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Mathematical Problems in Engineering
Volume 2015, Article ID 258265, 13 pages
http://dx.doi.org/10.1155/2015/258265
Review Article

Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

1College of Aerospace Science and Engineering, National University of Defense Technology, Changsha 410073, China
2Science and Technology on Space Physics Laboratory, Beijing 100076, China
3School of Computer Science, National University of Defense Technology, Changsha 410073, China
4Department of Engineering Science, University of Oxford, Oxford OX2 0ES, UK

Received 13 June 2014; Revised 6 August 2014; Accepted 9 September 2014

Academic Editor: Guido Maione

Copyright © 2015 Chunye Gong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.