  In this paper we investigate the porous medium equation with a fractional
temporal derivative. We justify that the resulting equation emerges when we
consider the waiting-time (or trapping) phenomenon that can happen in the
medium. Our deterministic derivation is dual to the stochastic CTRW framework
and can include nonlinear effects. With the use of the previously developed
method we approximate the investigated equation along with a constant flux
boundary conditions and obtain a very accurate solution. Moreover, we
generalize the approximation method and provide explicit formulas which can be
readily used in applications. The subdiffusive anomalies in some porous media
such as construction materials have been recently verified by experiment. Our
simple approximate solution of the time-fractional porous medium equation fits
accurately a sample data which comes from one of these experiments.
