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Q. Liu, F. Liu, I. Turner and V. Anh, “Approximation of the Lévy-Feller Advection-Dispersion Process by Random Walk and Finite Difference Method,” Journal of Computational Physics, Vol. 222, No. 1, 2007, pp. 57-70. http://dx.doi.org/10.1016/j.jcp.2006.06.005

has been cited by the following article:

• TITLE: Explicit Approximation Solutions and Proof of Convergence of the Space-Time Fractional Advection Dispersion Equations

  AUTHORS: E. A. Abdel-Rehim

  KEYWORDS: Advection-Dispersion Processes; Grünwald-Letnikov Scheme; Explicit Difference Schemes; Caputo Time-Fractional Derivative; Inverse Riesz Potential; Random Walk with and without a Memory; Convergence in Distributions; Fourier-Laplace Domain

  JOURNAL NAME: Applied Mathematics, Vol.4 No.10, October 14, 2013

  ABSTRACT: The space-time fractional advection dispersion equations are linear partial pseudo-differential equations with spatial fractional derivatives in time and in space and are used to model transport at the earth surface. The time fractional order is denoted by β∈ and is devoted to the space fractional order. The time fractional advection dispersion equations describe particle motion with memory in time. Space-fractional advection dispersion equations arise when velocity variations are heavy-tailed and describe particle motion that accounts for variation in the flow field over entire system. In this paper, I focus on finding the precise explicit discrete approximate solutions to these models for some values of with , while the Cauchy case as and the classical case as with are studied separately. I compare the numerical results of these models for different values of and and for some other related changes. The approximate solutions of these models are also discussed as a random walk with or without a memory depending on the value of . Then I prove that the discrete solution in the Fourierlaplace space of theses models converges in distribution to the Fourier-Laplace transform of the corresponding fractional differential equations for all the fractional values of and .