An attempt to provide a physical interpretation of fractional transport in heterogeneous domains

Séminaire de Potamologie

Résumé:

Vaughan R Voller, Civil Engineering, University of Minnesota


To first order, transport in many natural systems can be adequately
described by diffusion processes. When heterogeneities are present in
the system, however, a simple linear, integer diffusion treatment may
not be sufficient to account for experimental and filed observations.
For example in sedimentary deposition experiments, the observed
fluvial surfaces are much “flatter” than those predicted with a linear
diffusion model. If the length scales of heterogeneities in the
system are power law distributed and occur up to the scale of the
system itself, then a case can be made to model the transport with a
fractional diffusion equation.

A significant part of this talk some time will be taken in explaining
what fractional diffusion is, why it is a good model in heterogeneous
systems, and how it represents non-local behavior. Particular emphasis
is placed on developing discrete analog models (inspired by
observation) that provide a physical interpretation of fractional
diffusion. Following these developments a simple steady state
diffusion problem is used to illustrate the connections between a
mathematical, probabilistic, and discrete non-local treatment of a
fractional derivative. These ideas are then carried forward into to
the development of a fractional diffusion models that are able to
describe observations taken from experimental scale sediment transport
systems and field moisture infiltration measurements.