Vestnik KRAUNC. Fiz.-Mat. Nauki. 2022. vol. 40. no. 3. pp. 53–63. ISSN 2079-6641
Solution of the boundary problem for the generalized Laplace equation with a fractional derivative
О. Kh. Masaeva
Institute of Applied Mathematics and Automation, 360000, Kabardino-Balkarian Republic, Nalchik, st. Shortanova, 89a, Russia
In this paper, we study the Dirichlet boundary value problem in the upper halfplane for a second-order partial differential equation containing a composition of Riemann-Liouville fractional differentiation operators with respect to one of two independent variables. The equation under consideration, for an integer value of the order of fractional differentiation, passes into the Laplace equation in two independent variables. An explicit representation of the solution of the problem under study (in terms of a function of the Mittag-Leffler type) is obtained by the method of the integral Fourier transform. Asymptotic estimates for a particular solution and its derivatives are found. Theorems on the existence and uniqueness of a regular solution are proved. The existence of a solution is proved in the class of continuous functions with weight in a closed half-plane. The uniqueness of the solution is proved in the class of continuously differentiable functions with respect to the spatial variable and having a corresponding continuous fractional derivative with weight with respect to the time variable in a closed half-plane.
Key words: fractional derivative, Mittag-Leffler type function, generalized Laplace equation with fractional derivative, Dirichlet problem.
Original article submitted: 10.10.2022
Revision submitted: 03.11.2022
For citation. Masaeva О. Kh. Solution of the boundary problem for the generalized Laplace equation with a fractional derivative. Vestnik KRAUNC. Fiz.-mat. nauki. 2022, 40: 3, 53-63. DOI: 10.26117/2079-6641-2022-40-3-53-63
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)
© Masaeva O. Kh., 2022
Funding. The work was carried out within the framework of the state tasks of the Ministry of Education and Science of the Russian Federation «Nonlinear singular integro-differential equations and boundary value problems» (project FEGS-2020-0001)
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Masaeva Olesya Khazhismelovna – Candidate of Physical and Mathematical Sciences, Researcher, Division of Fractional Calculus, Institute of Applied Mathematics and Automation of Kabardin-Balkar Scientific Centre of RAS, Nalchik, Russia, ORCID 0000-0002-0392-6189.