Fourier spectral method for higher order space fractional reaction–diffusion equations

•Fractional Fourier Transform method is proposed to solve fractional reaction – diffusion problems.•We investigate the linear stability analysis of the model, in our analysis results.•The dynamic richness of spatial pattern formations is explored in the sub-diffusive, diffusive and super-diffusive c...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation Vol. 40; pp. 112 - 128
Main Authors: Pindza, Edson, Owolabi, Kolade M.
Format: Journal Article
Language:English
Published: Elsevier B.V 01.11.2016
Subjects:
Derivatives
Evolution
Fourier analysis
Fourier transform
Fractional exponential integrators
Fractional reaction–diffusion system
Mathematical analysis
Mathematical models
Pattern formation
Reaction-diffusion equations
Spectral methods
Three dimensional
Turing instability
65L05
Pattern formation
35A05
93C10
Turing instability
Fractional reaction–diffusion system
65M06
Fractional exponential integrators
Fourier transform
35K57
ISSN:1007-5704, 1878-7274
Online Access:Get full text
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Summary:•Fractional Fourier Transform method is proposed to solve fractional reaction – diffusion problems.•We investigate the linear stability analysis of the model, in our analysis results.•The dynamic richness of spatial pattern formations is explored in the sub-diffusive, diffusive and super-diffusive cases.•Results show that the proposed method is reliable, efficient and achieves higher-order of accuracy. Evolution equations containing fractional derivatives can provide suitable mathematical models for describing important physical phenomena. In this paper, we propose a fast and accurate method for numerical solutions of space fractional reaction–diffusion equations. The proposed method is based on an exponential integrator scheme in time and the Fourier spectral method in space. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency, and a completely straightforward extension to high spatial dimensions. Although, in general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives, we introduce them to describe fractional hyper-diffusions in reaction diffusion. The scheme justified by a number of computational experiments, this includes two and three dimensional partial differential equations. Numerical experiments are provided to validate the effectiveness of the proposed approach.
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ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2016.04.020