Vestnik КRAUNC. Fiz.-Mat. nauki. 2023. vol. 43. no. 2. P. 87-110. ISSN 2079-6641

Research Article
Full text in English
MSC 34A08, 65Y05, 65M06

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Parallelization of a Numerical Algorithm for Solving the Cauchy Problem for a Nonlinear Differential Equation of Fractional Variable Order Using OpenMP Technology

D. A. Tverdyi¹², R. I. Parovik¹²^\ast, A. R. Hayotov³, A.K Boltaev³

¹Vitus Bering Kamchatka State University, Russia, 683032, Petropavlovsk-Kamchatsky, Pogranichnaya st., 4.
²Institute for Cosmophysical Research and Radio Wave Propagation FEB RAS, Russia, 684034, Paratunka, Mirnaya st., 7.
³V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences,100174, Tashkent, University st., 9.

Abstract. The article presents a software implementation of a parallel efficient and fast computational algorithm for solving the Cauchy problem for a nonlinear differential equation of a fractional variable order. The computational algorithm is based on a non-local explicit finite-difference scheme, taking into account the approximation of the Gerasimov-Caputo fractional derivative VO included in the main differential equation. The algorithms for parallelization of the non-local explicit finite difference scheme were implemented as functions of the user library of the C programming language using the OpenMP technology. The OpenMP technology allows implementing parallel algorithms for working with the CPU computing node using its multithreading. The C language was chosen because of its versatility and lack of strict restrictions on memory handling. Further in the paper, the efficiency of the parallel algorithm is investigated. Efficiency is understood as the optimal ratio in coordinates: acceleration of calculations – the amount of RAM memory occupied, in comparison with the sequential version of the algorithm. The average computation time is analyzed in terms of: running time, acceleration, efficiency and cost of the algorithm. These algorithms were run on two different computing systems: a gaming laptop and a computing server. For a non-local explicit scheme, a significant performance increase of 3-5 times is shown for various methods of software implementation.

Key words: fractional derivatives, heredity, memory effect, finite difference schemes, parallel computing, OpenMP

Received: 02.06.2023; Revised: 09.06.2023; Accepted: 20.06.2023; First online: 30.06.2023

For citation. Tverdyi D. A., Parovik R. I., Hayotov A. R., Boltaev A. K. Parallelization of a numerical algorithm for
solving the Cauchy problem for a nonlinear differential equation of fractional variable order using OpenMP technology.
Vestnik KRAUNC. Fiz.-mat. nauki. 2023, 43: 2, 87-110. EDN: WFDGQO.

Funding. This research was funded by grant of the President of the Russian Federation grant number MD-758.2022.1.1 on the topic “Development of mathematical models of fractional dynamics in order to study oscillatory processes and processes with saturation”.

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., Hayotov A. R., Boltaev A. K., 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 of the International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky, Russia, ORCID 0000-0001-6983-5258.

Parovik Roman Ivanovich – D. Sci. (Phys. & Math.), Associate Professor, Head of the International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, Petropavlovsk-Kamchatsky, Russia, ORCID 0000-0002-1576-1860.

Hayotov Abdullo Rahmonovich – D. Sci. (Phys. & Math.), Associate Professor, Laboratory manager Scientific laboratory of computational mathematics Institute of Mathematics named after V.I. Romanovsky AS RUz, 683023, Tashkent, Uzbekistan, 0000-0002-2756-9542.

Boltayev Aziz Qo‘ziyevich – Ph. D. (Phys. & Math.), Senior Researcher Scientific laboratory of computational mathematics Institute of Mathematics named after V.I. Romanovsky AS RUz, 683023, Tashkent, Uzbekistan, 0000-0002-8329-4440.