On the remarkable nonlinear diffusion equation (∂/∂<i>x</i>)[<i>a</i> (<i>u</i>+<i>b</i>)−2(∂<i>u</i>/∂<i>x</i>)]−(∂<i>u</i>/∂<i>t</i>)=0

Abstract

We study the invariance properties (in the sense of Lie–Bäcklund groups) of the nonlinear diffusion equation (∂/∂x)[C (u)(∂u/∂x)]−(∂u/∂t) =0. We show that an infinite number of one-parameter Lie–Bäcklund groups are admitted if and only if the conductivity C (u) =a (u+b)−2. In this special case a one-to-one transformation maps such an equation into the linear diffusion equation with constant conductivity, (∂2ū/∂x̄2)−(∂ū/∂t̄) =0. We show some interesting properties of this mapping for the solution of boundary value problems.