Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations

We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value $ \partial_t^{\alpha} u(x, t) = -Au(x, t) $, where $ -A = \sum_{i, j = 1}^d \partial_i(a_{ij}(x) \partial_j) + \sum_{j = 1}^d b_j(x) \partial_j + c(x) $. We establish the uniqueness for an inverse problem of determining an order $ \alpha $ of fractional derivatives by data $ u(x_0, t) $ for $ 0<t<T $ at one point $ x_0 $ in a spatial domain $ \Omega $. The uniqueness holds even under assumption that $ \Omega $ and $ A $ are unknown, provided that the initial value does not change signs and is not identically zero. The proof is based on the eigenfunction expansions of finitely dimensional approximating solutions, a decay estimate and the asymptotic expansions of the Mittag-Leffler functions for large time.