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.
Highlights
The special functions play a fundamental role in all fields of Applied Mathematics and Mathematical Physics because any analytical results are expressed in terms of some of these functions
The Gaussian function must be generalized in a suitable way in the framework of partial differential equations of non-integer order for describing the anomalous diffusion and the transition from fractional diffusion to wave propagation. Their usefulness and meaningfulness extends to other topics. These functions and their Laplace Transforms can be applied in electromagnetic problems, see the 1958 paper by Ragab [1] and the recent 2020 paper by Stefanski and Gulgowski [2]
The use of the Wright functions of the second kind in time fractional diffusion-wave equations has appeared in several papers for a variety of different purposes, see, e.g., Bazhlekova [27], D’Ovidio [28], Gorenflo, Luchko and Mainardi [29], Mentrelli and Pagnini [30], Mosley and Ansari [31], Pagnini [32], Povstenko [33], and references therein
Summary
The special functions play a fundamental role in all fields of Applied Mathematics and Mathematical Physics because any analytical results are expressed in terms of some of these functions. The Gaussian function (known as the normal probability distribution) must be generalized in a suitable way in the framework of partial differential equations of non-integer order for describing the anomalous diffusion and the transition from fractional diffusion to wave propagation Their usefulness and meaningfulness extends to other topics. This analysis leads to generalizing the known results r of the standard diffusion equation in the one-dimensional case that is recalled in Appendix A by means of auxiliary functions as particular cases of the Wright functions of the second kind known as M-Wright or Mainardi functions.
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