(Difference between revisions)
 Revision as of 22:56, 18 November 2012 (view source)Nestor (Talk | contribs)← Older edit Latest revision as of 23:13, 18 November 2012 (view source)Nestor (Talk | contribs) Line 13: Line 13: - {{See also|Drift-diffusion equations}} + == See also == + + * [[Drift-diffusion equations]] == References == == References ==

## Latest revision as of 23:13, 18 November 2012

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. $K(y)=\mathbb{1}_{\{y\geq 0\}}|y|^{-1-\alpha}$).