UTA Department of Mathematics

Applied Mathematics Seminar

Date/Time/Room: Friday (11/12/2004) at 2:00pm in 304 Pickard Hall

Speaker: Professor Aleksey Telyakovskiy, Department of Mathematics and Statistics, University of Nevada at Reno

"Approximate solutions to the Boussinesq and the porous medium equations"

Abstract: In this talk we focus on two equations describing groundwater flows: the Boussinesq and the porous medium equations. The Boussinesq equation models unconfined groundwater flow under the Dupuit assumption that the equipotential lines are vertical, making the flowlines horizontal. The porous medium equation, as the Boussinesq equation, is a nonlinear diffusion equation that describes the laminar filtration of a polytropic gas through porous media. Also, it models the motion of moisture through the porous media in case of a power-law diffusivity. Solving these equations usually requires numerical approximations, but for certain classes of initial and boundary conditions there are approximate analytical solution techniques. This work focuses on the latter approach, using the scaling properties of the equation.

We consider one-dimensional semi-infinite initially empty aquifer with boundary conditions at the inlet. Solutions would propagate with the finite speed. We analyze the effect of different conditions on the form of the (approximate) scaling function, and we compare approximate solutions with the direct numerical solutions obtained by using the scaling properties of the equations. Also, we mention a number of a few potential research problems.