Approximate solutions of some nonlinear diffusion equations

Approximate analytic solutions are given for several equations obtained from a similarity analysis of the nonlinear diffusion equation ${\rho }_t-\overrightarrow{\nabla }·\left(D\overrightarrow{\nabla }\rho \right)=0\mathrm{}$, where the diffusion coefficient $D$ is of the form $D\sim {\rho }^n$. The solution technique is also demonstrated for a case where $D$ is inhomogeneous as well as nonlinear ($D\sim r^{\gamma }{\rho }^n$, where $r$ is the radial coordinate). The approach is general, algebraically simple, and flexible enough to allow applications to many related problems. The analytic predictions are compared to previous numerical results for some cases and shown to yield good agreement.