Fractional diffusion equations have recently been used to model problems in physics, hydrology, biology and other areas of application. In this paper, we consider a two-dimensional time fractional diffusion equation (2D-TFDE) on a finite domain. An implicit difference approximation for the 2D-TFDE is presented. Stability and convergence of the method are discussed using mathematical induction. Finally, a numerical example is given. The numerical result is in excellent agreement with our theoretical analysis.
AgrawalO.P., Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain, J. Nonlinear Dynamics, 29 (2002), pp. 145–155.
2.
AnhV.V.LeonenkoN.N., Spectral analysis of fractional kinetic equations with random data, J.Stat.Pgys., Vol. 104 (2001), pp. 1349–1387.
3.
OrsingherEnzoBeghinLuisa, Time-fractional telegraph equations and telegraph processes with Brownian time, Probab. Theory Related Fields, Vol. 128, No. 1, (2004), pp. 141–160.
4.
FixG. J. and RoopJ. P., Least squares finite element solution of a fractional order two-point boundary value problem. Computers Math. Applic., Vol. 48 (2004), pp. 1017–1033.
5.
GinoaM.CerbelliS. and RomanH.E., Fractional diffusion equation and relaxation in complex viscoelastic materials, Physica A, 191 (1992), pp. 449–453.
6.
GorenfloR.LuchkoYu. and MainardiF., Wright function as scale-invariant solutions of the diffusion-wave equation, J. Comp. Appl. Math., 118 (2000), pp. 175–191.
7.
GorenfloR.MainardiF.MorettiD. and ParadisiP., Time Fractional Diffusion: A Discrete RandomWalk Approach [J], Nonlinear Dynamics, 29 (2002), pp. 129–143.
8.
HuangF. and LiuF., The time fractional diffusion and advection-dispersion equation, ANZIAM J., 46 (2005), pp. 1–14.
9.
LanglandsT.A.M. and HenryB.I., The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comp. Phys., 205 (2005), pp. 719–736.
10.
LiuF.AnhV.TurnerI. and ZhuangP., Time fractional advection dispersion equation, J. Appl. Math. Computing, Vol. 13 (2003), pp. 233–245.
11.
LiuV. AnhTurnerI., Numerical solution of space fractional Fokker-Planck equation, J. Comp. and APPL. Math., 166 (2004), pp. 209–219.
12.
LiuF.AnhV.TurnerI. and ZhuangP., Numerical simulation for solute transport in fractal porous media, ANZIAM J., 45(E), (2004), pp. 461–473.
13.
LiuF.ShenS.AnhV. and TurnerI., Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J., 46(E), (2005), pp. 488–504.
14.
LuisaB. and EnzoO., The telegraph processes stopped at stable-distributed times and its connection with the fractional telegraph equation, Fract. Calc. Appl. Anal., Vol. 6, No.2, (2003), pp. 187–204.
15.
MeerschaertM., and TadjeranC., Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Num. Math. (2005) in press.
16.
MeerschaertM. and TadjeranC., Finite difference approximations for fractional advection-dispersion flow equations, J. Comp. and Appl. Math. (2005) in press.
17.
MeerschaertM., and TadjeranC., Finite difference methods for two-dimensional fractional disperse equations, J. Comp. Phys. (2005) in press.
18.
MetzlerR. and KlafterJ., The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), pp. 1–77.
SchneiderW.R. and WyssW., Fractional diffusion and wave equations, J. Math. Phys., 30 (1989), pp. 134–144.
21.
WyssW., The fractional diffusion equation, J. Math. Phys., 27 (1986), pp. 2782–2785.
22.
YusteS.B. and AcedoL., An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., Vol. 42, No. 5, (2005), pp. 1862–1874.
