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 $$\frac{\partial^{\alpha}}{\partial e^{\alpha}} u(x) = c_{n,\alpha}\int_0^\infty \frac{u(x)-u(x-ye)}{y}y^{-\alpha}dy\ \ \text{and}\ \ \frac{\partial^{\alpha}}{\partial (-e)^{\alpha}} u(x) = c_{n,\alpha}\int_0^\infty \frac{u(x)-u(x+ye)}{y}y^{-\alpha}dy .$$ 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

1. Meerschaert, M.M.; Benson, D. J.; Bäumer, B. (1999), "Multidimensional advection and fractional dispersion", Physical Review E (APS) 59 (5): 5026, ISSN 1539-3755
2. Benson, D. J.; Wheatcraft, S.W.; Meerschaert, M.M. (2000), "Application of a fractional advection-dispersion equation", Water Resources Research 36 (6): 1403–1412
3. 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
4. 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