Analytic solutions for irregular diffusion equations with concentration dependent diffusion coefficients

TL;DR

This paper derives analytic solutions for nonlinear diffusion equations with concentration-dependent diffusion coefficients, using self-similar and traveling wave approaches, including formulas with hypergeometric functions.

Contribution

It provides explicit analytic formulas for solutions of nonlinear diffusion equations with power-law concentration dependence, expanding the understanding of such equations.

Findings

Analytic implicit formulas for power-law cases

Solutions expressed with hypergeometric functions

Detailed analysis for specific parameter sets

Abstract

We investigate diffusion equations which have concentration dependent diffusion coefficients with physically two relevant Ans\"atze, the self-similar and the traveling wave Ansatz. We found that for power-law concentration dependence some of the results can be expressed with a general analytic implicit formulas for both trial functions. For the self-similar case some of the solutions can be given with a formula containing the hypergeometric function. For the traveling wave case different analytic formulas are given for different exponents. For some physically reasonable parameter sets the direct solutions are given and analyzed in details.