Subdiffusion equation with Caputo fractional derivative with respect to another function

TL;DR

This paper introduces a generalized subdiffusion model using Caputo fractional derivatives with respect to another function, capturing evolving media structures and transitions from subdiffusion to ultraslow diffusion.

Contribution

It presents a novel subdiffusion equation with a fractional derivative relative to a function, enabling modeling of media with evolving structures and complex aging processes.

Findings

Model describes transition from subdiffusion to ultraslow diffusion.

Method for solving the g-subdiffusion equation is provided.

Analysis of aging effects in evolving media.

Abstract

We show an application of a subdiffusion equation with Caputo fractional time derivative with respect to another function $g$ to describe subdiffusion in a medium having a structure evolving over time. In this case a continuous transition from subdiffusion to other type of diffusion may occur. The process can be interpreted as "ordinary" subdiffusion with fixed subdiffusion parameter (subdiffusion exponent) $\alpha$ in which time scale is changed by the function $g$. As example, we consider the transition from "ordinary" subdiffusion to ultraslow diffusion. The function $g$ generates the additional aging process superimposed on the "standard" aging generated by "ordinary" subdiffusion. The aging process is analyzed using coefficient of relative aging of $g$--subdiffusion with respect to "ordinary" subdiffusion. The method of solving the $g$-subdiffusion equation is also presented.