Journal
CHAOS SOLITONS & FRACTALS
Volume 93, Issue -, Pages 89-98Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.chaos.2016.10.005
Keywords
Asymptotically stable; Exponential time differencing methods; Fractional reaction-diffusion; Pattern formation; Nonlinear time-dependent PDEs; Stability analysis; Super-diffusion
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The aim of this paper is to examine pattern formation in the sub- and super-diffusive scenarios and compare it with that of classical or standard diffusive processes in two-component fractional reaction-diffusion systems that modeled a predator-prey dynamics. The focus of the work concentrates on the use of two separate mathematical techniques, we formulate a Fourier spectral discretization method as an efficient alternative technique to solve fractional reaction-diffusion problems in higher-dimensional space, and later advance the resulting systems of ODEs in time with the adaptive exponential time-differencing solver. Obviously, the fractional Fourier approach is able to achieve spectral convergence up to machine precision regardless of the fractional order a, owing to the fact that our approach is able to give full diagonal representation of the fractional operator. The complexity of the dynamics in this system is theoretically discussed and graphically displayed with some examples and numerical simulations in one, two and three dimensions. (C) 2016 Elsevier Ltd. All rights reserved.
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