Fractional Advection Dispersion Equation

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The Fractional Advection Dispersion (fADE) was introduced (and further developed and justified ) 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. $K(y)=\mathbb{1}_{\{y\geq 0\}}|y|^{-1-\alpha}$).