[1]Baeumer, B., Meerschaert, M.M., Benson, D.A. and Wheatcraft, S.W., Subordinated advection-dispersion equation for contaminant transport, Water Resources Res. 37, 1543–1550 (2001).

[2]Barkai, E., Metzler, R. and Klafter, J., From continuous time random walks to the fractional Fokker-Planck equation, Phys. Rev. E 61, 132–138 (2000).

[3]Benson, D., Wheatcraft, S. and Meerschaert, M.M., The fractional-order governing equation of Levy motions, Water Resources Res. 36, 1413–1424 (2000).

[4]Blumen, A., Zumofen, G. and Klafter, J., Transport aspect in anomalous diffusion: Levy walks, Phys. Rev. A 40, 3964 (1989).

[5]Chaves, A., Fractional diffusion equation to describe Levy flights, Phys. Lett. A 239 13–16, (1998).

[6]Chen, M. and Deng, W., Fourth order difference approximations for space Riemann-Liouville derivatives based on weighted and shifted Lubich difference operators, Commun. Comp. Phys. 16, 516–540 (2014).

[7]Concezzi, M. and Spigler, R., Numerical solution of two-dimensional FDE by a high-order ADI method, Comm. Appl. Ind. Math. 3, No. 2, e-421 (2012).

[8]Cui, M., Compact finite difference method for the fractional diffusion equation, J. Comp. Phys. 228, 7792–7804 (2009).

[9]Deng, Z., Singh, V.P. and Bengtsson, L., Numerical solution of fractional advection-dispersion equation, J. Hydraulic Eng. 130, 422–431 (2004).

[10]Ervin, V.J., Heuer, N. and Roop, J.P., Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Num. Anal. 45, 572–591 (2007).

[11]Ervin, V.J. and Roop, J.P., Variational formulation for the stationary fractional advection dispersion equation, Num. Meth. Part. Diff. Eqns. 22, 558–576 (2005).

[12]Evans, D.J. and Abdullah, A.R.B., Group explicit methods for parabolic equations, Int. J. Comp. Math. 14, 73–105 (1983).

[13]Evans, D.J. and Abdullah, A.R.B., A new explicit methods for the diffusion-convection equation, Comp. & Math. with App. 11, 145–154 (1985).

[14]Ji, X. and Tang, H., High-Order accurate Runge-Kutta (local) discontinuous Galerkin methods for one- and two-dimensional fractional diffusion equations, Num. Math. Theor. Meth. App. 5, 333–358 (2012).

[15]Liu, F., Anh, V. and Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math. 166, 209–219 (2004).

[16]Hilfer, R., Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000).

[17]Isaacson, E. and Keller, H.B., Analysis of Numerical Methods, Wiley, New York (1966).

[18]Kirchner, J. W., Feng, X. and Neal, C., Fractal stream chemistry and its implications for contaminant transport in catchments, Nature 403, 524–526 (2000).

[19]Langlands, T.A.M. and Henry, B.I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comp. Phys. 205, 719–736 (2005).

[20]Li, X. and Xu, C., The existence and uniqueness of the week solution of the space-time fractional diffusion equation and a spectral method approximation, Comm. Comp. Phys. 8, 1016–1051 (2010).

[21]Lin, Y. and Xu, C., Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comp. Phys. 225, 1533–1552 (2007).

[22]Lin, R., Liu, F., Anh, V. and Turner, I., Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation, Appl. Math. Comp. 212, 435–445 (2009).

[23]Liu, F. and Anh, V., Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math. 166, 209–219 (2004).

[24]Lynch, V.E., Carreras, B.A., Del-Castillo-Negrete, D., Ferreria-Mejias, K.M. and Hicks, H.R., Numerical methods for the solution of partial differential equations of fractional order, J. Comp. Phys. 192, 406–421 (2003).

[25]Meerschaert, M.M., Scheffler, H.P. and Tadjeran, C., Finite difference methods for the two-dimensional fractional dispersion equation, J. Comp. Phys. 211, 249–261 (2006).

[26]Meerschaert, M.M. and Tadjeran, C., Finite difference approximations for fractional advection-dispersion flow equation, J. Comp. Appl. Math. 172, 65–77 (2004).

[27]Meerschaert, M. and Tadjeran, C., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Numer. Math. 56, 80–90 (2006).

[28]Podlubny, I., Fractional Differential Equations, Academic Press, New York (1999).

[29]Raberto, M., Scalas, E. and Mainardi, F., Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314, 749–755 (2001).

[30]Sabatelli, L., Keating, S., Dudley, J. and Richmond, P., Waiting time distributions in financial markets, Eur. Phys. J. B 27, 273–275 (2002).

[31]Saulyev, V.K., Integration of Equations of Parabolic Type by the Method of Nets, Pergamon Press, New York (1964).

[32]Scalas, E., Gorenflo, R. and Mainardi, F., Fractional calculus and continuous-time finance, Phys. A 284, 376–384 (2000).

[33]Schumer, R., Benson, D.A., Meerschaert, M.M. and Baeumer, B., Multiscaling fractional advection-dispersion equations and their solutions, Water Resources Res. 39,1022–1032 (2003).

[34]Schumer, R., Benson, D.A., Meerschaert, M.M. and Wheatcraft, S.W., Eulerian derivation of the fractional advection-dispersion equation, J. Contaminant Hydrol. 48, 69–88 (2001).

[35]Sokolov, I.M., Klafter, J. and Blumen, A., Fractional kinetics, Physics Today 55, 28–53 (2002).

[36]Sousa, E., Finite difference approximates for a fractional advection diffusion problem, J. Comp. Phys. 228, 4038–4054 (2009).

[37]Su, L., Wang, W. and Yang, Z., Finite difference approximations for the fractional advection-diffusion equation, Phys. Lett. A 373, 4405–4408 (2009).

[38]Tadjeran, C., Meerschaert, M.M. and Scheffler, H.P., A second-order accurate numerical approximation for the fractional diffusion equation, J. Comp. Phys. 213, 205–213 (2006).

[39]Xu, M. and Tan, W., Theoretical analysis of the velocity field and vortex sheet of generalized second order fluid with fractional anomalous diffusion, Sci. China Ser. A: Math. 44, 1387–1399 (2001).

[40]Ye, X. and Xu, C., Spectral optimization methods for the time fractional diffusion inverse problem, Num. Math. Theor. Meth. Appl. 6, 499–519 (2013).

[41]Zaslavsky, G., Fractional kinetic equation for Hamiltonian Chaotic advection, tracer dynamics and turbulent dispersion, Physica D 76, 110–122 (1994).

[42]Zhang, B., Alternating difference block methods and their difference graphs, Science China Technological Sc. 41, 482–487(1998).

[43]Zhang, B., Difference graphs of block ADI method, SIAM J. Num. Anal. 38, 742–752 (2000).