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A novel implicit finite difference method for the one-dimensional fractional percolation equation

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Abstract

In this paper, a novel implicit finite difference method for the one-dimensional fractional percolation equation is proposed. Consistency, stability and convergence of the method are established. Some numerical examples are given. The numerical results demonstrate the effectiveness of theoretical analysis.

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References

  1. Huang, A.X.: A new decomposition for solving percolation equations in porous media. In: 3rd. Int. Symp. on Aerothermodynamics of Internal Flows, Beijing, China, pp. 417–420 (1996)

  2. Thusyanthan, N.I., Madabhushi, S.P.G.: Scaling of seepage flow velocity in centrifuge models. CUED/D-SOILS/TR326 (2003)

  3. Petford, N., Koenders, M.A.: Seepage flow and consolidation in a deforming porous medium. Geophys. Res. Abstr. 5, 13329 (2003)

    Google Scholar 

  4. Chou, H., Lee, B., Chen, C.: The transient infiltration process for seepage flow from cracks. In: Advances in Subsurface Flow and Transport: Eastern and Western Approaches III (2006)

  5. He, J.-H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998)

    Article  MATH  Google Scholar 

  6. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  7. Podlubny, I.: Fractional Differential Equations. Academic (1999)

  8. Samko, S., Kilbas, A., Marichev, O.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach (1993)

  9. Huang, F., Liu, F.: The fundamental solution of the space-time fractional advection-dispersion equation. J. Appl. Math. Comput. 19, 233–245 (2005)

    Article  Google Scholar 

  10. Liu, F., Anh, V., Turner, I., Zhuang, P.: Time fractional advection dispersion equation. J. Appl. Math. Comput. 13, 233–245 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker-Planck equation. J. Comp. Appl. Math. 166, 209–219 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, F., Anh, V., Turner, I., Zhuang, P.: Numerical simulation for solute transport in fractal porous media. ANZIAM J. 45(E), 461–473 (2004)

    MathSciNet  Google Scholar 

  13. Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. J. Appl. Math. Comput. 191, 12–21 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu, Q., Liu, F., Turner, I., Anh, V.: Approximation of the Lévy–Feller advection-dispersion process by random walk and finite difference method. J. Comput. Phys. 222, 57–70 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. J. Appl. Numer. Math. 56, 80–90 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fix, G.J., Roop, J.P.: Least squares finite element solution of a fractional order two-point boundary value problem. J. Comput. Math. Appl. 48, 1017–1033 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ervin, V.S., Roop, J.P.: Variational solution of fractional advection dispersion equations on bounded domains in R d. Numer. Methods for Partial Differ. Equ. 22, 558–576 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Momani, S., Odibat, Z.: Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method. J. Appl. Math. Comput. 177, 488–494 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Varga, R.: Matrix Iterative Analysis. Prentice-Hall (1962)

  21. Isaacson, E., Keller, H.B.: Analysis of Numerical Methods. Wiley, New York (1966)

    MATH  Google Scholar 

  22. Meerschaert, M.M., Scheffler, H.P., Tadjeran, C.: Finite difference methods for two-dimensional fractional dispersion equation. J. Comput. Phys. 211, 249–261 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Chen, S., Liu, F. & Anh, V. A novel implicit finite difference method for the one-dimensional fractional percolation equation. Numer Algor 56, 517–535 (2011). https://doi.org/10.1007/s11075-010-9402-0

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  • DOI: https://doi.org/10.1007/s11075-010-9402-0

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