# Fractional Advection Dispersion Equation

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The Fractional Advection Dispersion (fADE) was introduced ^{[1]} (and further developed and justified ^{[2]} ^{[3]} ^{[4]}) 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 operators with non-symmetric kernels (in this case, e.g. $K(y)=\mathbb{1}_{\{y\geq 0\}}|y|^{-1-\alpha}$).

## References

- ↑ Meerschaert, M.M.; Benson, D. J.; Bäumer, B. (1999), "Multidimensional advection and fractional dispersion",
*Physical Review E*(APS)**59**(5): 5026, ISSN 1539-3755 - ↑ Benson, D. J.; Wheatcraft, S.W.; Meerschaert, M.M. (2000), "Application of a fractional advection-dispersion equation",
*Water Resources Research***36**(6): 1403–1412 - ↑ Meerschaert, M.M.; Benson, D. J.; Baeumer, B. (2001), "Operator Lévy motion and multiscaling anomalous diffusion",
*Physical Review E*(APS)**63**(2): 021112, ISSN 1539-3755 - ↑ Benson, D. J.; Schumer, R.; Meerschaert, M.M.; Wheatcraft, S.W. (2001), "Fractional dispersion, Lévy motion, and the MADE tracer tests",
*Transport in Porous Media*(Berlin, New York: Springer-Verlag)**42**(1): 211–240