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

Abstract

We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value \begin{document}$ \partial_t^{\alpha} u(x, t) = -Au(x, t) $\end{document}, where \begin{document}$ -A = \sum_{i, j = 1}^d \partial_i(a_{ij}(x) \partial_j) + \sum_{j = 1}^d b_j(x) \partial_j + c(x) $\end{document}. We establish the uniqueness for an inverse problem of determining an order \begin{document}$ \alpha $\end{document} of fractional derivatives by data \begin{document}$ u(x_0, t) $\end{document} for \begin{document}$ 0 at one point \begin{document}$ x_0 $\end{document} in a spatial domain \begin{document}$ \Omega $\end{document}. The uniqueness holds even under assumption that \begin{document}$ \Omega $\end{document} and \begin{document}$ A $\end{document} 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.