Volume 10, Issue 4
Convergence of Finite Difference Method in Positive Time for Multi-Term Time Fractional Differential Equations

East Asian J. Appl. Math., 10 (2020), pp. 774-785.

Published online: 2020-08

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• Abstract

A multi-time fractional-order reaction-diffusion equation with the Caputo fractional derivative is considered. On a uniform grid, the problem is discretised by using the $L$1 formula. For the problem solutions with a singularity at time $t$ = 0, the convergence order is $\mathcal{O}$ (τα1). For any subdomain bounded away from $t$ = 0, the method has the convergence rate $\mathcal{O}$ (τ), which is better than the convergence rate $\mathcal{O}$ (τα1) for the whole time-space domain. Results of theoretical analysis are illustrated by numerical experiments.

• Keywords

Caputo fractional derivative, multi-term fractional differential equation, weak singularity, uniform mesh, L1 scheme.

65M06, 65M12, 65M15, 26A33

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• RIS
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@Article{EAJAM-10-774, author = {Haili Qiao , and Aijie Cheng , }, title = {Convergence of Finite Difference Method in Positive Time for Multi-Term Time Fractional Differential Equations}, journal = {East Asian Journal on Applied Mathematics}, year = {2020}, volume = {10}, number = {4}, pages = {774--785}, abstract = {

A multi-time fractional-order reaction-diffusion equation with the Caputo fractional derivative is considered. On a uniform grid, the problem is discretised by using the $L$1 formula. For the problem solutions with a singularity at time $t$ = 0, the convergence order is $\mathcal{O}$ (τα1). For any subdomain bounded away from $t$ = 0, the method has the convergence rate $\mathcal{O}$ (τ), which is better than the convergence rate $\mathcal{O}$ (τα1) for the whole time-space domain. Results of theoretical analysis are illustrated by numerical experiments.

}, issn = {2079-7370}, doi = {https://doi.org/10.4208/eajam.040220.020520}, url = {http://global-sci.org/intro/article_detail/eajam/17960.html} }
TY - JOUR T1 - Convergence of Finite Difference Method in Positive Time for Multi-Term Time Fractional Differential Equations AU - Haili Qiao , AU - Aijie Cheng , JO - East Asian Journal on Applied Mathematics VL - 4 SP - 774 EP - 785 PY - 2020 DA - 2020/08 SN - 10 DO - http://doi.org/10.4208/eajam.040220.020520 UR - https://global-sci.org/intro/article_detail/eajam/17960.html KW - Caputo fractional derivative, multi-term fractional differential equation, weak singularity, uniform mesh, L1 scheme. AB -

A multi-time fractional-order reaction-diffusion equation with the Caputo fractional derivative is considered. On a uniform grid, the problem is discretised by using the $L$1 formula. For the problem solutions with a singularity at time $t$ = 0, the convergence order is $\mathcal{O}$ (τα1). For any subdomain bounded away from $t$ = 0, the method has the convergence rate $\mathcal{O}$ (τ), which is better than the convergence rate $\mathcal{O}$ (τα1) for the whole time-space domain. Results of theoretical analysis are illustrated by numerical experiments.

Haili Qiao & Aijie Cheng. (2020). Convergence of Finite Difference Method in Positive Time for Multi-Term Time Fractional Differential Equations. East Asian Journal on Applied Mathematics. 10 (4). 774-785. doi:10.4208/eajam.040220.020520
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