  We study the fast diffusion equation (FDE) with a linear forcing term under
the Ricci flow on complete manifolds with bounded curvature and nonnegative
curvature operator. We prove Aronson-B\'enilan and Li-Yau-Hamilton type
differential Harnack estimates for positive solutions of the FDE. In addition,
we use similar method to prove certain Li-Yau-Hamilton estimates for the heat
equation and conjugate heat equation which extend those obtained by X. Cao and
R. Hamilton, X. Cao, and S. Kuang and Q. Zhang to noncompact setting.
