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