Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 Objectives -- Define anomalous diffusion -- Illustration of normal, sub and super diffusion --Outline some very basic elements of factional calculus -- Provide an explicit physical connection between the order of fractional derivatives and sub and super diffusion processes

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 Despite our best efforts we can not be in two places at once Hence our intuitive physical sense of the world is based on LOCAL information Local At a point slope At an instant rate

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 But many physical process are NON-LOCAL Holdup – release Depends on time scale Extensive developing literature that argues that these non local process Can be described by Fractional Derivatives !!! How on earth do we associate these constructs with our intuitively locally?

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 A physically incomplete but meaningful analogy to describe anomalous diffusion How will drop of colored water spread out on tissue after time t ?

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 How will drop of colored water spread out on tissue after time t ? r sub rnrn r sup If blot radius grows as Diffusion is said to be normal If the time exponent differs from ½ Diffusion is said to be anomalous Supper Diffusion Sub Diffusion

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 How will drop of colored water spread out on tissue after time t ? r sub rnrn r sup normal Supper Diffusion Sub Diffusion Described by Transient change in volume Divergence of flux (volume balance) How do the exponents in the frac. Dervs. relate to sub or supper diffusion non zero wait non local flux

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 How do the exponents in the frac. Dervs. relate to sub or supper diffusion ? To answer we consider a well known limit problem for flow in porous media fixed head sharp Moving Front between saturated and dry Water supply saturateddry Gov. equ. (mass con + Darcy) Note transient limits Extra volume balance condition at front

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 saturateddry Extra volume balance condition at front solution satisfying conditions sub here Darcy Flux assumes normal diffusion This results in advance of saturated region with characteristic time scale

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 saturateddry Now let us look at problem with fractional derivatives If our fractional derivatives are Caputo derivatives (see below) then we can easily solve this problem. The result is a wetting front that moves as So choice of fractional exponents allows us to Move from a sub to super diff. behavior

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 saturateddry To solve fractional derivative version of problem we use Caputo derivatives Can evaluate using Lapalce transform IF For solution of problem all you need to know is that gamma function

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 saturateddry Now let us look at problem with fractional derivatives solution satisfying conditions

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 saturateddry

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 saturateddry Observations:For appropriate choices of fractional derivatives Super  >.5, normal  =.5, or sub  <.5 diffusion can be realized If only flux fractional derivative is used 0 <  < 1,  = 1 then ONLY Super diff can be realized If only trans. fractional derivative is used 0 <  < 1,  = 1 then ONLY Sub diff can be realized

Non-local behavior in Geomorphology: Vaughan R Voller, NCED Summer Institute, 4:30-6:00 pm, August 19, 2009 Figure 1: Movement of the liquid/solid interface for choices of time fractional and flux fractional derivatives. This simple problem has allowed for a “trivial” solution to a Fractional PDE. A solution that has provided a clear physical connection between the order of the fractional derivative and the nature of the anomalous diffusion