Objectives -- Define anomalous diffusion -- Illustration of normal, sub and super diffusion

1. 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

2. Local At a point rate 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 At an instant slope

3. 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?

4. A physically incomplete but meaningful analogy to describe anomalous diffusion How will drop of colored water spread out on tissue after time t ?

5. How will drop of colored water spread out on tissue after time t ? rn rsub rsup If blot radius grows as If the time exponent differs from ½ Diffusion is said to be anomalous Sub Diffusion Supper Diffusion Diffusion is said to be normal

6. How will drop of colored water spread out on tissue after time t ? Sub Diffusion Supper Diffusion normal rn rsub rsup Described by non local flux non zero wait How do the exponents in the frac. Dervs. relate to sub or supper diffusion Transient change in volume Divergence of flux (volume balance)

7. 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 fixed head Water supply saturated dry sharp Moving Front between saturated and dry Gov. equ. (mass con + Darcy) Extra volume balance condition at front Note transient limits

8. saturated dry 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

9. Now let us look at problem with fractional derivatives saturated dry 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

10. gamma function saturated dry 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

11. Now let us look at problem with fractional derivatives saturated dry solution satisfying conditions

12. saturated dry

13. saturated dry Observations: For appropriate choices of fractional derivatives Super g >.5, normal g = .5 , or sub g <.5 diffusion can be realized If only flux fractional derivative is used 0 < a < 1, b = 1 then ONLY Super diff can be realized If only trans. fractional derivative is used 0 < b < 1, a = 1 then ONLY Sub diff can be realized

14. 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 . Figure 1: Movement of the liquid/solid interface for choices of time fractional and flux fractional derivatives