Similarity solutions of the nonlinear diffusion equation

The paper considers similarity solutions of the nonlinear diffusion equation of the form ${t^\alpha }f\left ( \eta \right )$ where $\eta = r{t^{ - \delta }}$ or $\exp \left ( {\alpha t} \right )f\left ( \eta \right )$ where $\eta = r\exp \left ( { - \delta t} \right )$. The novel feature of the paper is that the second-order differential equation for $f$ is reduced to a system of first-order equations and a phase plane analysis of one member of the system can be made. In this way we may discuss the existence and uniqueness of all the solutions for $f\left ( n \right )$. Restricting the discussion to plane geometry, we list all the continuous solutions to the basic problem on $0 \le \eta \le \infty$ with $f\left ( 0 \right ) = U \ge 0$ and $f\left ( \infty \right ) = 0$. Solutions of previous authors are identified as special cases.