On linear differential equations of fractional order

The processes involving 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 ct in time, with 0 < alpha < 1 or 1 < alpha < 2, leads to processes that, in mathematical physics, we may refer to as fractional phenomena. Our analysis, carried out by the Laplace transform, leads to certain special functions in one variable, the Mittag-Leffler and the Wright functions, which generalize in a straightforward way the characteristic functions of the basic phenomena, namely the exponential and the gaussian.