Numerical solution of the boundary value problem for the heat equation with fractional Riesz derivative
Vol 6, Issue 2, 2023
(Abstract)
Abstract
The work is devoted to the numerical solution of the initial boundary value problem for the heat equation with a fractional Riesz derivative. Explicit and implicit difference schemes are constructed that approximate the boundary value problem for the heat equation with a fractional Riesz derivative with respect to the coordinate. In the case of an explicit difference scheme, a condition is obtained for the time step at which the difference scheme converges. For an implicit difference scheme, a theorem on unconditional convergence is proved. An example of a numerical calculation using an implicit difference scheme is given. It has been established that when passing to a fractional derivative, the process of heat propagation slows down.
Keywords
Full Text:
PDFReferences
1. Uchaikin VV, Sibatov RT. Fractional differential kinetics of dispersive transport as the consequence of its self-similarity. JETP Letters 2007; 86(8): 512–516. doi: 10.1134/s0021364007200040
2. Reviznikov DL, Slastushenskiy YV. Numerical simulation of anomalous diffusion in polygonal billiard gas channel (Russian). Matematicheskoe Modelirovanie 2013; 25(5): 3–14.
3. Liu J, Xu M. Some exact solutions to Stefan problems with fractional differential equations. Journal of Mathematical Analysis and Applications 2009; 351(2): 536–542. doi: 10.1016/j.jmaa.2008.10.042
4. Meerschaert MM, Tadjeran C. Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computational and Applied Mathematics 2004; 172(1): 65–77. doi: 10.1016/j.cam.2004.01.033
5. Meerschaert ММ, Tadjeran С. Finite difference approximations for two-sides space-fractional partial differential equations. Applied Numerical Mathematic 2006; 56: 80–90. doi: 10.5555/1126893.1642837
6. Beybalaev VD, Abduragimov EI, Yakubov AZ, et al. Numerical research of non-isothermal filtration process in fractal medium with non-locality in time. Thermal Science 2021; 25(1, Part B): 465–475. doi: 10.2298/tsci190223328b
7. Sweis H, Shawagfeh N, Abu Arqub O. Fractional crossover delay differential equations of Mittag-Leffler kernel: Existence, uniqueness, and numerical solutions using the Galerkin algorithm based on shifted Legendre polynomials. Results in Physics 2022; 41: 105891. doi: 10.1016/j.rinp.2022.105891
8. Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives: theory and applications. Gordon and Breach Science Publishers; 1993. p. 371.
9. Samarsky AA, Gulin AV. Numerical Methods (Russian). Nauka; 1989. p. 240.
DOI: https://doi.org/10.24294/tse.v6i2.2082
Refbacks
- There are currently no refbacks.
Copyright (c) 2023 Vetlugin Beybalayev, Abutrab Aliverdiev
License URL: https://creativecommons.org/licenses/by-nc/4.0/
This site is licensed under a Creative Commons Attribution 4.0 International License.