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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References
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)
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)
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)
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)
Benson, D.A., Wheatcraft, S., Meerschaert, M.M.: Application of a fractional advection-dispersion equation. Water Resour. Res. 36, 1403–1412 (2000)
Briggs, W.L., Henson, V.E.: The DFT: an owner’s manual for the discrete Fourier transform. SIAM, Philadelphia (2000)
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)
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)
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)
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)
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)
Engler, H.: On the speed of spread for fractional reaction-diffusion equations. Int. J. Diff. Eqn. 315, 421 (2010)
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)
FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membranes. Biophys. J. 1, 445–466 (1961)
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)
Gray, P., Scott, S.K.: Sustained oscillations and other exotic patterns of behavior in isothermal reactions. J. Phys. Chem. 89, 22–32 (1985)
Hanert, E.: A comparison of three Eulerian numerical methods for fractional-order transport models. Environ. Fluid Mech. 10, 7–20 (2010)
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)
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)
Khader, M.M.: On the numerical solutions for the fractional diffusion equation. Commun. Nonlinear Sci. Numer. Simulat. 16, 2535–2542 (2010)
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
Lefèvre, J., Mangin, J.F.: A reaction-diffusion model of human brain development. PLoS Comput. Biol. 6, e1000749 (2010)
Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)
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)
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)
Lui, S.H.: Spectral domain embedding for elliptic PDEs in complex domains. J. Comput. Appl. Math. 225, 541–557 (2009)
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)
Meerschaert, M.M., Benson, D.A., Wheatcraft, S.W.: Subordinated advection-dispersion equation for contaminant transport. Water Resour. Res. 37, 1543–1550 (2001)
Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. App. Num. Math. 56, 80–90 (2006)
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)
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)
Nagumo, J., Animoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. Inst. Radio Eng. 50, 2061–2070 (1962)
Pang, H.K., Sun, H.W.: Multigrid method for fractional diffusion. J. Comp. Phys. 231, 693–703 (2012)
Pearson, J.E.: Complex patterns in a simple system. Science 261, 189–192 (1993)
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)
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)
Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous time finance. Phys. A 284, 376–384 (2000)
Trefethen, L.N.: Spectral methods in Matlab. SIAM, Philadelphia (2000)
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)
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)
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)
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)
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)
Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36, A40–A62 (2014)
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)
Zhou, K., Du, Q.: Mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions. SIAM J. Numer. Anal. 48, 1759–1780 (2010)
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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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DOI: https://doi.org/10.1007/s10543-014-0484-2