  In this article, we consider the reconstruction of $\rho(t)$ in the
(time-fractional) diffusion equation
$(\partial_t^\alpha-\triangle)u(x,t)=\rho(t)g(x)$ ($0<\alpha \le 1$) by the
observation at a single point $x_0$. We are mainly concerned with the situation
of $x_0 \notin$ supp g, which is practically important but has not been well
investigated in literature. Assuming the finite sign changes of $\rho$ and an
extra observation interval, we establish the multiple logarithmic stability for
the problem based on a reverse convolution inequality and a lower estimate for
positive solutions. Meanwhile, we develop a fixed point iteration for the
numerical reconstruction and prove its convergence. The performance of the
proposed method is illustrated by several numerical examples.
