  In this paper, we first establish a weak unique continuation property for
time-fractional diffusion-advection equations. The proof is mainly based on the
Laplace transform and the unique continuation properties for elliptic and
parabolic equations. The result is weaker than its parabolic counterpart in the
sense that we additionally impose the homogeneous boundary condition. As a
direct application, we prove the uniqueness for an inverse problem on
determining the spatial component in the source term in by interior
measurements. Numerically, we reformulate our inverse source problem as an
optimization problem, and propose an iteration thresholding algorithm. Finally,
several numerical experiments are presented to show the accuracy and efficiency
of the algorithm.
