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

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Abstract

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 \(\mathbb {R}^n\). 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.

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Authors and Affiliations

  1. Oxford Centre for Collaborative Applied Mathematics, University of Oxford, Oxford , OX1 3LB, UK

    Alfonso Bueno-Orovio

  2. Department of Computer Science, University of Oxford, Oxford , OX1 3QD, UK

    Alfonso Bueno-Orovio, David Kay & Kevin Burrage

  3. School of Mathematical Sciences, Queensland University of Technology, Brisbane , 4001, Australia

    Kevin Burrage

Authors
  1. Alfonso Bueno-Orovio
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  2. David Kay
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  3. Kevin Burrage
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Correspondence to Alfonso Bueno-Orovio.

Additional information

Communicated by Mechthild Thalhammer.

This publication is based on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).

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Bueno-Orovio, A., Kay, D. & Burrage, K. Fourier spectral methods for fractional-in-space reaction-diffusion equations. Bit Numer Math 54, 937–954 (2014). https://doi.org/10.1007/s10543-014-0484-2

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