[Paper Review] Analytical properties and applications of the Wright function

Rudolf Gorenflo, Yuri Luchko+1 more|ArXiv.org|Jan 28, 2007

Fractional Differential Equations Solutions7 references226 citations

TL;DR

This paper provides a comprehensive survey of the Wright function's analytical properties and its central role in fractional partial differential equations (FPDEs), particularly in representing Green's functions for time-fractional diffusion-wave equations. It establishes that the Wright function is an entire function of completely regular growth for ρ > -1 and demonstrates its use in constructing scale-invariant solutions via Lie group symmetry methods, with explicit representations in terms of generalized Wright functions and special functions.

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.

Motivation & Objective

• To systematize and present the analytical properties of the Wright function φ(ρ, β; z) for ρ > -1 and β ∈ ℂ.
• To establish the Wright function’s role in solving boundary-value problems for fractional diffusion-wave equations.
• To extend Lie group methods to fractional PDEs by deriving group-invariant solutions in terms of the Wright and generalized Wright functions.
• To analyze the distribution of zeros, order, type, and indicator function of the Wright function, proving it is of completely regular growth.
• To unify applications across fractional calculus, integral transforms, and physical modeling through the Wright function framework.

Proposed method

• Derives integral representations and Laplace transform pairs involving the Wright function using classical results from Wright and others.
• Applies asymptotic analysis and special function theory to characterize the Wright function’s behavior, including its hypergeometric-type representations.
• Uses the Erdélyi-Kober fractional integral and differential operators to reduce time- and space-fractional PDEs to ODEs solvable via the Wright function.
• Applies scaling symmetry (group invariance) to transform the fractional PDE into a self-similar form, leading to solutions in terms of φ(ρ, β; z).
• Employs generalized hypergeometric functions and the generalized Wright function pΨq to express solutions for general α, β in the time- and space-fractional diffusion equation.
• Proves that the Wright function is an entire function of completely regular growth by analyzing its order, type, and indicator function.

Experimental results

Research questions

• RQ1 What are the complete analytical properties of the Wright function φ(ρ, β; z) for ρ > -1, including its asymptotics, zeros, and growth characteristics?
• RQ2 How can the Wright function be used to construct exact solutions—particularly Green’s functions—for fractional diffusion-wave equations?
• RQ3 In what way do Lie group symmetry methods extend to fractional PDEs, and how are the invariant solutions expressed via the Wright function?
• RQ4 What is the distribution of zeros of the Wright function, and how does this relate to its order, type, and indicator function?
• RQ5 How do generalized Wright functions and special functions of hypergeometric type emerge in the solutions of time- and space-fractional PDEs?

Key findings

• The Wright function φ(ρ, β; z) is an entire function of completely regular growth for every ρ > -1.
• The Green’s function for the time-fractional diffusion-wave equation with 0 < α ≤ 2 is expressible in terms of the Wright function φ(−α/2, 1; z).
• Scale-invariant solutions of the time- and space-fractional PDE ∂t^α u = D ∂x^β u are given by u(x,t) = t^γ ∑ C_j v_j(x t^{-α/β}), where v_j are expressed via the generalized Wright function.
• For β = 2 and 1 < α < 2, the solutions reduce to combinations of φ(−α/2, 1 + γ; ±y/√D), explicitly showing the Wright function’s role in symmetric solutions.
• In the case β = 1, the scale-invariant solution is u(x,t) = t^γ φ(−α, 1 + γ; x t^{-α}/D), confirming the Wright function’s direct applicability in first-order space-fractional models.
• The distribution of zeros of the Wright function is shown to be regular, supporting its classification as a function of completely regular growth.