Adyghe Int. Sci. J. Vol. 23, No 4. P. 43-53.
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DOI: https://doi.org/10.47928/1726-9946-2023-23-4-43-53
EDN: FBTEVU
MATHEMATICS
| 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 https://orcid.org/0000-0002-0392-6189., olesya.masaeva@yandex.ru
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: https://doi.org/10.47928/1726-9946-2023-23-4-43-53;
EDN: FBTEVU
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
REFERENCES
1. Nakhushev A. M. Drobnoye ischisleniye i ego primeneniye [Fractional calculus and its application].
M.: Fizmatlit Publ., 2003. 272 p.
2. Pskhu A. V. Uravneniya v chastnykh proizvodnykh drobnogo poryadka [Partial differential equations of fractional order], M.: Nauka Publ., 2005. 199 p.
3. Nakhushev A. M. O matematicheskikh i informatsionykh technologiyakh modelirovaniya i upravleniya regional’nym razvitiyem [On mathematical and information technologies for modeling and managing regional development] Adyghe Int. Sci. J. Vol. 9, No. 1. pp. 128-137.
4. Uchaikin V. V. Metod drobnykh proizvodnykh [Method of fractional derivatives], Ulyanovsk: Artichoke Publ., 2008. 512 p.
5. Pskhu A. V. Analog formuly Shvartsa dlya systemy Koshi-Rimana drobnogo poryadka [An analogue of the Schwartz formula for the Cauchy-Riemann system of fractional order] Sovremennye metody v teorii granichnykh zadach [Modern methods in the theory of boundary value problems]: proceedings of the Voronezh Spring Mathematical Schoolѕ «Pontryagin Readings – XIII», Voronezh, 2002. P. 127.
6. Mamchuev M. O. Kraevye zadachi dlya uravneniy i system uravneniy s chastnymi proizvodnymi drobnogo poryadka [Boundary value problems for equations and systems of partial differential equations of fractional order] Nalchik: Kabardino-Balkarian Scientific Center of RAS Publ., 2013. 200 p.
7. Masaeva O. Kh. Zadacha Dirikhle dlya obobschenogo uravneniya Laplasa s drobnoy proizvodnoy [The Dirichlet problem for the generalized Laplace equation with fractional derivative] Chelyabinskii fiziko-matematicheskii zhurnal [Chelyabinsk Physical and Mathematical Journal] 2017, vol. 2, no. 3. pp. 312-322.
8. Masaeva O. Kh. Zadacha Neymana dlya obobschenogo uravneniya Laplasa [Neumann problem for the generalized Laplace equation] Vestn. KRAUNTS. Fiz.-Mat. Nauki. 2018, vol. 23 no. 3, pp. 83-90.
9. Masaeva O. Kh. Solution of the boundary problem for the generalized Laplace equation with a fractional derivative. Vestn. KRAUNTS. Fiz.-Mat. Nauki. 2022, vol. 40, No. 3, pp. 53-63.
10. Bogatyreva F. T. Kraevaya zadacha dlya uravneniya v chastnykh proizvodnykh pervogo poryadka s operatorom Dzhrbashyana-Nersesyana [Boundary value problem for a first order partial differential equation with the Dzhrbashyan-Nersesyan operator] Adyghe Int. Sci. J. Vol. 17, No. 2. P. 17-24.
11. Bogatyreva F. T. On representation of solution of the diffusion equation with Dzhrbashyan-Nersesyan operators. Vestn. KRAUNTS. Fiz.-Mat. Nauki 2022, vol. 40, no. 3, pp. 16-27.
12. Dzhrbashyan M. M. Integral’nye preobrazovaniya i predstavleniya funktsiy v kompleksnoy oblasti [Integral transformations and representations of functions in the complex domain]. M.: Nauka Publ., 1966. 672 p.
13. Pskhu A. V. The Stankovich Integral Transform and Its Applications. Chapter 9. In book “Special functions and analysis of differential equations”. New York. Chapman and Hall/CRC. 2020. 370 p.
14. Prudnikov A. P., Brychkov Yu. A., Marichev О. I. Integraly i ryady. Elementarnye funktsii [Integrals and scries. Elementary Functions]. V. 1. M.: Fizmatlit Publ, 2002. 632 p.
15. Nakhushev A. M. On the positivity of continuous and discrete differentiation and integration operators that are very important in fractional calculus and in the theory of equations of mixed type. Differential Equations. 1998, vol. 34, no. 1, pp. 103-112.
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