Some Random Directions in Fractional Derivatives Vaughan Voller

Some Random Directions in Fractional Derivatives
Vaughan Voller
Work with Chris Paola. Efi Foufoula, Vamsi Ganti and Dam Zielinski
The objective of this talk is to present two recent excursions into solving problems with
fractional derivatives. The first topic—introduced briefly at the pervious STRESS meeting—will
outline how a random-walk solution for fractional diffusion equations can be generated using
what we refer to as a "random domain shifting" technique. The second topic looks at non-local
model of sediment transport in the source—to –sink long profile. Form previous STRESS related
meetings it is observed that models of upland erosional systems use fractional diffusion
treatments with a non-local direction only directed upstream, whereas models of depositional
systems employ a downstream only non-local direction. With a simple long profile model, it will
be demonstrated here, that physical realistic solutions for the fluvial surface shape can only be
predicted if this bifurcation between erosional and depositional systems is enforced. A
discussion of why this might be true and its possible consequences for fractional derivative
models of non-locality in land-scape dynamics will be discussed.