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    Fourier spectral methods for fractional-in-space reaction-diffusion equations

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    Type
    Article
    Authors
    Bueno-Orovio, Alfonso
    Kay, David
    Burrage, Kevin
    KAUST Grant Number
    KUK-C1-013-04
    Date
    2014-04-01
    Online Publication Date
    2014-04-01
    Print Publication Date
    2014-12
    Permanent link to this record
    http://hdl.handle.net/10754/598359
    
    Metadata
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    Abstract
    © 2014, Springer Science+Business Media Dordrecht. Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains of ℝ. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.
    Citation
    Bueno-Orovio A, Kay D, Burrage K (2014) Fourier spectral methods for fractional-in-space reaction-diffusion equations. Bit Numer Math 54: 937–954. Available: http://dx.doi.org/10.1007/s10543-014-0484-2.
    Sponsors
    This publication is based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).
    Publisher
    Springer Nature
    Journal
    BIT Numerical Mathematics
    DOI
    10.1007/s10543-014-0484-2
    ae974a485f413a2113503eed53cd6c53
    10.1007/s10543-014-0484-2
    Scopus Count
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