Analytical studies of a time-fractional porous medium equation. Derivation, approximation and applications

• We provide a deterministic derivation of the nonlinear fractional diffusion equation.
• We devise analytic formulas for approximations of the exact solution.
• Our analytical results are very accurate, what is confirmed by numerical analysis.

In this paper we investigate the porous medium equation with a time-fractional derivative. We justify that the resulting equation emerges when we consider a waiting-time (or trapping) phenomenon that can have its place 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 generalise 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.