# Generalised fractional diffusion equations for subdiffusion on arbitrarily growing domains

## TL;DR

This paper derives new fractional partial differential equations to model subdiffusion on arbitrarily growing domains, incorporating domain growth dynamics and a novel comoving fractional derivative, based on a Continuous Time Random Walk framework.

## Contribution

It introduces a new fractional derivative and formulates generalized equations for subdiffusion on growing domains, addressing challenges of history dependence.

## Key findings

- Derived fractional PDEs for subdiffusion on growing domains
- Introduced a new comoving fractional derivative
- Framework applicable to various physical phenomena involving domain growth

## Abstract

Many physical phenomena occur on domains that grow in time. When the timescales of the phenomena and domain growth are comparable, models must include the dynamics of the domain. A widespread intrinsically slow transport process is subdiffusion. Many models of subdiffusion include a history dependence. This greatly confounds efforts to incorporate domain growth. Here we derive the fractional partial differential equations that govern subdiffusion on a growing domain, based on a Continuous Time Random Walk. This requires the introduction of a new, comoving, fractional derivative.