Uniqueness in determination of the fractional order in the TFDE using one measurement

This article deals with an inverse problem of identifying the fractional order in the 1D time fractional diffusion equation (TFDE in short) using the measurement at one space-time point. Based on the expression of the solution to the forward problem, the inverse problem is transformed to a nonlinear algebraic equation. By choosing suitable initial values and the measured point, the nonlinear equation has a unique solution by the monotonicity of the Mittag-Lellfer function. Theoretical testifications are presented to demonstrate the unique solvability of the inverse problem.