  This work is concerned with numerically recovering multiple parameters
simultaneously in the subdiffusion model from one single lateral measurement on
a part of the boundary, while in an incompletely known medium. We prove that
the boundary measurement corresponding to a fairly general boundary excitation
uniquely determines the order of the fractional derivative and the polygonal
support of the diffusion coefficient, without knowing either the initial
condition or the source. The uniqueness analysis further inspires the
development of a robust numerical algorithm for recovering the fractional order
and diffusion coefficient. The proposed algorithm combines small-time
asymptotic expansion, analytic continuation of the solution and the level set
method. We present extensive numerical experiments to illustrate the
feasibility of the simultaneous recovery. In addition, we discuss the
uniqueness of recovering general diffusion and potential coefficients from one
single partial boundary measurement, when the boundary excitation is more
specialized.
