Advanced self-similar solutions of regular and irregular diffusion equations

TL;DR

This paper derives new self-similar solutions for both regular and irregular diffusion equations by transforming PDEs into ODEs, emphasizing physically meaningful solutions with time-dependent diffusion coefficients expressed via special functions.

Contribution

It introduces a method to obtain analytic self-similar solutions for diffusion equations with time-dependent coefficients, expanding the class of solvable diffusion problems.

Findings

Solutions expressed with Kummer's and Whittaker functions

Physically reasonable solutions for infinite horizon cases

Analysis of time-dependent diffusion phenomena

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 Ansat\"atze 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.