The fractional-order governing equation of Levy motion

Abstract

A governing equation of stable random walks is developed in one dimension. This Fokker-Planck equation is similar to, and contains as a subset, the second-order advection dispersion equation (ADE) except that the order (α) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Levy's α-stable densities that resemble the Gaussian except that they spread proportional to time(1/α) have heavier tails, and incorporate any degree of skewness. The measured variance of a plume undergoing Levy motion would grow faster than Fickian plume, at a rate of time(2/α) where 0 < α ≤ 2. The equation is parsimonious since the parameters are not functions of time or distance. The scaling behavior of plumes that undergo Levy motion is accounted for by the fractional derivatives, which are appropriate measures of fractal functions. In real space the fractional derivatives are integrodifferential operators, so the fractional ADE describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space.