Implicit difference approximation of the Galilei invariant fractional advection diffusion equation

Abstract

A Galilei invariant fractional advection diffusion equation with initial-boundary conditions is considered. An implicit difference approximation for solving the Galilei invariant fractional advection diffusion equation is presented. We introduce a new Fourier method for analyzing the stability and convergence of the implicit difference approximation. Finally, some numerical examples are given. The numerical results are in good agreement with our theoretical analysis. This method and supporting theoretical techniques can also be extended to a larger class of fractional integro-differential equations. References T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comp. Phys., 205 (2005), 719--736. doi:10.1016/j.jcp.2004.11.025 F. Liu, V. Anh, I. Turner, Numerical Solution of the Space Fractional Fokker--Planck Equation, J. Comp. Appl. Math., 166 (2004), 209--219. doi:10.1016/j.cam.2003.09.028 Ch. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), 704--719. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Reports, 339,(2000), 1--77. doi:10.1016/S0370-1573(00)00070-3 M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comp. Appl. Math., 172 (2004), 65--77. doi:10.1016/j.cam.2004.01.033 I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. http://people.tuke.sk/igor.podlubny/fde.html I. M. Sokolov, J. Klafter and A. Blumen, Fractional Kinetics, Physics Today, 55(11) (2002) 48--54. S. B. Yuste and L. Acedo, An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42(5) (2005), 1862--1874. P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comp., 22(3) (2006) 87--99. http://jamc.net/contents/table_contents_view.php?idx=580