Volume 21, Issue 3
Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations

Shidong Jiang, Jiwei Zhang, Qian Zhang & Zhimin Zhang

Commun. Comput. Phys., 21 (2017), pp. 650-678.

Published online: 2018-04

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  • Abstract

The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total $\mathcal{O}$($N_S$$N_T$) memory and $\mathcal{O}$(($N_S$$N^2_T$) work, where $N_T$ and $N_S$ represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative $^C_0$$D^α_t$$f(t)$ of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel $t^{−1−α}$ on the interval [∆t,T] with a uniform absolute error ε. We give the theoretical analysis to show that the number of exponentials $N_{exp}$ needed is of order $\mathcal{O}$($log$$N_T$) for T≫1 or $\mathcal{O}$($log^2N_T$) for T≈1 for fixed accuracy ε. The resulting algorithm requires only $\mathcal{O}$($N_SN_{exp}$) storage and $\mathcal{O}$($N_SN_TN_{exp}$) work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

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@Article{CiCP-21-650, author = {}, title = {Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations}, journal = {Communications in Computational Physics}, year = {2018}, volume = {21}, number = {3}, pages = {650--678}, abstract = {

The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total $\mathcal{O}$($N_S$$N_T$) memory and $\mathcal{O}$(($N_S$$N^2_T$) work, where $N_T$ and $N_S$ represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative $^C_0$$D^α_t$$f(t)$ of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel $t^{−1−α}$ on the interval [∆t,T] with a uniform absolute error ε. We give the theoretical analysis to show that the number of exponentials $N_{exp}$ needed is of order $\mathcal{O}$($log$$N_T$) for T≫1 or $\mathcal{O}$($log^2N_T$) for T≈1 for fixed accuracy ε. The resulting algorithm requires only $\mathcal{O}$($N_SN_{exp}$) storage and $\mathcal{O}$($N_SN_TN_{exp}$) work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2016-0136}, url = {http://global-sci.org/intro/article_detail/cicp/11254.html} }
TY - JOUR T1 - Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations JO - Communications in Computational Physics VL - 3 SP - 650 EP - 678 PY - 2018 DA - 2018/04 SN - 21 DO - http://dor.org/10.4208/cicp.OA-2016-0136 UR - https://global-sci.org/intro/article_detail/cicp/11254.html KW - AB -

The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total $\mathcal{O}$($N_S$$N_T$) memory and $\mathcal{O}$(($N_S$$N^2_T$) work, where $N_T$ and $N_S$ represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative $^C_0$$D^α_t$$f(t)$ of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel $t^{−1−α}$ on the interval [∆t,T] with a uniform absolute error ε. We give the theoretical analysis to show that the number of exponentials $N_{exp}$ needed is of order $\mathcal{O}$($log$$N_T$) for T≫1 or $\mathcal{O}$($log^2N_T$) for T≈1 for fixed accuracy ε. The resulting algorithm requires only $\mathcal{O}$($N_SN_{exp}$) storage and $\mathcal{O}$($N_SN_TN_{exp}$) work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

Shidong Jiang, Jiwei Zhang, Qian Zhang & Zhimin Zhang. (2020). Fast Evaluation of the Caputo Fractional Derivative and Its Applications to Fractional Diffusion Equations. Communications in Computational Physics. 21 (3). 650-678. doi:10.4208/cicp.OA-2016-0136
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