Vestnik КRAUNC. Fiz.-Mat. nauki. 2024. vol. 46. no. 1. P. 103-117. ISSN 2079-6641

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

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Application of High-Performance Computing to Solve the Cauchy problem with the Fractional Riccati Equation Using an Nonlocal Implicit Finite-Difference Scheme

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

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

Abstract. The article presents a study of the computational efficiency of a parallel version of a numerical algorithm for solving the Riccati equation with a fractional variable order derivative of the Gerasimov-Caputo type. The numerical algorithm is a nonlocal implicit finite-difference scheme, which reduces to a system of nonlinear algebraic equations and is solved using a modified Newton method. The nonlocality of the numerical scheme creates a high computational load on computing resources, which creates the need to implement efficient parallel algorithms for solving them. The numerical algorithm studied for efficiency is implemented in the C language due to its versatility when working with memory. Parallelization was carried out using OpenMP technology. A series of computational experiments are being carried out on the NVIDIA DGX STATION computing server (Institute of Mathematics named after V.I. Romanovsky, Tashkent, Uzbekistan) and the HP Pavilion Gaming Laptop Z270X, where the Cauchy problem for the fractional Riccati equation with non-constant coefficients was solved. Based on the average computation time, the speedup, efficiency and cost of the algorithm are calculated. From the data analysis it is clear that the OpenMP parallel software implementation of the non-local implicit finite-difference scheme shows an acceleration of 9-12 times, depending on the number of CPU cores involved.

Keywords: parallel computing, OpenMP, implicit finite difference schemes, Newton’s method, fractional
derivatives, memory effect, non-locality, non-linearity

Received: 18.01.2024; Revised: 18.02.2024; Accepted: 07.03.2024; First online: 07.03.2024

For citation. Tverdyi D. A., Parovik R. I. Application of high-performance computing to solve the Cauchy problem with the fractional Riccati equation using an nonlocal implicit finite-difference scheme. Vestnik KRAUNC. Fiz.-mat. nauki. 2024, 46: 1, 103-117. EDN: GNJWJM.

Funding. The research was carried out within the framework of the RSF grant № 22-11-00064 on the topic “Modelling of dynamic processes in 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., 2024

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


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Tverdyi Dmitrii Alexsandrovich – Ph. D. (Phys. & Math.), Researcher, Laboratory of Electromagnetic Radiation, 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 at the Laboratory for Modeling Physical Processes, Institute of Cosmophysical Research and Radio Wave Propagation, FEB RAS, Paratunka, Russia, ORCID 0000-0002-1576-1860.