# Uniqueness of determining the variable fractional order in variable-order time-fractional diffusion equations

## TL;DR

This paper proves the uniqueness of determining the variable fractional order in a one-dimensional time-fractional diffusion equation from solution observations, based on well-posedness and smoothing properties.

## Contribution

It establishes a novel uniqueness result for an inverse problem involving variable-order fractional derivatives in diffusion equations.

## Key findings

- Uniqueness of the variable fractional order from solution data
- Well-posedness of the initial-boundary value problem
- Smoothing properties depend on initial-time behavior of the variable order

## Abstract

We study an initial-boundary value problem of variable-order time-fractional diffusion equations in one space dimension. Based on the wellposedness of the proposed model and the smoothing properties of its solutions, which are shown to be determined by the behavior of the variable order at the initial time, a uniqueness result for an important inverse problem of determination of the variable order in the time-fractional derivative contained in the proposed model from observations of its solutions is obtained.