverse problems

For citation. Tverdyi D.A., Parovik R.I. Inverse problem on determining variable-order fractional derivative in mathematical model of anomalous variations in radon volumetric activity. News of the Kabardino-Balkarian Scientific Center of RAS. 2025. Vol. 27. No. 6. Pp. 77–88. DOI: 10.35330/1991-6639-2025-27-6-77-88

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Information about the authors

Dmitrii A. Tverdyi, Candidate of Physics and Mathematics, Researcher, Institute of Cosmophysical Research and Radio Wave Propagation Far Eastern Branch of the Russian Academy of Sciences;
7, Mirnaya street, Paratunka village, 684034, Russia;
dimsolid95@gmail.com, ORCID: https://orcid.org/0000-0001-6983-5258, SPIN-code: 2849-4814
Roman I. Parovik, Doctor of Physics and Mathematics, Professor of the Far Eastern Branch of the Russian Academy of Sciences, Leading Researcher, Institute of Cosmophysical Research and Radio Wave Propagation Far Eastern Branch of the Russian Academy of Sciences;
7, Mirnaya street, Paratunka village, 684034, Russia;
parovik@ikir.ru, ORCID: https://orcid.org/0000-0002-1576-1860, SPIN-code: 4295-6894