Year: 2012
Author: Na Zhang, Weihua Deng, Yujiang Wu
Advances in Applied Mathematics and Mechanics, Vol. 4 (2012), Iss. 4 : pp. 496–518
Abstract
We present the finite difference/element method for a two-dimensional modified fractional diffusion equation. The analysis is carried out first for the time semi-discrete scheme, and then for the full discrete scheme. The time discretization is based on the $L1$-approximation for the fractional derivative terms and the second-order backward differentiation formula for the classical first order derivative term. We use finite element method for the spatial approximation in full discrete scheme. We show that both the semi-discrete and full discrete schemes are unconditionally stable and convergent. Moreover, the optimal convergence rate is obtained. Finally, some numerical examples are tested in the case of one and two space dimensions and the numerical results confirm our theoretical analysis.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.10-m1210
Advances in Applied Mathematics and Mechanics, Vol. 4 (2012), Iss. 4 : pp. 496–518
Published online: 2012-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Modified subdiffusion equation finite difference method finite element method stability convergence rate.
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https://doi.org/10.1016/j.enganabound.2022.11.003 [Citations: 9] -
A block‐centered finite difference method for fractional Cattaneo equation
Li, Xiaoli | Rui, Hongxing | Liu, ZhengguangNumerical Methods for Partial Differential Equations, Vol. 34 (2018), Iss. 1 P.296
https://doi.org/10.1002/num.22198 [Citations: 3] -
Three-point combined compact difference schemes for time-fractional advection–diffusion equations with smooth solutions
Gao, Guang-Hua | Sun, Hai-WeiJournal of Computational Physics, Vol. 298 (2015), Iss. P.520
https://doi.org/10.1016/j.jcp.2015.05.052 [Citations: 28] -
Time second-order finite difference/finite element algorithm for nonlinear time-fractional diffusion problem with fourth-order derivative term
Liu, Nan | Liu, Yang | Li, Hong | Wang, JinfengComputers & Mathematics with Applications, Vol. 75 (2018), Iss. 10 P.3521
https://doi.org/10.1016/j.camwa.2018.02.014 [Citations: 39] -
Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation
Dehghan, Mehdi | Abbaszadeh, Mostafa | Mohebbi, AkbarApplied Mathematical Modelling, Vol. 40 (2016), Iss. 5-6 P.3635
https://doi.org/10.1016/j.apm.2015.10.036 [Citations: 70] -
Fast difference schemes for solving high-dimensional time-fractional subdiffusion equations
Zeng, Fanhai | Zhang, Zhongqiang | Karniadakis, George EmJournal of Computational Physics, Vol. 307 (2016), Iss. P.15
https://doi.org/10.1016/j.jcp.2015.11.058 [Citations: 61] -
Spectral direction splitting methods for two-dimensional space fractional diffusion equations
Song, Fangying | Xu, ChuanjuJournal of Computational Physics, Vol. 299 (2015), Iss. P.196
https://doi.org/10.1016/j.jcp.2015.07.011 [Citations: 34] -
Numerical Solutions of Fractional Differential Equations Arising in Engineering Sciences
Jannelli, Alessandra
Mathematics, Vol. 8 (2020), Iss. 2 P.215
https://doi.org/10.3390/math8020215 [Citations: 22]