Two demonstrations of the theoretical and physical nature of fractional diffusion and
their possible role in explaining fluvial depositional profiles
Vaughan Voller1 and Chris Paola2
NCED, St Anthony Falls Laboratory,
1
Department of Civil Engineering, 2Deaprtment of Geology and Geophysics, University
of Minnesota
Two demonstrations that illustrate the theoretical and physical nature of Caputo fractional
diffusion equations are presented.
In the first instance a generalized, steady state, one-dimensional fraction diffusion
equation with Dirichlet boundary conditions is considered. Borrowing from classical
numerical heat transfer methods a Monte-Carlo solution is sought. In this case, however,
the step sizes in the simulation are chosen from an appropriately skewed Levy
distribution. This approach, across a full range of heavy tailed weightings, accurately
recovers the analytical solution of the fractional diffusion equation. Thus demonstrating
the link between the Caputo derivative and the Levy distribution.
In the second demonstration a closed form solution is developed for a limit case Stefan
problem governed by a transient fractional diffusion equation. The limit problem studied
corresponds to the liquid filling of an initially dry horizontal porous tube under a fixed
head condition at the origin. When the standard integer diffusion equation is used, the
analytical solution shows that the assumed sharp front between saturated and dry regions
moves with the classic square root of time dependence. The introduction of a fractional
transient term, however, results in a sub-diffusion front movement where the time
exponent is less that one-half. On the other hand, the introduction of a fractional diffusion
term results in a super-diffusion front movement, where the time exponent is greater that
one-half. Thus, the analytical solution provides an explicit linking between the fractional
derivatives in the diffusion equation and the resulting nature and scale of the anomalous
diffusion.
Together these partial advances in the authors understanding are used in an attempt to
explain observations related to the shape of fluvial depositional profiles. At field scales a
“typical” profile will have a constant curvature. At laboratory scales, however, observed
profiles tend to have little or no curvature. This leads to our current question of interest—
can fractional calculus, which may allow fro non-local behavior to be important at small
scales, explain the observed lack of curvature in experimental sediment deposits?