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

Document Type

Article

Publication Date

3-1-2017

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. 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.

Identifier

85012305634 (Scopus)

Publication Title

Communications in Computational Physics

External Full Text Location

https://doi.org/10.4208/cicp.OA-2016-0136

e-ISSN

19917120

ISSN

18152406

First Page

650

Last Page

678

Issue

3

Volume

21

Grant

1418918

Fund Ref

National Science Foundation

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