An Efficient Iteration Method for Toeplitz-Plus-Band Triangular Systems Generated from Fractional Ordinary Differential Equation

Gong, Chunye; Bao, Weimin; Tang, Guojian; Min, Changwan; Liu, Jie

doi:10.1155/2014/194249

Cited by 3 publications

(2 citation statements)

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“…The interesting technologies are the adjustable matrix bandwidth and solving fractional ordinary differential equations with iteration method. The experimental results show that the presented efficient iteration method is 4.25 times faster than the regular solution [10].…”

Section: Fft Based Solutionmentioning

confidence: 98%

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

Gong

Bao

Tang

et al. 2015

Mathematical Problems in Engineering

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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 areO(N2M),O(NM2), andO(NM(M+N)) compared withO(MN) for the classical partial differential equations with finite difference methods, whereM,Nare 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.

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Section: Fft Based Solutionmentioning

confidence: 98%

“…FDEs may be divided into two fundamental types: time fractional differential equations and space fractional differential equations. For the fractional ordinary equations and fractional order control systems are also studied [9,10]. The stability of fractional order control systems attracts many attentions [11,12].…”

Section: Introductionmentioning

confidence: 99%