# Finite-velocity diffusion on a comb

## TL;DR

This paper analyzes a Cattaneo equation on a comb structure, deriving solutions for fractional diffusion that describe the transition from normal diffusion to subdiffusion due to the geometry.

## Contribution

It provides a rigorous analysis of fractional diffusion equations on a comb, with solutions expressed via Fox H-functions, capturing the transition from normal to subdiffusive behavior.

## Key findings

- Solutions in Fox H-function form for probability distribution
- Mean square displacement expressed as an infinite series
- Transition from normal diffusion to subdiffusion due to geometry

## Abstract

A Cattaneo equation for a comb structure is considered. We present a rigorous analysis of the obtained fractional diffusion equation, and corresponding solutions for the probability distribution function are obtained in the form of the Fox $H$-function and its infinite series. The mean square displacement along the backbone is obtained as well in terms of the infinite series of the Fox $H$-function. The obtained solutions describe the transition from normal diffusion to subdiffusion, which results from the comb geometry.