Approximate solutions to the Boussinesq equation

The movement of water in unconfined aquifers is often modeled by the Boussinesq equation. It is a nonlinear diffusion equation which was derived under the Dupuit assumption of essentially horizontal flows. There are multiple usages of the Boussinesq equation in baseflow studies, catchment hydrology and agricultural drainage problems as well as in other applications. Solutions of the Boussinesq equation can propagate with a finite speed, unlike solutions to the linear diffusion equation. In this work we obtain approximate closed-form solutions to the one-dimensional Boussinesq equation. With similarity transformations a number of initial-boundary-value problems for the Boussinesq equation can be reduced to boundary-value problems for an ordinary differential equation in terms of a similarity variable and scaling function, and we construct approximate polynomial solutions. Our solutions reproduce known exact solutions and compare favorably with the numerical results. Moreover our solutions preserve scaling properties of the problems and require small computational effort. This approach can be extended to the porous medium equation, which generalizes the Boussinesq equation.