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

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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 .

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

*Applied Mathematics*,

**4**, 1427-1440. doi: 10.4236/am.2013.410193.

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