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

Bueno-Orovio, Alfonso, Kay, David, & Burrage, Kevin (2014) Fourier spectral methods for fractional-in-space reaction-diffusion equations. Bit (Lisse): numerical mathematics, 54(4), pp. 937-954.

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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 ofRn. 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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39 citations in Scopus
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17 citations in Web of Science®

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ID Code: 88502
Item Type: Journal Article
Refereed: Yes
Keywords: Fractional calculus, Fractional laplacian, Spectral methods, Reaction-diffusion equations
DOI: 10.1007/s10543-014-0484-2
ISSN: 0006-3835
Divisions: Current > Schools > School of Mathematical Sciences
Current > QUT Faculties and Divisions > Science & Engineering Faculty
Deposited On: 20 Oct 2015 05:40
Last Modified: 20 Oct 2015 05:40

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