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

Alfonso Bueno‐Orovio1, David Kay2, Kevin Burrage2
1Oxford Centre for Collaborative Applied Mathematics, University of Oxford, Oxford , OX1 3LB, UK
2Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK

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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 $${C}^\infty $$ C ∞ 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 $${R}^2$$ 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 $$O(N \log ^2 N)$$ 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 $$O(N \log ^2 N)$$ 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)