Fractional diffusion equations interpolate between damping and waves

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.