Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by a Caputo fractional derivative, and the second order space derivative by a symmetric fractional derivative. First, a method of separating variables expresses the analytical solution of the TSS-FDE in terms of the Mittag--Leffler function. Second, we propose two numerical methods to approximate the Caputo time fractional derivative: the finite difference method; and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims. References D.A. Benson, S.W. Wheatcraft and M.M. Meerschaert. Application of a fractional advection-dispersion equation. Water Resources Research , 36(6):1403--1412, 2000. doi:10.1029/2000WR900031 R. Gorenflo and F. Mainardi. Random walk models for space-fractional diffusion processes. Fractional Calculus and Applied Analysis , 1(2):167--191, 1998. http://www.ge. infm.it/ecph/papers/mainardipapers/mainardipap.html . R. Hilfer. Applications of Fractional Calculus in Physics . World Scientific, Singapore, 2000. M. Ilic, F. Liu, I. Turner, and V. Anh. Numerical approximation of a fractional-in-space diffusion equation (II)---with nonhomogeneous boundary conditions. Fractional Calculus and Applied Analysis , 9(4):333--349, 2006. Y. Lin and C. Xu. Finite difference/spectral approximations for the time-fractional diffusion equation. Journal of Computational Physics , 225(2):1533--1552, 2007. doi:10.1016/j.jcp. 2007.02. 001 R. Metzler and J. Klafter. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Physics Reports , 339:1--77, 2000. http://www.tau. ac.il/ klafter1/258.pdf M.M. Meerschaert, D.A. Benson, H.P. Scheffler, and B. Baeumer. Stochastic solution of space-time fractional diffusion equations. Physical Review E , 65:041103, 2002. http://link.aps. org/abstract/PRE/v65/e041103 M.M. Meerschaert and E. Scalas. Coupled continuous time random walks in finance. Physica A , 370:114-118, 2006. http://arxiv.org/abs/physics/0608281 M.M. Meerschaert, H.P. Scheffler, and C. Tadjeran. Finite difference methods for two-dimensional fractional dispersion equation. Journal of Computational Physics , 211(1):249-261, 2006. doi:10.1016/j.jcp. 2005.05. 017 M.M. Meerschaert and C. Tadjeran. Finite difference approximations for two-sided space-fractional partial differential equations. Applied Numerical Mathematics , 56(1):80-90, 2006. doi:10.1016/j.apnum. 2005.02. 008 M.M. Meerschaert and C. Tadjeran. Finite difference approximations for fractional advection-dispersion flow equations. Journal of Computational and Applied Mathematics ,172(1):65-77, 2004 doi:10.1016/j.cam. 2004.01. 033 I. Podlubny. Fractional Differential Equations . Academic Press, New York, 1999. S.G. Samko, A.A. Kilbas, and O.I. Marichev. Fractional Integrals and Derivatives: Theory and Applications . Gordon and Breach Science Publishers, Amsterdam, 1993. Q. Yang, F. Liu, and I. Turner. Computationally efficient numerical methods for the fractional partial differential equations with the Riesz space fractional derivatives. Appl. Math. Model. , Submitted, 2008. G.M. Zaslavsky. Fractional kinetic equation for Hamiltonian chaos. Physica D: Nonlinear Phenomena , 76:110--122, 1994. G.M. Zaslavsky. Chaos, fractional kinetics, and anomalous transport. Physics Reports , 371(6):461--580, 2002.