Fractional relaxation-oscillation and fractional diffusion-wave phenomena

The processes involving the basic phenomena of relaxation, diffusion, oscillations and wave propagation are of great relevance in physics; from a mathematical point of view they are known to be governed by simple differential equations of order 1 and 2 in time. The introduction of fractional derivatives of order α in time, with 0 < α < 1 or 1 < α < 2, leads to processes that, in mathematical physics, we may refer to as fractional phenomena. The objective of this paper is to provide a general description of such phenomena adopting a mathematical approach to the fractional calculus that is as simple as possible. The analysis carried out by the Laplace transform leads to certain special functions in one variable, which generalize in a straightforward way the characteristic functions of the basic phenomena, namely the exponential and the gaussian.