  This paper is concerned with an inverse problem of recovering a potential
term and fractional order in a one-dimensional subdiffusion problem, which
involves a Djrbashian-Caputo fractional derivative of order $\alpha\in(0,1)$ in
time, from the lateral Cauchy data. In the model, we do not assume a full
knowledge of the initial data and the source term, since they might be
unavailable in some practical applications. We prove the unique recovery of the
spatially-dependent potential coefficient and the order $\alpha$ of the
derivation simultaneously from the measured trace data at one end point, when
the model is equipped with a boundary excitation with a compact support away
from $t=0$. One of the initial data and the source can also be uniquely
determined, provided that the other is known. The analysis employs a
representation of the solution and the time analyticity of the associated
function. Further, we discuss a two-stage procedure, directly inspired by the
analysis, for the numerical identification of the order and potential
coefficient, and illustrate the feasibility of the recovery with several
numerical experiments.
