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Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations
Zhang, Jia Li; Fang, Zhi Wei; Sun, Hai Wei
2022-02
Source PublicationNumerical Mathematics
ISSN1004-8979
Volume15Issue:1Pages:200-226
Abstract

In this paper, we propose a fast second-order approximation to the variable-order (VO) Caputo fractional derivative, which is developed based on L2-1 formula and the exponential-sum-approximation technique. The fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional derivative. This fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme (F L2-1 scheme) for the multi-dimensional VO time-fractional sub-diffusion equations. Theoretically, F L2-1 scheme is proved to fulfill the similar properties of the coefficients as those of the well-studied L2-1 scheme. Therefore, F L2-1 scheme is strictly proved to be unconditionally stable and convergent. A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.

KeywordConvergence Exponential-sum-approximation Method Fast Algorithm Stability Time-fractional Sub-diffusion Equation Variable-order Caputo Fractional Derivative
DOI10.4208/NMTMA.OA-2021-0148
URLView the original
Indexed BySCIE
Language英語English
WOS Research AreaMathematics
WOS SubjectMathematics, Applied ; Mathematics
WOS IDWOS:000734870800001
Scopus ID2-s2.0-85125723826
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Cited Times [WOS]:1   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
Corresponding AuthorSun, Hai Wei
AffiliationDepartment of Mathematics, University of Macau, Macao
First Author AffilicationUniversity of Macau
Corresponding Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Zhang, Jia Li,Fang, Zhi Wei,Sun, Hai Wei. Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations[J]. Numerical Mathematics,2022,15(1):200-226.
APA Zhang, Jia Li,Fang, Zhi Wei,&Sun, Hai Wei.(2022).Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations.Numerical Mathematics,15(1),200-226.
MLA Zhang, Jia Li,et al."Fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations".Numerical Mathematics 15.1(2022):200-226.
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