BIT Numerical Mathematics

, Volume 54, Issue 4, pp 937–954 | Cite as

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

  • Alfonso Bueno-OrovioEmail author
  • David Kay
  • Kevin Burrage


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.


Fractional calculus Fractional laplacian Spectral methods  Reaction-diffusion equations 

Mathematics Subject Classification (2010)

35R11 65M70 34L10 65T50 35K57 


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Alfonso Bueno-Orovio
    • 1
    • 2
    Email author
  • David Kay
    • 2
  • Kevin Burrage
    • 2
    • 3
  1. 1.Oxford Centre for Collaborative Applied MathematicsUniversity of OxfordOxford UK
  2. 2.Department of Computer ScienceUniversity of OxfordOxford UK
  3. 3.School of Mathematical SciencesQueensland University of TechnologyBrisbane Australia

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