BIT Numerical Mathematics

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

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

Article

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.

Keywords

Fractional calculus Fractional laplacian Spectral methods  Reaction-diffusion equations 

Mathematics Subject Classification (2010)

35R11 65M70 34L10 65T50 35K57 

References

  1. 1.
    Adams, E.E., Gelhar, L.W.: Field study of dispersion in a heterogeneous aquifer: 2. Spatial moment analysis. Water Resour. Res. 28, 3293–3307 (1992)CrossRefGoogle Scholar
  2. 2.
    Alexandrescu, A., Bueno-Orovio, A., Salgueiro, J.R., Pérez-García, V.M.: Mapped Chebyshev pseudospectral method to study multiple scale phenomena. Comput. Phys. Commun. 180, 912–919 (2009)CrossRefMATHGoogle Scholar
  3. 3.
    Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. Mater. 27, 1085–1095 (1979)CrossRefGoogle Scholar
  4. 4.
    Becker-Kern, P., Meerschaert, M.M., Scheffler, H.P.: Limit theorem for continuous time random walks with two time scales. J. App. Prob. 41, 455–466 (2004)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Benson, D.A., Wheatcraft, S., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)CrossRefGoogle Scholar
  6. 6.
    Briggs, W.L., Henson, V.E.: The DFT: an owner’s manual for the discrete Fourier transform. SIAM, Philadelphia (2000)Google Scholar
  7. 7.
    Bueno-Orovio, A.: Fourier embedded domain methods: periodic and \({C}^\infty \) extension of a function defined on an irregular region to a rectangle via convolution with Gaussian kernels. App. Math. Comp. 183, 813–818 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bueno-Orovio, A., Cherry, E.M., Fenton, F.H.: Minimal model for human ventricular action potentials in tissue. J. Theor. Biol. 253, 554–560 (2008)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bueno-Orovio, A., Pérez-García, V.M.: Spectral smoothed boundary methods: the role of external boundary conditions. Numer. Meth. Part. Differ. Equ. 22, 435–448 (2006)CrossRefMATHGoogle Scholar
  10. 10.
    Bueno-Orovio, A., Pérez-García, V.M., Fenton, F.H.: Spectral methods for partial differential equations in irregular domains: the spectral smoothed boundary method. SIAM J. Sci. Comput. 28, 886–900 (2006)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Burrage, K., Hale, N., Kay, D.: An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput. 34, A2145–A2172 (2012)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Engler, H.: On the speed of spread for fractional reaction-diffusion equations. Int. J. Diff. Eqn. 315, 421 (2010)Google Scholar
  13. 13.
    Feng, W.M., Yu, P., Hu, S.Y., Liu, Z.K., Du, Q., Chen, L.Q.: Spectral implementation of an adaptive moving mesh method for phase-field equations. J. Comput. Phys. 220, 498–510 (2006)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membranes. Biophys. J. 1, 445–466 (1961)CrossRefGoogle Scholar
  15. 15.
    Gray, P., Scott, S.K.: Autocatalytic reactions in the isothermal, continuous stirred tank reactor. Isolas and other forms of multistability. Chem. Eng. Sci. 38, 29–43 (1983)CrossRefGoogle Scholar
  16. 16.
    Gray, P., Scott, S.K.: Sustained oscillations and other exotic patterns of behavior in isothermal reactions. J. Phys. Chem. 89, 22–32 (1985)CrossRefGoogle Scholar
  17. 17.
    Hanert, E.: A comparison of three Eulerian numerical methods for fractional-order transport models. Environ. Fluid Mech. 10, 7–20 (2010)Google Scholar
  18. 18.
    Ilić, M., Liu, F., Turner, I., Anh, V.: Numerical approximation of a fractional-in-space diffusion equation. I. Frac. Calc. App. Anal. 8, 323–341 (2005)MATHGoogle Scholar
  19. 19.
    Ilić, M., Turner, I.W.: Approximating functions of a large sparse positive definite matrix using a spectral splitting method. ANZIAM J. 46, C472–C487 (2005)Google Scholar
  20. 20.
    Khader, M.M.: On the numerical solutions for the fractional diffusion equation. Commun. Nonlinear Sci. Numer. Simulat. 16, 2535–2542 (2010)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Khader, M.M., Sweilam, N.H.: Approximate solutions for the fractional advection-dispersion equation using Legendre pseudo-spectral method. Comp. Appl. Math. doi:10.1007/s40314-013-0091-x
  22. 22.
    Lefèvre, J., Mangin, J.F.: A reaction-diffusion model of human brain development. PLoS Comput. Biol. 6, e1000749 (2010)CrossRefGoogle Scholar
  23. 23.
    Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Li, X., Xu, C.: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8, 1016–1051 (2010)MathSciNetGoogle Scholar
  25. 25.
    Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the finite difference method for the space-time fractional advection-diffusion equation. App. Math. Comp. 191, 12–20 (2007)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Lui, S.H.: Spectral domain embedding for elliptic PDEs in complex domains. J. Comput. Appl. Math. 225, 541–557 (2009)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Magin, R.L., Abdullah, O., Baleanu, D., Zhou, X.J.: Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation. J. Magn. Reson. 190, 255–270 (2008)CrossRefGoogle Scholar
  28. 28.
    Meerschaert, M.M., Benson, D.A., Wheatcraft, S.W.: Subordinated advection-dispersion equation for contaminant transport. Water Resour. Res. 37, 1543–1550 (2001)CrossRefGoogle Scholar
  29. 29.
    Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. App. Num. Math. 56, 80–90 (2006)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Mulholland, L.S., Huang, W.Z., Sloan, D.M.: Pseudospectral solution of near-singular problems using numerical coordinate transformations based on adaptivity. SIAM J. Sci. Comput. 19, 1261–1289 (1998)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Nagumo, J., Animoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. Inst. Radio Eng. 50, 2061–2070 (1962)Google Scholar
  33. 33.
    Pang, H.K., Sun, H.W.: Multigrid method for fractional diffusion. J. Comp. Phys. 231, 693–703 (2012)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Pearson, J.E.: Complex patterns in a simple system. Science 261, 189–192 (1993)CrossRefGoogle Scholar
  35. 35.
    Roop, J.: Computational aspects of FEM approximations of fractional advection dispersion equations on bounded domains on \({R}^2\). J. Comp. Appl. Math. 193, 243–268 (2005)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Sabetghadam, F., Sharafatmandjoor, S., Norouzi, F.: Fourier spectral embedded boundary solution of the Poisson’s and Laplace equations with Dirichlet boundary conditions. J. Comput. Phys. 228, 55–74 (2009)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous time finance. Phys. A 284, 376–384 (2000)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Trefethen, L.N.: Spectral methods in Matlab. SIAM, Philadelphia (2000)CrossRefMATHGoogle Scholar
  39. 39.
    Turner, I., Ilić, M., Perr, P.: The use of fractional-in-space diffusion equations for describing microscale diffusion in porous media. In: 11th International Drying Conference, Magdeburg, Germany (2010)Google Scholar
  40. 40.
    Wang, H., Wang, K.: An \(O(N \log ^2 N)\) alternating-direction finite difference method for two-dimensional fractional diffusion equations. J. Comput. Phys. 230, 7830–7839 (2011)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Wang, H., Wang, K., Sircar, T.: A direct \(O(N \log ^2 N)\) finite difference method for fractional diffusion equations. J. Comput. Phys. 229, 8095–8104 (2010)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. App. Num. Mod. 34, 200–218 (2010)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Yang, Q., Turner, I., Liu, F., Ilić, M.: Novel numerical methods for solving the time-space fractional diffusion equation in 2D. SIAM J. Sci. Comp. 33, 1159–1180 (2011)CrossRefMATHGoogle Scholar
  44. 44.
    Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36, A40–A62 (2014)CrossRefMATHMathSciNetGoogle Scholar
  45. 45.
    Zhang, Y., Benson, D.A., Reeves, D.M.: Time and space nonlocalities underlying fractional-derivative models: distinction and literature review of field applications. Adv. Water Res. 32, 561–581 (2009)CrossRefGoogle Scholar
  46. 46.
    Zhou, K., Du, Q.: Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48, 1759–1780 (2010)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Alfonso Bueno-Orovio
    • 1
    • 2
  • 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

Personalised recommendations