  In physics, phenomena of diffusion and wave propagation have great relevance;
these physical processes are governed in the simplest cases by partial
differential equations of order 1 and 2 in time, respectively. By replacing the
time derivatives in the above standard equations with pseudo-differential
operators interpreted as derivatives of non integer order (nowadays misnamed as
of fractional order) we are lead to generalized processes of diffusion that may
be interpreted as slow diffusion and interpolating between diffusion and wave
propagation. In mathematical physics we may refer these interpolating processes
to as fractional diffusion-wave phenomena. In this work we analyze and simulate
both the situations in which the input function is a Dirac delta generalized
function and a box function, restricting ourselves to the Cauchy problem. In
the firsrst case we get the fundamental solutions (or Green functions) of the
problem whereas in the latter case the solutions are obtained by a space
convolution of the Green function with the input function. In order to clarify
the matter for the non-specialist readers, we briefly recall the basic and
essential notions of the fractional calculus (the mathematical theory that
regards the integration and differentiation of noninteger order) with a look at
the history of this discipline.
