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

Bueno-Orovio, Alfonso; Kay, David; Burrage, Kevin

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

Type

Article

Authors

Bueno-Orovio, Alfonso
Kay, David
Burrage, Kevin

KAUST Grant Number

KUK-C1-013-04

Online Publication Date

2014-04-01

Print Publication Date

2014-12

Date

2014-04-01

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.

Acknowledgements

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