# A Spatial Structural Derivative Model for Ultraslow Diffusion

## TL;DR

This paper introduces a novel spatial structural derivative model using an exponential function to describe ultraslow diffusion, providing analytical solutions and numerical analysis of mean squared displacement.

## Contribution

It proposes a new local structural derivative diffusion equation with exponential spatial function as an alternative model for ultraslow diffusion.

## Key findings

- Analytical solution is a Biexponential distribution.
- Mean squared displacement increases slower than logarithmic in time.
- Model offers a new physical and mathematical approach to ultraslow diffusion.

## Abstract

This study investigates the ultraslow diffusion by a spatial structural derivative, in which the exponential function exp(x)is selected as the structural function to construct the local structural derivative diffusion equation model. The analytical solution of the diffusion equation is a form of Biexponential distribution. Its corresponding mean squared displacement is numerically calculated, and increases more slowly than the logarithmic function of time. The local structural derivative diffusion equation with the structural function exp(x)in space is an alternative physical and mathematical modeling model to characterize a kind of ultraslow diffusion.