Adyghe Int. Sci. J. Vol. 23, No 4. P. 34-42.
Read article Contents of this issue
DOI: https://doi.org/10.47928/1726-9946-2023-23-4-34-42
EDN: OQCHTN
MATHEMATICS
| MSC 35A02, 35C15 | Original Article |
Green’s function of the problem with local displacement
for the fractional telegraph equation
MamchuevMurat Osmanovich
D. Sci. (Phys. & Math.), Head of the Department of Fractional Calculus, Institute of Applied Mathematics and Automation, KBSC RAS, Nalchik, Russia; Professor of the Department of Computer Technologies and Artificial Intelligence, Kabardino-Balkarian State University named after H.M. Berbekov, Nalchik, Russia; ORCID https://orcid.org/0000-0002-7986-456X, mamchuev@rambler.ru
Abstract. In this paper, we study a nonlocal problem with local displacement for the fractional telegraph equation with the Riemann–Liouville derivative. The correctness of this problem is established and its solution is constructed in terms of the Green’s function.
Keywords: Fractional telegraph equation, Riemann–Liouville derivative, Fourier problem, local displacement problem, Green’s function.
Acknowledgments: the author are thankful to the anonymous reviewer for his valuable remakes.
For citation. Mamchuev M. O. Green’s function of the problem with local displacement for the fractional telegraph equation. Adyghe Int. Sci. J. 2023. Vol. 23, No. 4. P. 34–42.
DOI: https://doi.org/10.47928/1726-9946-2023-23-4-34-42; EDN: OQCHTN
The author has read and approved the final version of the manuscript.
Submitted 28.11.2023; approved after reviewing 15.12.2023; aeeepted for publication 21.12.2023.
© Mamchuev M. O., 2023
REFERENCES
1. Nakhushev A.M. Fractional Calculus and Its Application. Moscow: Fizmatlit, 2003. 272 p. (In Russian).
2. Pskhu A.V. Fractional partial differential equations. Moscow: Nauka, 2005. 199 p. (In Russian).
3. Eckstein E.C., Goldstein J.A., Leggas M. The analytical solutions and asymptotic analyticity. Electron. J. Differ. Eq. Conf. 3 (1999) 39-50.
4. Cascaval R. C., Eckstein E. C., Frota C. L., Goldstein J. A. Fractional telegraph equations. Journal of Mathematical Analysis and Applications, 2002. Vol. 276, No. 1. P. 145-159.
5. Orsinger E., Beghin L. Time-fractional telegraph equations and telegraph processes with brownian time. Probability Theory and Related Fields, 2004. Vol. 128, No. 1. P. 141–160.
6. Orsinger E., Zhao X. The space-fractional telegraph equation and the related fractional telegraph process. Chinese Annals of Mathematics Series B, 2003. Vol. 24, No. 1. P. 45–56.
7. Mamchuev M. O. Fundamental solution of a loaded second-order parabolic equation with constant coefficients. Differential Equations, 2015. Vol. 51, No. 5. P. 620-–629.
8. Mamchuev M. O. Modified Cauchy problem for a loaded second-order parabolic equation with constant coefficients. Differential Equations, 2015. Vol. 51, No. 9. P. 1137—1144.
9. Mamchuev M. O. Solutions of the main boundary value problems for a loaded second-order parabolic equation with constant coefficients. Differential Equations, 2016. Vol. 52, No. 6. P. 789-
–797.
10. Bogatyreva F. T. On representation of solution of the diffusion equation with Dzhrbashyan-Nersesyan operators. Vestnik KRAUNC. Fiziko-Matematicheskie Nauki, 2022. Vol. 40, No. 3. P. 16-–27.
11. Pshibikhova R. A. Analog of the Goursat problem for a generalized fractional telegraph equation. Differential Equations, 2014. V. 50, No. 6. P. 840–845.
12. Pshibikhova R. A. The Goursat problem for the fractional telegraph equation with Caputo derivatives. Mathematical Notes, 2016. V. 99, No. 3-4. P. 552–555.
13. Nakhushev A.M. Problems with a shift for partial differential equations. Moscow: Nauka, 2006. 287 p. (in Russian).
14. Mamchuev M.O., Mamchuev A. M. Fourier problem for fractional diffusion-wave equation. Lobachevskii Journal of Mathematics, 2023. V. 44, No. 2. P. 620-–628.
15. Wright E. M. On the coefficients of power series having exponential singularities. J. London Math. Soc, 1933. V. 8, No. 29. P. 71–79.
16. Wright E. M. The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. Ser. II, 1934. V. 38. P. 257–270.
This work is licensed under a Creative Commons Attribution 4.0 License.