The stretched exponential behavior and its underlying dynamics. The phenomenological approach

TL;DR

This paper explores the connection between anomalous diffusion equations with fractional derivatives and Le9vy stable distributions, providing explicit solutions and highlighting the underlying dynamics of stretched exponential behavior.

Contribution

It establishes a link between fractional differential equations and Le9vy distributions, offering explicit solutions and a phenomenological understanding of stretched exponential dynamics.

Findings

Fractional derivatives relate to Le9vy stable distributions.

Explicit solutions for various fractional orders are provided.

The approach clarifies the dynamics behind stretched exponential behavior.

Abstract

We show that the anomalous diffusion equations with a fractional derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the L\'evy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution operator is presented as integral transforms whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and different initial conditions.