A Series Solution to the Porous Medium Equation

The Porous Medium Equation is a generalization of the Boussinesqequation, when the diffusivity is a power-law function of thehydraulic head, not only a linear function as in the case of theBoussinesq equation. We consider the case of a one-dimensionalaquifer, initially dry, and of semi-infinite extent. At theboundary representing a fluid source, the boundary condition isspecified as a power-law function of time. Following Barenblatt'sapproach, self-similar variables can be introduced. This reducesthe original initial-boundary value problem for the partialdifferential equation to a boundary value problem for a nonlinearordinary differential equation. The boundary representing thewetting front is not known, and must be found in the process ofsolution. A power series solution is found for this nonlinear ODE.We construct a recurrence relation for the coefficients of theseries, and show the convergence of the series. Results arecompared against a highly accurate numerical solution of this ODEas well as the results of a lab-scale experiment