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

• Published in: Mathematical problems in engineering 2015-01, Vol.2015 (2015), p.1-13
• Main Authors: Jiang, Yuewen, Tang, Guojian, Bao, Weimin, Gong, Chunye, Liu, Jie
• Format: Article
• Language: English
• Subjects: Alternating direction implicit methods; Approximation; Calculus; Code reuse; Computation; Computational efficiency; Computer memory; Computer science; Computer simulation; Computing costs; Derivatives; Differential equations; Fast Fourier transformations; Finite difference method; Finite volume method; Fourier transforms; Fractional calculus; Implicit methods; Iterative methods; Mathematical analysis; Mathematical models; Mathematical problems; Monte Carlo methods; Monte Carlo simulation; Multigrid methods; Numerical analysis; Operators (mathematics); Parallel processing; Partial differential equations; Semiconductor research; Velocity
• ISSN: 1024-123X; 1563-5147
• DOI: 10.1155/2015/258265

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.