Abstract.
In this paper, an implicit numerical method for the three-dimensional fractional percolation equation without the assumption of continued and rigid body motion is proposed. Consistency, stability and convergence of the implicit finite difference method are established. Finally, some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis.
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References
A.X. Huang, et al., 3rd. Int. Symp, On Aerothermodynamics of Internal Flows (Beijing, China 1996), p. 417
N.I. Thusyanthan, S.P.G. Madabhushi, Scaling of seepage flow velocity in centrifuge models, CUED/D-SOILS/TR326 (2003)
N. Petford, M.A. Koenders, Geophys. Res. Abstr. 5, 13329 (2003)
H. Chou, B. Lee, C. Chen, Adv. subsurface flow and transport: Eastern and Western Approaches III (2006)
D. Swartzendruber, Soil Sci. 93, 22 (1962)
l. Schreyer Bennethum, M.A. Murad, J.H. Cushman, Comp. Geotechn. 20, 245 (1997)
A.K. Das, J. Canad. Petrol. Technol. 36, (1997)
He Ji-Huan, J. Computer Meth. Appl. Mech. Eng. 167, 57 (1998)
K. Miller, B. Ross, An introduction to the Fractional Calculus and Fractional differential Equations (Wiley, New York, 1993)
I. Podlubny, Fractional Differential Equations (Academic Press, 1999)
S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon Breach, 1993)
F. Huang, F. Liu, J. Appl. Math. Computing 19, 233 (2005)
F. Liu, V. Anh, I. Turner, P. Zhuang, J. Appl. Math. Computing 13, 233 (2003)
F. Liu, V. Anh, I. Turner, J. Comp. APPl. Math. 166, 209 (2004)
F. Liu, V. Anh, I. Turner, P. Zhuang, ANZIAM J. 45, 461 (2004)
Q. Liu, F. Liu, I. Turner, V. Anh, J. Phys. Comp. 22, 57 (2007)
M.M. Meerschaert, C. Tadjeran, J. Comp. Appl. Math. 172, 65 (2004)
M.M. Meerschaert, C. Tadjeran, J. Appl. Numer. Math. 56, 80 (2006)
S. Chen, F. Liu, I. Turner, M. Cox, Appl. Math. Comp. (2010) in press.
G.J. Fix, J.P. Roop, J. Comput. Math. Appl. 48, 1017 (2004)
V.S. Ervin, J.P. Roop, Numer. Meth. P.D.E. 22, 558 (2006)
S. Momani, Z. Odibat, J. Appl. Math. Computing. 177, 488 (2006)
R. Varga, Matrix Itertive Anslysis (Prentice-Hall, 1962)
R.D. Richtmyer, K.W. Morton, Difference methods for initial-value problems (Krieger Publishing, Malabar, FLorida, 1994)
M.M. Meerschaert, H.P. Scheffler, C. Tadjeran, J. Comput. Phys. 211, 249 (2006)
S. Chen, F. Liu, I. Turner, V. Anh, Numerical Algorithms (2010) (accepted)
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Chen, S. A novel implicit finite difference method for the three-dimensional fractional percolation equation. Eur. Phys. J. Spec. Top. 193, 205–220 (2011). https://doi.org/10.1140/epjst/e2011-01392-4
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DOI: https://doi.org/10.1140/epjst/e2011-01392-4