  In this article we study inverse problems of recovering a space-time
dependent source component from the lateral boundary observation in a
subidffusion model. The mathematical model involves a Djrbashian-Caputo
fractional derivative of order $\alpha\in(0,1)$ in time, and a second-order
elliptic operator with time-dependent coefficients. We establish a
well-posedness and a conditional stability result for the inverse problems
using a novel perturbation argument and refined regularity estimates of the
associated direct problem. Further, we present an algorithm for efficiently and
accurately reconstructing the source component, and provide several
two-dimensional numerical results showing the feasibility of the recovery.
