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R. Gorenflo and F. Mainardi, “Fractional Calculus: Integral and Differential Equations of Fractional Order,” In: A. Carpinteri and F. Mainardi, Eds., Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien and New York, 1997, pp. 223-276. http://www.fracalmo.org

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 .