Similarity solutions of the nonlinear diffusion equation

Abstract

The paper considers similarity solutions of the nonlinear diffusion equation of the form t α f ( η ) {t^\alpha }f\left ( \eta \right ) where η = r t − δ \eta = r{t^{ - \delta }} or exp ⁡ ( α t ) f ( η ) \exp \left ( {\alpha t} \right )f\left ( \eta \right ) where η = r exp ⁡ ( − δ t ) \eta = r\exp \left ( { - \delta t} \right ) . The novel feature of the paper is that the second-order differential equation for f 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 ( n ) f\left ( n \right ) . Restricting the discussion to plane geometry, we list all the continuous solutions to the basic problem on 0 ≤ η ≤ ∞ 0 \le \eta \le \infty with f ( 0 ) = U ≥ 0 f\left ( 0 \right ) = U \ge 0 and f ( ∞ ) = 0 f\left ( \infty \right ) = 0 . Solutions of previous authors are identified as special cases.