  In this paper we consider a telegraph equation with time-dependent
coefficients, governing the persistent random walk of a particle moving on the
line with a time-varying velocity $c(t)$ and changing direction at instants
distributed according to a non-stationary Poisson distribution with rate
$\lambda(t)$. We show that, under suitable assumptions, we are able to find the
exact form of the probability distribution. We also consider the
space-fractional counterpart of this model, finding the characteristic function
of the related process. A conclusive discussion is devoted to the potential
applications to run-and-tumble models.
