The porous medium equation, also called the nonlinear heat equation, is a nonlinear partial differential equation taking the form:[1]
where is the Laplace operator. It may also be put into its equivalent divergence form:
Despite being a nonlinear equation, the porous medium equation may be solved exactly using separation of variables or a similarity solution. However, the separation of variables solution is known to blow up to infinity at a finite time.[2]
The similarity approach to solving the porous medium equation was taken by Barenblatt[3] and Kompaneets/Zeldovich,[4] which for was to find a solution satisfying:
The porous medium equation has been found to have a number of applications in gas flow, heat transfer, and groundwater flow.[5]
The porous medium equation name originates from its use in describing the flow of an ideal gas in a homogeneous porous medium.[6] We require three equations to completely specify the medium's density , flow velocity field , and pressure : the continuity equation for conservation of mass; Darcy's law for flow in a porous medium; and the ideal gas equation of state. These equations are summarized below:
Using Fourier's law of heat conduction, the general equation for temperature change in a medium through conduction is: