# Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 40. no. 3. pp. 16–27. ISSN 2079-6641

**MSC 35R11, 35A08 **

**Research Article**

**On representation of solution of the diffusion equation with Dzhrbashyan-Nersesyan operators**

**F. T. Bogatyreva**

Institute of Applied Mathematics and Automation KBSC RAS, 89 A, Shortanova str., 360000, Nalchik, Russia

E-mail: fatima_bogatyreva@bk.ru

The paper investigates a parabolic partial differential equation with fractional differentiation with respect to one of two independent variables associated with time. Such equations are usually referred to the class of fractional diffusion equations. The fractional differentiation operator is a linear combination of two Dzhrbashyan-Nersesyan operators. The main result of the work is a theorem on the general representation of regular solutions of the equation under study in an infinite strip. A fundamental solution is constructed in terms of the Wright function and its main properties are studied. In particular, formulas for fractional differentiation are proved, the asymptotic behavior is investigated, and estimates are obtained for the fundamental solution and its derivatives for large and small values of the self-similar variable, and its positiveness is proved. To construct a general solution, the Green’s function method adapted to equations containing Dzhrbashyan-Nersesyan operators is used. Particular cases of the equation under consideration include equations with Riemann-Liouville and Gerasimov-Caputo derivatives. Therefore, the results obtained remain valid for equations with these fractional differentiation operators and their combinations.

*Key words: fractional diffusion equation, Dzhrbashyan-Nersesyan operators, fractional derivative, Wright function.*

DOI: 10.26117/2079-6641-2022-40-3-16-27

**Original article submitted:** 16.10.2022

**Revision submitted:** 25.11.2022

**For citation.** Bogatyreva F. T. On representation of solution of the diffusion equation with Dzhrbashyan-Nersesyan operators. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 40: 3, 16-27.

DOI: 10.26117/2079-6641-2022-40-3-16-27

**Competing interests.** The authors declare that 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.

The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)

© Bogatyreva F. T., 2022

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**Bogatyreva Fatima Takhirovna** – Junior Researcher, Department of Fractional Calculus, Institute of Applied Mathematics and Automation, KBSC RAS, Nalchik, Russia, ORCID 0000-0003-1765-066X.