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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<α<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(τ+h 2 ), where τ and h are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.
65R20Integral equations (numerical methods)
45J05Integro-ordinary differential equations
26A33Fractional derivatives and integrals (real functions)
35K05Heat equation
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
[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 · doi:10.1023/A:1016539022492
[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 · doi:10.1023/A:1010474332598
[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 · doi:10.1007/s00440-003-0309-8
[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 · doi:10.1016/j.camwa.2004.10.003
[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.
[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 · doi:10.1016/S0377-0427(00)00288-0
[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 · doi:10.1023/A:1016547232119
[8]F. Huang and F. Liu,The time fractional diffusion and advection-dispersion equation, ANZIAM J.46 (2005), 1–14. · Zbl 1072.35218 · doi:10.1017/S1446181100008282
[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 · doi:10.1016/j.cam.2003.09.028
[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 · doi:10.1007/BF02936089
[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.
[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.
[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.
[14]F. Mainardi,The fundamental solutions for the fractional diffusiona-wave equation, Appl. Math.9(6) (1996), 23–28.
[15]M. Meerschaert and C. Tadjeran,Finite difference approximations for two-sided spacefractional partial differential equations, (2005), to appear.
[16]M. Meerschaert and C. Tadjeran,Finite difference approximations for fractional advection-dispersion flow equations, J. Comp. and Appl. Math. (2005), (in press).
[17]I. Podlubny,Fractional Differential Equations, Academic Press, 1999.
[18]W. R. Schneider and W. Wyss,Fractional diffusion and wave equations, J. Math. Phys.30 (1989), 134–144. · Zbl 0692.45004 · doi:10.1063/1.528578
[19]W. Wyss,The fractional diffusion equation, J. Math. Phys.27 (1986), 2782–2785. · Zbl 0632.35031 · doi:10.1063/1.527251