Recovering Multiple Fractional Orders in Time-Fractional Diffusion in an Unknown Medium

In this work, we investigate an inverse problem of recovering multiple orders in time-fractional diffusion type problems from the data observed at one single point on the boundary. We prove the unique recovery of the orders together with their weights, which does not require a full knowledge of the domain or medium properties, e.g., the diffusion and potential coefficients, initial condition and source in the model. The proof is based on Laplace transform and asymptotic expansion. Further, inspired by the analysis, we propose a numerical procedure for recovering these parameters based on a nonlinear least-squares fitting with either fractional polynomials or rational approximations as the model function, and provide numerical experiments to illustrate the approach at small time $t$.