  In the continuous time random walk model, the time-fractional operator
usually expresses an infinite waiting time probability density. Different from
that usual setting, this work considers the tempered time-fractional operator,
which reflects a finite waiting time probability density. Firstly, we analyse
the solution of a tempered benchmark problem, which shows a weak singularity
near the initial time. The L1 scheme on graded mesh and the WSGL formula with
correction terms are adapted to deal with the non-smooth solution, in which we
compare these two methods systematically in terms of the convergence and
consumed CPU time. Furthermore, a fast calculation for the time tempered Caputo
fractional derivative is developed based on a sum-of-exponentials
approximation, which significantly reduces the running time. Moreover, the
tempered operator is applied to the Bloch equation in nuclear magnetic
resonance and a two-layered problem with composite material exhibiting distinct
memory effects, for which both the analytical (or semi-analytical) and
numerical solutions are derived using transform techniques and finite
difference methods. Data fitting results verify that the tempered
time-fractional model is much effective to describe the MRI data. An important
finding is that, compared with the fractional index, the tempered operator
parameter could further accelerate the diffusion. The tempered model with two
parameters $\alpha$ and $\rho$ are more flexible, which can avoid choosing a
too small fractional index leading to low regularity and strong heterogeneity.
