  We address the now classical problem of a diffusion process that crosses over
from a ballistic behavior at short times to a fractional diffusion (sub- or
super-diffusion) at longer times. Using the standard non-Markovian diffusion
equation we demonstrate how to choose the memory kernel to exactly respect the
two different asymptotics of the diffusion process. Having done so we solve for
the probability distribution function (pdf) as a continuous function which
evolves inside a ballistically expanding domain. This general solution agrees
for long times with the pdf obtained within the continuous random walk approach
but it is much superior to this solution at shorter times where the effect of
the ballistic regime is crucial.
