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Adyghe Int. Sci. J. Vol. 23, No 4. P. 43-53

Adyghe Int. Sci. J. Vol. 23, No 4. P. 43-53. 

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MSC 35A02, 35C15 Original Article

On the Dirichlet problem for the generalized

Laplace equation

Olesya Khazhismelovna Masaeva
D. in Physics and Mathematics, Research Associate, Fractional Calculus Department, Institute of Applied Mathematics and Automation KBNC RAS, (360017, Russia, Nalchik, 89 A Shortanov St.), ORCID,

Abstract. This paper considers the Dirichlet problem for a second-order partial differential equation with the Riemann-Liouville derivative with respect to one of two independent order variables, less than two, in the upper half-plane. The equation under study turns into a two-dimensional Laplace equation if the order of the fractional derivative coincides with an integer. The main result of this work is the proof of theorems on the existence and uniqueness of a solution to the problem posed. An explicit form of representation of the solution is obtained. Сorresponding asymptotic estimates аre given.

Keywords: upper half-plane, two-dimensional Laplace equation, Dirichlet problem, uniqueness of solution, Riemann-Liouville fractional derivative.

Acknowledgments: the author are thankful to the anonymous reviewer for his valuable remakes.

Funding: The research was carried out within the framework of the state assignment of the Ministry of Education and Science of the Russian Federation (project № FEGS-2020-0001).

For citation. Masaeva O. Kh. On the Dirichlet problem for the generalized Laplace equation. Adyghe Int. Sci. J. 2023. Vol. 23, No. 4. P. 43–53. DOI:;

The author has read and approved the final version of the manuscript.
Submitted 11.10.2023; approved after reviewing 15.11.2023; aeeepted for publication 20.11.2023.

© Masaeva O. Kh., 2023


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