Time-fractional diffusion equation and its applications in physics

In physics, process involving the phenomena of diffusion and wave propagation have great relevance; these physical process are governed, from a mathematical point of view, by differential equations of order 1 and 2 in time. By introducing a fractional derivatives of order $\alpha$ in time, with $0 < \alpha < 1$ or $1 <= \alpha <= 2$, we lead to process in mathematical physics which we may refer to as fractional phenomena; this is not merely a phenomenological procedure providing an additional fit parameter. The aim of this thesis is to provide a description of such phenomena adopting a mathematical approach to the fractional calculus. The use of Fourier-Laplace transform in the analysis of the problem leads to certain special functions, scilicet transcendental functions of the Wright type, nowadays known as M-Wright functions. We will distinguish slow-diffusion processes ($0 < \alpha < 1$) from intermediate processes ($1 <=\alpha <= 2$), and we point out the attention to the applications of fractional calculus in certain problems of physical interest, such as the Neuronal Cable Theory.