Analytical solution of the space-time fractional reaction-diffusion equation with variable coefficients
- Authors: Mahmoud E.I.1
-
Affiliations:
- RUDN University
- Issue: Vol 69, No 3 (2023)
- Pages: 430-444
- Section: Articles
- URL: https://journals.rudn.ru/CMFD/article/view/36489
- DOI: https://doi.org/10.22363/2413-3639-2023-69-3-430-444
- EDN: https://elibrary.ru/FKQFNA
Cite item
Full Text
Abstract
In this paper, we solve the problem of an inhomogeneous one-dimensional fractional differential reaction-diffusion equation with variable coefficients (1.1)-(1.2) by the method of separation of variables (the Fourier method). The Caputo derivative and the Riemann-Liouville derivative are considered in the time and space directions, respectively. We prove that the obtained solution of the boundary-value problem satisfies the given boundary conditions. We discuss the convergence of the series defining the proposed solution.
About the authors
E. I. Mahmoud
RUDN University
Author for correspondence.
Email: ei_abdelgalil@yahoo.com
Moscow, Russia
References
- Алероев Т.С., Алероева Х.Т. Об одном классе несамосопряженных операторов, сопутствующих дифференциальным уравнениям дробного порядка// Укр. мат. вiсн.- 2015.- 12, № 3.-С. 293-310.
- Нахушев А.М. Дробное исчисление и его применение. -М.: Физматлит, 2003.
- Aleroev T.S. Solving the boundary value problems for differential equations with fractional derivatives by the method of separation of variables// Mathematics.- 2020.- 8.- 1877.
- Aleroev T.S., Elsayed A.M., Mahmoud E.I. Solving one dimensional time-space fractional vibration string equation// Conf. Ser. Mater. Sci. Eng.- 2021.-1129.-С. 20-30.
- Aleroev T.S., Kirane M., Malik S.A. Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition// Electron. J. Differ. Equ. -2013.-270.- С. 1-16.
- Curtiss D.R. Recent extensions of Descartes’ rule of signs// Ann. Math.- 1918.- 19, № 4.- С. 251-278.
- Gorenflo R., Kilbas A.A., Mainardi F., Rogosin S.V. Mittag-Leffler Functions Related Topics and Applications.- New York: Springer, 2014.
- Gorenflo R., Mainardi F. Random walk models for space fractional diffusion processes// Fract. Calc. Appl. Anal. -1998.-1.- С. 167-191.
- Hu Z., Liu W., Liu J. Boundary value problems for fractional differential equations// Tijdschrift voor Urologie.-2014.- 2014, № 1.-С. 1-11.
- Luchko Y., Gorenflo R. An operational method for solving fractional differential equations// Acta Math.- 1999.-24.-С. 207-234.
- Plociniczak L. Eigenvalue asymptotics for a fractional boundary-value problem// Appl. Math. Comput.- 2014.-241.- С. 125-128.
- Samko S.G., Kilbas A.A., Marichev O.I. Fractional Integrals and Derivatives. Theory and Applications.- New York: Gordon and Breach, 1993.
