Approximate similarity solution to a nonlinear diffusion equation with spherical symmetry
J. Mortensen 1, S. Olsen 1, J. Parlange 2, A. Telyakovskiy 11 Department of Mathematics and Statistics, University of Nevada, Reno, NV 89557, USA
2 Department of Biological and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA
Received by the editors May 9, 2009 and, in revised form, October 21, 2009.
In this article we construct an approximate similarity solution to a nonlinear diffusion equation in spherical coordinates. In hydrology this equation is known as the Boussinesq equation when written in planar or cylindrical coordinates. Recently Li et al.  obtained an approximate similarity solution to the Boussinesq equation in cylindrical coordinates. Here we consider the same problem in spherical coordinates with the prescribed power law point source boundary condition. The resulting scaling function has a power law singularity at the origin versus a logarithmic singularity in the cylindrical case.AMS subject classifications: 34B15, 35K20, 76S05, 80A20
Key words: Approximate solutions, similarity solutions, Boussinesq equation, nonlinear diffusion.