Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 45. no. 4. P. 24-35. ISSN 2079-6641

MATHEMATICAL MODELLING
https://doi.org/10.26117/2079-6641-2023-45-4-24-35
Research Article
Full text in Russian
MSC 34A08, 34A34

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Fractional Model of Geoacoustic Emission

R. I. Parovik^\ast

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

Abstract. In this work, a fractional dynamic system that describes high-frequency geoacoustic emission with heredity was proposed and investigated. The model is a system of two connected linear oscillators with nonconstant coefficients and Gerasimov-Caputo fractional order derivatives. Each oscillator describes a dislocation source of geoacoustic emission. The model is built on the assumption that interaction between sources occurs only through radiation. The presence of heredity indicates a change in the intensity of such interaction. For a fractional dynamic model with Gerasimov-Caputo derivatives, local initial conditions are valid, i.e. the Cauchy problem is posed. Further in the work, based on the Gerasimov-Caputo approximation of fractional derivatives, a nonlocal explicit finite-difference scheme is constructed for the numerical solution of the Cauchy problem. The numerical solution is visualized. Oscillograms and phase trajectories were constructed using a numerical algorithm for various values of the orders of fractional derivatives in the Maple computer algebra environment. Some interpretation of the simulation results is given.

Key words: mathematical modeling, fractional dynamic system, geoacoustic emissions, oscillogram, phase
trajectory, explicit finite-difference, Maple

Received: 07.11.2023; Revised: 18.11.2023; Accepted: 25.11.2023; First online: 12.12.2023

For citation. Parovik R. I. Fractional model of geoacoustic emission. Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 45: 4, 24-35. EDN: CJMOVH. https://doi.org/10.26117/2079-6641-2023-45-4-24-35.

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. The author participated in the writing of the article and is fully responsible for submitting the final version of the article to the press.

^\astCorrespondence: E-mail: parovik@ikir.ru

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

© Parovik R. I., 2023

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

References

1. Dolgikh G. I., et al. Deformation and acoustic precursors of earthquakes, Doklady Earth Sciences, 2007, 413:1, 281–285,(In Russian)
2. Marapulets Y. V., et. al. Geoacoustic emission response to deformation processes activation during earthquake preparation, Russian Journal of Pacific Geology, 2012, 6:6, 457–464.
3. Kissin I. G. Flyuidy v zemnoy kore: Geofizicheskie i tektonicheskie aspekty [Fluids in the Earth’s Crust: Geophysical and Tectonic Aspects]. Moscow, Nauka, 2015, 328 (In Russian)
4. Morgunov V. A., Lyuboshevskij M. N., Fabricius V. Z., Fabricius Z. E. Geoakusticheskij predvestnik Spitakskogo zemletryaseniya [Geoacoustic precursor of the Spitak earthquake], Vulkanologiya i sejsmologiya, 1991, 4, 104–106. (In Russian)
5. Gregori G.P., et al. «Storms of crustal stress»and AE earthquake precursors, Natural Hazards and Earth System Sciences, 2010, 10, 319–337.
6. Iskakov B. A., et al. Geoacoustic emission when high-energy muons of cosmic origin pass through the earth’s crust. Vestnik Perm. Univer. Seriya: Fizika, 2021, 1. 5-11(In Russian)
7. Bokov V. N., Vorob’yev V. N. Variability of geoacoustic emissions and changes in atmospheric circulation. Uchenyye zapiski Rossiyskogo gosudarstvennogo gidrometeorologicheskogo universiteta, 2013, 32, 43-54(In Russian)
8. Gavrilov V.A., Morozova Yu.V., Storcheus A.V. Variations in the level of geoacoustic emission in the deep well G-1 (Kamchatka) and their relationship with seismic activity. Volcanology and seismology. 2006. 1. pp. 52–67(In Russian)
9. Marapulets Yu. V. High frequency acoustic emission effect. Vestnik. KRAUNC. Fiz.-mat. nauki. 2015, 10:1, 44–53,(In Russian)
10. Krylov V. V., Landa P. S., Robsman V. A. Model’ razvitiya akusticheskoy emissii kak khaotizatsiya perekhodnykh protsessov v svyazannykh nelineynykh ostsillyatorakh. Akusticheskiy zhurnal, 1993, 39:1, 108-122,(In Russian)
11. Gapeev M., Solodchuk A., Parovik R. Coupled oscillators as a model of high-frequency geoacoustic emission. Vest. KRAUNC. Fiz.-mat. nauki. 2022, 40:3, 88-100,(In Russian)
12. Tristanov A., et al. Improvement of methods for sparse model identification of pulsed geophysical signals. IEEE, 2019, 256–260 DOI: 10.23919/SPA.2019.8936817
13. Rabotnov Yu. N. Elementy nasledstvennoy mekhaniki tvordykh tel [Elements of hereditary mechanics of solids]. Мoscow. Nauka, 1977. 384 p.(In Russian)
14. Volterra V. Functional theory, integral and integro-differential equations. N-Y. Dover Publications, 2005. 288 p.
15. Nakhushev A. M. Drobnoye ischisleniye i yego primeneniye [Fractional calculus and its applications]. Moscow. Fizmatlit. 2003. 272 p.(In Russian)
16. Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations. Amsterdam. Elsevier. 2006. 523 p.
17. Mingazova D., Parovik R. Some aspects of the qualitative analysis of the high-frequency geoacoustic emission model. Vest. KRAUNC. Fiz.-mat. nauki. 2023, 42:1, 191-206.(In Russian)
18. Parovik R. I. Haoticheskiye i regulyarnyye rezhimy drobnykh ostsillyatorov [Chaotic and regular modes of fractional oscillators]. P-K. KAMCHATPRESS, 2019. 132 p. (In Russian)

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