## Implicit difference approximation for the time fractional diffusion equation.(English)Zbl 1140.65094

Author’s summary: We consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $$0<\alpha<1$$). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $$O(\tau+h^2)$$, where $$\tau$$ and $$h$$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

### MSC:

 65R20 Numerical methods for integral equations 45J05 Integro-ordinary differential equations 26A33 Fractional derivatives and integrals 35K05 Heat equation 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Full Text:

### References:

 [1] O. P. Agrawal,Solution for a Fractional Diffusion-Wave Equation Defined in a Bounded Domain, J. Nonlinear Dynamics29 (2002), 145–155. · Zbl 1009.65085 [2] V. V. Anh and N. N. Leonenko,Spectral analysis of fractional kinetic equations with random data, J. Stat. Pgys.104 (2001), 1349–1387. · Zbl 1034.82044 [3] Orsingher, Enzo, Beghin, Luisa,Time-fractional telegraph equations and telegraph processes with Brownian time, Probab. Theory Related Fields128(1) (2004), 141–160. · Zbl 1049.60062 [4] G. J. Fix and J. P. Roop,Least squares finite element solution of a fractional order two-point boundary value problem, Computers Math. Applic.48 (2004), 1017–1033. · Zbl 1069.65094 [5] R. Gorenflo, A. Iskenderov and Yu. Luchko,Maping between solusions of fractional diffusion-wave equations, Fract. Calculus and Appl. Math.3 (2000), 75–86. · Zbl 1033.35161 [6] R. Gorenflo, Yu. Luchko and F. Mainardi,Wright function as scale-invariant solutions of the diffusion-wave equation, J. Comp. Appl. Math.118 (2000), 175–191. · Zbl 0973.35012 [7] R. Gorenflo, F. Mainardi, D. Moretti and P. Paradisi,Time Fractional Diffusion: A Discrete Random Walk Approach [J], Nonlinear Dynamics29 (2002), 129–143. · Zbl 1009.82016 [8] F. Huang and F. Liu,The time fractional diffusion and advection-dispersion equation, ANZIAM J.46 (2005), 1–14. · Zbl 1072.35218 [9] Liu, V. Anh, I. Turner,Numerical solution of space fractional Fokker-Planck equation J. Comp. and Appl. Math.166 (2004), 209–219. · Zbl 1036.82019 [10] F. Liu, V. Anh, I. Turner and P. Zhuang,Time fractional advection dispersion equation, J. Appl. Math. & Computing13 (2003), 233–245. · Zbl 1068.26006 [11] 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. · Zbl 1123.76363 [12] F. Liu, S. Shen, V. Anh and I. Turner,Analysis of a discrete non-Markovian random walk approximation for the time fractional diffusion equation, ANZIAM J.46(E) (2005), 488–504. · Zbl 1082.60511 [13] B. Luisa and O. Enzo,The telegraph processes stopped at stable-distributed times and its connection with the fractional telegraph equation, Fract. Calc. Appl. Anal.6(2) (2003), 187–204. · Zbl 1083.60039 [14] F. Mainardi,The fundamental solutions for the fractional diffusiona-wave equation, Appl. Math.9(6) (1996), 23–28. · Zbl 0879.35036 [15] M. Meerschaert and C. Tadjeran,Finite difference approximations for two-sided spacefractional partial differential equations, (2005), to appear. · Zbl 1086.65087 [16] M. Meerschaert and C. Tadjeran,Finite difference approximations for fractional advection-dispersion flow equations, J. Comp. and Appl. Math. (2005), (in press). · Zbl 1126.76346 [17] I. Podlubny,Fractional Differential Equations, Academic Press, 1999. · Zbl 0924.34008 [18] W. R. Schneider and W. Wyss,Fractional diffusion and wave equations, J. Math. Phys.30 (1989), 134–144. · Zbl 0692.45004 [19] W. Wyss,The fractional diffusion equation, J. Math. Phys.27 (1986), 2782–2785. · Zbl 0632.35031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.