The Wright functions of the second kind in Mathematical Physics

TL;DR

This review highlights the significance of Wright functions of the second kind in mathematical physics, especially their role as solutions to fractional diffusion-wave equations, aiding in modeling non-Gaussian stochastic processes.

Contribution

It emphasizes the analytical properties and applications of Wright functions of the second kind as fundamental solutions in fractional diffusion-wave equations.

Findings

Wright functions of the second kind serve as fundamental solutions to fractional diffusion-wave equations.

They provide a framework for describing non-Gaussian stochastic processes.

The paper discusses the transition from sub-diffusion to wave propagation using these functions.

Abstract

In this review paper we stress the importance of the higher transcendental Wright functions of the second kind in the framework of Mathematical Physics.We first start with the analytical properties of the classical Wright functions of which we distinguish two kinds. We then justify the relevance of the Wright functions of the second kind as fundamental solutions of the time-fractional diffusion-wave equations. Indeed, we think that this approach is the most accessible point of view for describing non-Gaussian stochastic processes and the transition from sub-diffusion processes to wave propagation. Through the sections of the text and suitable appendices we plan to address the reader in this pathway towards the applications of the Wright functions of the second kind. Keywords: Fractional Calculus, Wright Functions, Green's Functions, Diffusion-Wave Equation,