Advanced self-similar solutions of regular and irregular diffusion equations

Abstract

We study the diffusion equation with an appropriate change of variables. This equation is in general a partial differential equation (PDE). With the self-similar and related Ansatätze we transform the PDE of diffusion to an ordinary differential equation. The solutions of the PDE belong to a family of functions which are presented for the case of infinite horizon. In the presentation we accentuate the physically reasonable solutions. We also study time dependent diffusion phenomena, where the spreading may vary in time. To describe the process we consider time dependent diffusion coefficients. The obtained analytic solutions all can be expressed with Kummer's or Whittaker-type of functions.

arXiv comment28 pages, 5 figures

References in corpus (4)

• Transition from anomalous kinetics towards Fickian diffusion for Si dissolution into amorphous Ge
• General self-similar solutions of diffusion equation and related constructions
• Spiral modes in the diffusion of a granular particle on a vibrating surface
• Geometrical origin of chaoticity in the bouncing ball billiard