Uniqueness in determining fractional orders of derivatives and initial values

Abstract

For initial boundary value problems of time-fractional diffusion-wave equations with the zero Dirichlet boundary value, we consider an inverse problem of determining orders of the fractional derivatives and initial values by three kinds of measurement data on a time interval: (i) solution in a subdomain (ii) Neumann data on a subboundary (iii) spatial averaged values. We prove that the order and initial values are uniquely determined by the above data if the solutions do not identically vanish. Our main results show that the uniqueness for the fractional order holds even if the coefficients of the fractional diffusion-wave equations and source terms within some class are unknown. Moreover we consider pointwise data for the uniqueness in the inverse problem. The proof is based on the eigenfunction expansions and the asymptotic expansions of the Mittag–Leffler functions for large time.