  The behaviour of the solutions of the time-fractional diffusion equation,
based on the Caputo derivative, is studied and its dependence on the fractional
exponent is analysed. The time-fractional convection-diffusion equation is also
solved and an application to Pennes bioheat model is presented. Generically, a
wave-like transport at short times passes over to a diffusion-like behaviour at
later times.
