  In this work, we investigate an inverse problem of recovering multiple orders
in a time-fractional diffusion model from the data observed at one single point
on the boundary. We prove the unique recovery of the orders together with their
weights, which does not require a full knowledge of the domain or medium
properties, e.g., diffusion and potential coefficients, initial condition and
source in the model. The proof is based on Laplace transform and asymptotic
expansion. Further, inspired by the analysis, we propose a numerical procedure
for recovering these parameters based on a nonlinear least-squares fitting with
either fractional polynomials or rational approximations as the model function,
and provide numerical experiments to illustrate the approach for small time
$t$.
