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