  We study a model that intermediates among the wave, heat, and transport
equations. The approach considers the propagation of initial disturbances in a
one-dimensional medium that can vibrate. The medium is nonlinear in such a form
that nonlocal differential expressions are required to describe the
time-evolution of solutions. Non-locality is modeled with a space-time
fractional differential equation of order $1 \leq \alpha \leq 2$ in time, and
order $1\leq \beta \leq 2$ in space. We adopt the notion of Caputo for
time-derivative and the Riesz pseudo-differential operator for
space-derivative. The corresponding Cauchy problem is solved for zero initial
velocity and initial disturbance represented by either the Dirac delta or the
Gaussian distributions. Well known results for the partial differential
equations of wave propagation, diffusion and (modified) transport processes are
recovered as particular cases. In addition, regular solutions are found for the
partial differential equation that arises from $\alpha=2$ and $\beta=1$. Unlike
the above cases, the latter equation permits the presence of nodes in its
solutions.
