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Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations

Published online by Cambridge University Press:  07 February 2017

Shidong Jiang*
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
Jiwei Zhang*
Affiliation:
Beijing Computational Science Research Center, Beijing 100093, China
Qian Zhang*
Affiliation:
Beijing Computational Science Research Center, Beijing 100093, China
Zhimin Zhang*
Affiliation:
Beijing Computational Science Research Center, Beijing 100093, China Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
*
*Corresponding author. Email addresses:[email protected] (S. Jiang), [email protected] (J. Zhang), [email protected] (Q. Zhang), [email protected] (Z. Zhang)
*Corresponding author. Email addresses:[email protected] (S. Jiang), [email protected] (J. Zhang), [email protected] (Q. Zhang), [email protected] (Z. Zhang)
*Corresponding author. Email addresses:[email protected] (S. Jiang), [email protected] (J. Zhang), [email protected] (Q. Zhang), [email protected] (Z. Zhang)
*Corresponding author. Email addresses:[email protected] (S. Jiang), [email protected] (J. Zhang), [email protected] (Q. Zhang), [email protected] (Z. Zhang)
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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 memory and work, where NT and NS 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 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 [Δt, T] with a uniform absolute error ε. We give the theoretical analysis to show that the number of exponentials Nexp needed is of order for T≫1 or for TH1 for fixed accuracy ε. The resulting algorithm requires only storage and 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.

Type
Research Article
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

Communicated by Tao Zhou

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