Numerical treatment for the fractional Fokker-Planck equation

We consider a space-time fractional Fokker--Planck equation on a finite domain. The space-time fractional Fokker--Plank equation is obtained from the general Fokker--Planck equation by replacing the first order time derivative by the Caputo fractional derivative, the second order space derivative by the left and right Riemann--Liouville fractional derivatives. We propose a computationally effective implicit numerical method to solve this equation. Stability and convergence of the numerical method are discussed. We prove that the implicit numerical method is unconditionally stable, and convergent. The error estimate is also given. Numerical result is in good agreement with theoretical analysis. References D. A. Benson, S. W. Wheatcraft, M.M Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res. , 36 (6) (2000) 1403--1412. doi:10.1029/2000WR900031 J. S. Ervin, J. P. Roop, Variational solution of fractional advection dispersion equations on bounded domains in $R^d$, Numer. Meth. P. D. E. , 23 (2) (2007) 256--281. doi:10.1002/num.20169 F. Huang, F. Liu, The fundamental solution of the space-time fractional advection-dispersion equation, J. Appl. Math. Computing , 18 (2005) 233--245. http://jamc.net/contents/table_contents_view.php?idx=223 F. Liu, V. Anh, I. Turner and P. Zhuang, Time fractional advection dispersion equation, J. Appl. Math. Computing , 13 (2003) 233--245. http://jamc.net/contents/table_contents_view.php?idx=74 F. Liu, V.Anh, I. Turner, Numerical solution of space fractional Fokker--Planck equation, J. Comp. and Appl. Math. , 166 (2004) 209--219. doi:10.1016/j.cam.2003.09.028 F. Liu, V. Anh, I. Turner and P. Zhuang, Numerical simulation for solute transport in fractal porous media, ANZIAM J. , 45 (E) (2004) 461--473. http://anziamj.austms.org.au/V45/CTAC2003/Liuf Q. Liu, F. Liu, I. Turner and V. Anh, Approximation of the Levy--Feller advection-dispersion process by random walk and finite difference method, J. Phys. Comp. , 222 (2007) 57--70. doi:10.1016/j.jcp.2006.06.005 M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comp. and Appl. Math. , 172 (2004) 65--77. doi:10.1016/j.cam.2004.01.033 R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000) 1--77. doi:10.1016/S0370-1573(00)00070-3 K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: John Wiley, 1993. K. B. Oldham, J. Spanier, The fractional calculus , New York and London: Academic Press, 1974. I. Podlubny, Fractional Differential Equations , Academic, Press, New York, 1999. H. Risken, The Fokker--Planck Equations , Springer, Berlin, 1988. S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: theory and applications , USA: Gordon and Breach Science Publishers, 1993. V. V. Yanovsky, A. V. Chechkin, D. Schertzer and A. V. Tur, Levy anomalous diffusion and fractional Fokker-Planck equation, Physica A , 282 , (2000), 13--34. doi:10.1016/S0378-4371(99)00565-8 Q. Yu, F. Liu, V. Anh and I. Turner, Solving linear and nonlinear space-time fractional reaction-diffusion equations by a domian decomposition method, International J. for Numer. Meth. In Eng. , (2007), in press.