Hyperbolic versus parabolic equation with fractional derivative to describe subdiffusion in a membrane system

TL;DR

This paper compares hyperbolic and parabolic fractional derivative equations to model subdiffusion in membrane systems, deriving solutions and analyzing differences under various boundary conditions.

Contribution

It introduces solutions for both hyperbolic and parabolic fractional subdiffusion equations in membrane systems, highlighting their differences under specific boundary conditions.

Findings

Solutions for hyperbolic and parabolic models are derived.

Differences between models depend on boundary conditions.

The study enhances understanding of subdiffusion dynamics in membranes.

Abstract

We use the parabolic and hyperbolic equation with fractional time derivative to describe the subdiffusion in a system with thin membrane. We find the Green's function and solutions of the equation for the system where the homogeneous solution is separated by a thin membrane from the pure solvent. The solutions were obtained for two boundary conditions where the ratio of the concentrations at the membrane surfaces does not change in time and where the flux flowing through the membrane is proportional to the concentration difference between membrane surfaces. We discuss the difference between the solutions for parabolic and hyperbolic subdiffusion equations obtained for both boundary conditions.