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; f rom a mathematical point of view they are known to be governed by simp le differential equations of order 1 and 2 in time. The introduction o f fractional derivatives of order alpha in time, with 0 < alpha < 1 or 1 < alpha < 2, leads to processes that, in mathematical physics, we m ay refer to as fractional phenomena. The objective of this paper is to provide a general description of such phenomena adopting a mathematic al approach to the fractional calculus that is as simple as possible. The analysis carried out by the Laplace transform leads to certain spe cial functions in one variable, which generalize in a straightforward way the characteristic functions of the basic phenomena, namely the ex ponential and the gaussian. Copyright (C) 1996 Elsevier Science Ltd.