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

Part of: General theory in ordinary differential equations Real functions Miscellaneous topics - Partial differential equations Computational aspects Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  07 February 2017

Shidong Jiang ,
Jiwei Zhang ,
Qian Zhang  and
Zhimin Zhang
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:shidong.jiang@njit.edu (S. Jiang), jwzhang@csrc.ac.cn (J. Zhang), qianzhang@csrc.ac.cn (Q. Zhang), zmzhang@csrc.ac.cn (Z. Zhang)
*Corresponding author. Email addresses:shidong.jiang@njit.edu (S. Jiang), jwzhang@csrc.ac.cn (J. Zhang), qianzhang@csrc.ac.cn (Q. Zhang), zmzhang@csrc.ac.cn (Z. Zhang)
*Corresponding author. Email addresses:shidong.jiang@njit.edu (S. Jiang), jwzhang@csrc.ac.cn (J. Zhang), qianzhang@csrc.ac.cn (Q. Zhang), zmzhang@csrc.ac.cn (Z. Zhang)
*Corresponding author. Email addresses:shidong.jiang@njit.edu (S. Jiang), jwzhang@csrc.ac.cn (J. Zhang), qianzhang@csrc.ac.cn (Q. Zhang), zmzhang@csrc.ac.cn (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.

Keywords

Fractional derivativeCaputo derivativesum-of-exponentials approximationfractional diffusion equationfast convolution algorithmstability analysis

MSC classification

Secondary: 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$ 33F05: Numerical approximation and evaluation 35Q40: PDEs in connection with quantum mechanics 35Q55: NLS-like equations (nonlinear Schrödinger) 34A08: Fractional differential equations 35R11: Fractional partial differential equations 26A33: Fractional derivatives and integrals
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 3 , March 2017 , pp. 650 - 678
Copyright
Copyright © Global-Science Press 2017 

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Footnotes

Communicated by Tao Zhou

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