Analytical properties and applications of the Wright function

TL;DR

This survey explores the Wright function's fundamental properties and its significant applications in fractional differential equations, including solutions and zero distribution analysis.

Contribution

It highlights the Wright function's role in representing solutions to fractional PDEs and extends Lie group methods to these equations, providing new insights.

Findings

Green function of fractional diffusion-wave equation expressed via Wright function

Group-invariant solutions of fractional PDEs derived using Wright functions

Recent results on zeros distribution, order, and type of the Wright function

Abstract

In this survey paper we consider some applications of the Wright function with special emphasis of its key role in the partial differential equations of fractional order. It was found that the Green function of the time-fractional diffusion-wave equation can be represented in terms of the Wright function. Furthermore, extending the methods of Lie groups in partial differential equations to the partial differential equations of fractional order it was shown that some of the group-invariant solutions of these equations can be given in terms of the Wright and the generalized Wright functions.Finally, we discuss recent results about distribution of zeros of the Wright function, its order, type and indicator function.