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

TL;DR

This paper studies a time-fractional porous medium equation, deriving it from physical phenomena, approximating solutions with explicit formulas, and validating the approach with experimental data on subdiffusive media.

Contribution

It provides a deterministic derivation of the fractional porous medium equation, develops an accurate approximation method with explicit formulas, and demonstrates its effectiveness with experimental data.

Findings

Accurate approximation of the time-fractional porous medium equation.

Explicit formulas for solutions applicable in real-world scenarios.

Validation with experimental data showing subdiffusive behavior.

Abstract

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.