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Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey
Joint Authors
Liu, Jie
Tang, Guojian
Bao, Weimin
Gong, Chunye
Jiang, Yuewen
Source
Mathematical Problems in Engineering
Issue
Vol. 2015, Issue 2015 (31 Dec. 2015), pp.1-13, 13 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2015-03-26
Country of Publication
Egypt
No. of Pages
13
Main Subjects
Abstract EN
We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science.
The computational complexities of time fractional, space fractional, and space-time fractional equations are O(N2M), O(NM2), and O(NM(M + N)) compared with O(MN) for the classical partial differential equations with finite difference methods, where M, N are the number of space grid points and time steps.
The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator), short memory principle, fast Fourier transform (FFT) based solutions, alternating direction implicit method, multigrid method, and preconditioner technology.
The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed.
The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods.
Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.
American Psychological Association (APA)
Gong, Chunye& Bao, Weimin& Tang, Guojian& Jiang, Yuewen& Liu, Jie. 2015. Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey. Mathematical Problems in Engineering،Vol. 2015, no. 2015, pp.1-13.
https://search.emarefa.net/detail/BIM-1073336
Modern Language Association (MLA)
Gong, Chunye…[et al.]. Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey. Mathematical Problems in Engineering No. 2015 (2015), pp.1-13.
https://search.emarefa.net/detail/BIM-1073336
American Medical Association (AMA)
Gong, Chunye& Bao, Weimin& Tang, Guojian& Jiang, Yuewen& Liu, Jie. Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey. Mathematical Problems in Engineering. 2015. Vol. 2015, no. 2015, pp.1-13.
https://search.emarefa.net/detail/BIM-1073336
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1073336