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Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian
Gu,Xian Ming1,2,3; Sun,Hai Wei2; Zhang,Yanzhi4; Zhao,Yong Liang5
2021-01-15
Source PublicationMathematical Methods in the Applied Sciences
ISSN0170-4214
Volume44Issue:1Pages:441-463
Abstract

In this paper, we develop two fast implicit difference schemes for solving a class of variable-coefficient time–space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded L1 formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the M-matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum-of-exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from (Formula presented.) by a direct solver to (Formula presented.) per preconditioned Krylov subspace iteration and a memory requirement from (MN) to (NN), where N and (N ≪) M are the number of spatial and temporal grid nodes, respectively. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods.

KeywordCaputo Derivative Circulant Preconditioner Fractional Diffusion Equations Integral Fractional Laplacian Krylov Subspace Solvers
DOI10.1002/mma.6746
URLView the original
Indexed BySCIE
Language英語English
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000553229700001
PublisherWILEY, 111 RIVER ST, HOBOKEN 07030-5774, NJ USA
Scopus ID2-s2.0-85088695939
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Cited Times [WOS]:2   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionUniversity of Macau
Corresponding AuthorZhao,Yong Liang
Affiliation1.School of Economic Mathematics/Institute of Mathematics,Southwestern University of Finance and Economics,Chengdu,611130,China
2.Department of Mathematics,University of Macau,Taipa,Avenida da Universidade,Macao
3.Bernoulli Institute for Mathematics,Computer Science and Artificial Intelligence,University of Groningen,Groningen,Nijenborgh 9, P.O. Box 407,9700 AK,Netherlands
4.Department of Mathematics and Statistics,Missouri University of Science and Technology,Rolla,65409-0020,United States
5.School of Mathematical Sciences,University of Electronic Science and Technology of China,Chengdu,611731,China
First Author AffilicationUniversity of Macau
Recommended Citation
GB/T 7714
Gu,Xian Ming,Sun,Hai Wei,Zhang,Yanzhi,et al. Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian[J]. Mathematical Methods in the Applied Sciences,2021,44(1):441-463.
APA Gu,Xian Ming,Sun,Hai Wei,Zhang,Yanzhi,&Zhao,Yong Liang.(2021).Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian.Mathematical Methods in the Applied Sciences,44(1),441-463.
MLA Gu,Xian Ming,et al."Fast implicit difference schemes for time-space fractional diffusion equations with the integral fractional Laplacian".Mathematical Methods in the Applied Sciences 44.1(2021):441-463.
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