Showing papers by "Francesco Mainardi published in 1996"
TL;DR: In this article, the authors introduced fractional derivatives of order α in time, with 0 for relaxation, diffusion, oscillations, and wave propagation, and showed that they are governed by simple differential equations of order 1 and 2 in time.
Abstract: 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
925 citations
TL;DR: In this paper, it was shown that the fundamental solutions of the basic Cauchy and signalling problems can be expressed in terms of an auxiliary function M(z;β), where z = |x| t β is the similarity variable.
623 citations
TL;DR: In this article, it was shown that the fundamental solutions of the basic Cauchy and signalling problems can be expressed in terms of an auxiliary function M (z; β), where z is the similarity variable.
Abstract: The time fractional diffusion-wave equation is obtained from the classical diffusion or wave equation by replacing the first- or second-order time derivative by a fractional derivative of order 2β with 0<β≤1/2 or 1/2<β≤1, respectively. Using the method of the Laplace transform, it is shown that the fundamental solutions of the basic Cauchy and signalling problems can be expressed in terms of an auxiliary function M (z; β), where z is the similarity variable. Such function, which reduces to the well-known Gaussian function for β=1/2 (ordinary diffusion), is proved to be an entire function of Wright type.
103 citations