The space-time fractional advection dispersion equations are linear partial pseudo-differential equation s 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. I n 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 Fourier laplace 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 .
KeywordsAdvection-Dispersion ProcessesGrünwald-Letnikov SchemeExplicit Difference SchemesCaputo Time-Fractional DerivativeInverse Riesz PotentialRandom Walk with and without a MemoryConvergence in DistributionsFourier-Laplace Domain
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