  In this work, we investigate the recovery of a parameter in a diffusion
process given by the order of derivation in time for some class of diffusion
equations, including both classical and time-fractional diffusion equations,
from the flux measurement observed at one point on the boundary. The
mathematical model for time-fractional diffusion equations involves a
Djrbashian-Caputo fractional derivative in time. We prove a uniqueness result
in an unknown medium (e.g., diffusion coefficients, obstacle, initial condition
and source) which can be seen as the recovery of the order of derivation in a
diffusion process having several pieces of unknown information. The proof
relies on the analyticity of the solution at large time, the asymptotic decay
behavior, the strong maximum principle of the elliptic problem and suitable
application of the Hopf lemma. Further we provide an easy-to-implement
reconstruction algorithm based on a nonlinear least-squares formulation, and
several numerical experiments are presented to complement the theoretical
analysis.
