Fractional Diffusion

• A completely separate mathematical technique has been found to describe Lévy-stable distributions.

• Time-fractional Diffusion Equation

  • d2Ýf / d2Ýt = D d2f / d2x
  • where D denotes positive constant with units of L2/T2Ý
  • Ý=1 wave equation; Ý=1/2 diffusion equation (Gauss)
  • Anomalous Diffusion
  • Ý<1/2 = slow diffusion (Cauchy); Ý>1/2 = fast diffusion
  • Solutions are Mittag-Leffler functions of order 2Ý, they are also Levy-stable pdf.
  • We can identify Ú-fcn with slow fractional diffusion