The fractionalorder governing equation of Lévy Motion
Abstract
A governing equation of stable random walks is developed in one dimension. This FokkerPlanck equation is similar to, and contains as a subset, the secondorder advection dispersion equation (ADE) except that the order (α) of the highest derivative is fractional (e.g., the 1.65th derivative). Fundamental solutions are Lévy's αstable 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 Lévy 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 Lévy 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.
 Publication:

Water Resources Research
 Pub Date:
 February 2000
 DOI:
 10.1029/2000WR900032
 Bibcode:
 2000WRR....36.1413B
 Keywords:

 Hydrology: Groundwater transport;
 Hydrology: Stochastic processes;
 Mathematical Geophysics: Modeling;
 Mathematical Geophysics: Fractals and multifractals