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

E. A. Abdel-Rehim

Department of Mathematics and Computer Science, Faculty of Science, Suez Canal University, Ismailia, Egypt.

DOI: 10.4236/am.2013.410193                

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 .

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

Conflicts of Interest

The authors declare no conflicts of interest.

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