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

Abstract

<p style='text-indent:20px;'>We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value <inline-formula><tex-math id="M1">\begin{document}$ \partial_t^{\alpha} u(x, t) = -Au(x, t) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ -A = \sum_{i, j = 1}^d \partial_i(a_{ij}(x) \partial_j) + \sum_{j = 1}^d b_j(x) \partial_j + c(x) $\end{document}</tex-math></inline-formula>. We establish the uniqueness for an inverse problem of determining an order <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> of fractional derivatives by data <inline-formula><tex-math id="M4">\begin{document}$ u(x_0, t) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ 0<t<T $\end{document}</tex-math></inline-formula> at one point <inline-formula><tex-math id="M6">\begin{document}$ x_0 $\end{document}</tex-math></inline-formula> in a spatial domain <inline-formula><tex-math id="M7">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. The uniqueness holds even under assumption that <inline-formula><tex-math id="M8">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ A $\end{document}</tex-math></inline-formula> 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.</p>