The fractional‐order governing equation of Lévy Motion

TLDR

The fractional‑order Fokker‑Planck equation generalizes the ADE by replacing the second‑order derivative with a fractional derivative, yielding Lévy α‑stable solutions that spread faster than Gaussian, have heavy tails, and are spatially nonlocal yet analytically solvable. The study develops a one‑dimensional governing equation for stable random walks. The equation uses fractional derivatives to capture the scaling behavior of Lévy‑motion plumes. The variance of a plume undergoing Lévy motion grows as time^(2/α), faster than the Fickian rate.

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 Lévy'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 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.