TY - JOUR
T1 - Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations
AU - Jiang, Shidong
AU - Zhang, Jiwei
AU - Zhang, Qian
AU - Zhang, Zhimin
N1 - Publisher Copyright:
© Copyright Global-Science Press 2017.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.
PY - 2017/3/1
Y1 - 2017/3/1
N2 - 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. We give the theoretical analysis to show that the number of exponentials N exp needed is of order for T1 or for TH1 for fixed accuracy. 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.
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 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. We give the theoretical analysis to show that the number of exponentials N exp needed is of order for T1 or for TH1 for fixed accuracy. 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.
KW - Caputo derivative
KW - Fractional derivative
KW - fast convolution algorithm
KW - fractional diffusion equation
KW - stability analysis
KW - sum-of-exponentials approximation
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U2 - 10.4208/cicp.OA-2016-0136
DO - 10.4208/cicp.OA-2016-0136
M3 - Article
AN - SCOPUS:85012305634
VL - 21
SP - 650
EP - 678
JO - Communications in Computational Physics
JF - Communications in Computational Physics
SN - 1815-2406
IS - 3
ER -