Analytical properties and applications of the Wright function

By Rudolf Gorenflo, Yuri Luchko, Francesco Mainardi. 2007

293 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/18b083373d6efad9d951f1776e160b4d75244e4d

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In Applied Mathematics

In Modeling and Simulation

In Statistical and Nonlinear Physics

Citations in the corpus

A Note on Hadamard Fractional Differential Equations with Varying Coefficients and Their Applications in Probability (2017, background): “Recall that, in the general form, the Wright function is defined as follows (we refer to [31] for a good survey):”
Semilinear Caputo time-fractional pseudo-parabolic equations (2021, unknown): “The following equality holds (for proof, see [26]) Eα,1(−z) = ∫ ∞”
An Explicit Finite Difference Method and a New von Neumann-Type Stability Analysis for Fractional Diffusion Equations (2003, unknown): “Some exact analytical solutions for a few cases, although important, are already known in terms of special functions such as the Wright function and Fox’s Hfunction [6, 7, 15, 16].”
The fundamental solution of the space-time fractional diffusion equation (2007, unknown): “We have, see e.g. [39], tD βf(t) := dm dtm [ 1 Γ(m− β) ∫ t 0 f(τ) dτ (t− τ)β+1−m ] , m− 1 β m, dm dtm f(t) , β = m. (2.14) When the order is not integer, Gorenflo and Mainardi have shown the following relationships between the two fractional derivatives (when both of them exist), see e.g. [21”
The Fox–Wright functions and Laguerre multiplier sequences (2006, background): “The specific entire functions we will consider in the sequel are memb the class of generalized Fox–Wright functions which recently have played an increa significant role in various types of applications (see [12,13,36,37]).”
On fractional powers of generators of fractional resolvent families (2010, background): “For more properties of the Mittag-Leffler function and function of Wright type, we refer to [12, 13].”
Time-fractional Diffusion of Distributed Order (2007, background): “Before considering the general case of time-fractional diffusion of distributed order, we prefer to briefly recall the determination of the fundamental solution u(x, t) (common for both the R-L and C forms) for the single order case.”
Attractivity for fractional evolution equations with almost sectorial operators (2018, background): “Called to be “Wright-type”, the function Mα(θ) can be defined through the Wright function (see [7, 15])”
A tutorial on inverse problems for anomalous diffusion processes (2015, unknown): “This is an entire function of order 1 / (1 + ρ ) [30, Theorem 2.4.1].”