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 .