View source for Fractional Advection Dispersion Equation

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The Fractional Advection Dispersion (fADE) was introduced <ref name="MBB1999"/> (and further developed and justified <ref name="BMW"/> <ref name="MBB2001"/> <ref name ="BMW2001"/>) to better account for super-diffusive spreading of tracer particles in aquifers.  It has quite a few variations at this point, but a couple of characteristic examples are

\[
u_t = c\frac{\partial u}{\partial x} + p\frac{\partial^\alpha u}{\partial x^\alpha} + q\frac{\partial^\alpha u}{\partial (-x)^\alpha} ,
\]
where the fractional derivatives are defined as 
\begin{equation}
	\frac{\partial^{\alpha}}{\partial e^{\alpha}} u(x) = c_{n,\alpha}\int_0^\infty \frac{u(x)-u(x-ye)}{y}y^{-\alpha}dy\ \ \text{and}\ \ \frac{\partial^{\alpha}}{\partial (-e)^{\alpha}} u(x) = c_{n,\alpha}\int_0^\infty \frac{u(x)-u(x+ye)}{y}y^{-\alpha}dy .
\end{equation}
It is important to point out that these fractional derivatives are examples of one-dimensional [[Linear integro-differential operator | linear integro-differential operators]] with non-symmetric kernels (in this case, e.g. $K(y)=\mathbb{1}_{\{y\geq 0\}}|y|^{-1-\alpha}$).


== See also ==

* [[Drift-diffusion equations]]

== References ==
{{reflist|refs=
<ref name="MBB1999">{{Citation | last1=Meerschaert | first1=M.M. | last2=Benson | first2=D. J. | last3=Bäumer | first3=B. | title=Multidimensional advection and fractional dispersion | publisher=APS | year=1999 | journal=[[Physical Review|Physical Review E]] | issn=1539-3755 | volume=59 | issue=5 | pages=5026}}</ref>
<ref name="BMW">{{Citation | last1=Benson | first1=D. J. | last2=Wheatcraft | first2=S.W. | last3=Meerschaert | first3=M.M. | title=Application of a fractional advection-dispersion equation | year=2000 | journal=Water Resources Research | volume=36 | issue=6 | pages=1403–1412}}</ref>
<ref name="MBB2001">{{Citation | last1=Meerschaert | first1=M.M. | last2=Benson | first2=D. J. | last3=Baeumer | first3=B. | title=Operator Lévy motion and multiscaling anomalous diffusion | publisher=APS | year=2001 | journal=[[Physical Review|Physical Review E]] | issn=1539-3755 | volume=63 | issue=2 | pages=021112}}</ref>
<ref name="BMW2001">{{Citation | last1=Benson | first1=D. J. | last2=Schumer | first2=R. | last3=Meerschaert | first3=M.M. | last4=Wheatcraft | first4=S.W. | title=Fractional dispersion, Lévy motion, and the MADE tracer tests | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=2001 | journal=Transport in Porous Media | volume=42 | issue=1 | pages=211–240}}</ref>
}}