Mittag-Leffler Functions and Their Applications

H. J. Haubold; A. M. Mathai; R. K. Saxena

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Mittag-Leffler Functions and Their Applications

H. J. Haubold, A. M. Mathai, R. K. Saxena

arXiv:0909.0230v2Sep 1, 2009math.CAcond-mat.stat-mech

TL;DR

The paper addresses the need for a unified account of Mittag-Leffler functions, their generalizations, properties, and applications. It surveys definitions, representations, fractional-calculus relations, and applications, reporting solutions and structural results across several models. The paper also identifies corrections needed in previously reported statistical-distribution results.

Problem
The paper surveys a broad and fragmented literature on Mittag-Leffler functions and their applications across physical and applied sciences.
Method
The authors provide a unified survey of Mittag-Leffler functions, generalized and type functions, their properties, integral representations, fractional-calculus relations, and applications.
Results
The survey presents theorem-based solutions for kinetic, time-fractional diffusion, fractional-space diffusion, and reaction-diffusion equations, alongside generalized-function results.
Takeaways & Limitations
Mittag-Leffler functions provide representations and solutions used in fractional kinetic, diffusion, reaction-diffusion, and statistical-distribution settings.
Takeaways & Limitations
The paper states that earlier representations and logarithmic expected-value relations for Mittag-Leffler variables require correction.

Abstract

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Motivated essentially by the success of the applications of the Mittag-Leffler functions in many areas of science and engineering, the authors present in a unified manner, a detailed account or rather a brief survey of the Mittag- Leffler function, generalized Mittag-Leffler functions, Mittag-Leffler type functions, and their interesting and useful properties. Applications of Mittag-Leffler functions in certain areas of physical and applied sciences are also demonstrated. During the last two decades this function has come into prominence after about nine decades of its discovery by a Swedish mathematician G.M. Mittag-Leffler, due its vast potential of its applications in solving the problems of physical, biological, engineering and earth sciences etc. In this survey paper, nearly all types of Mittag-Leffler type functions existing in the literature are presented. An attempt is made to present nearly an exhaustive list of references concerning the Mittag-Leffler functions to make the reader familiar with the present trend of research in Mittag-Leffler type functions and their applications.

1. Introduction

The section introduces Mittag-Leffler functions, their historical development, special cases, and relevance to fractional-calculus applications. It also presents integral-transform representations and related properties.

Mittag-Leffler functions arise naturally in fractional integral and differential equations, including models of kinetics, random walks, Lévy flights, super-diffusive transport, and complex systems.
Their ordinary and generalized forms interpolate between purely exponential and power-law-like behavior in ordinary and fractional kinetic phenomena.
The functions have broad applications in fluid flow, rheology, diffusive transport, electric networks, probability, and statistical distribution theory.
Special cases include rational-order functions, incomplete-gamma representations, error-function forms, and explicit formulas for half-integer orders.
Integral and inverse-Laplace representations connect Mittag-Leffler functions with Lévy one-sided stable distributions and facilitate analysis of Eα(−x).

5. Recurrence relations

This section develops derivative and recurrence formulas for Mittag-Leffler functions and connects their asymptotic behavior to fractional physical models. The results are stated under explicit parameter and sector conditions.

Series-based identities provide recurrence and derivative relations for Eα,β(z), with higher-order derivative formulas obtained under ℜ(α) > 0 and ℜ(β) > 0.
Corollaries specialize the derivative theorem to r = 2, 3, 4 under the same positivity conditions on ℜ(α) and ℜ(β).
The asymptotic expansion of Eα(z) is derived from an integral representation and is organized by α-dependent angular sectors as |z| →∞.
The asymptotic behavior is relevant to interpreting fractional reaction, relaxation, diffusion, and reaction-diffusion problems in complex systems.
Separate asymptotic expansions are given for 0 < α < 2 and α ≥ 2, with the sector condition µ ≤ |arg z| ≤ π stated for the former cases.

7. Integral representations

The section collects integral representations for Mittag-Leffler functions and relates them to beta/gamma methods, H-functions, generalized Wright functions, and fractional-diffusion problems.

Several integrals associated with Mittag-Leffler functions are established using beta and gamma function formulas and related techniques.
The listed representations include formulas under conditions such as |z| < 1 and ℜ(α) > 0, with additional forms involving α, β, s, and a.
Equation (7.7) can be used to compute numerical coefficients of the leading term in the asymptotic expansion of Eα(−x).
The H-function and its special cases: The H-function is defined by a Mellin-Barnes integral whose contour separates specified gamma-function poles under stated validity conditions.
The H-function and its special cases: The generalized Wright function contains the Mittag-Leffler functions and reduces to a generalized hypergeometric function when Ai = Bj = 1.
The H-function and its special cases: Special cases of the generalized Wright function occur widely in fractional-diffusion problems and connect with H-function representations.

9. Mellin-Barnes integrals for Mittag-Leffler functions

This section presents Mellin-Barnes and related integral representations for Mittag-Leffler and generalized Mittag-Leffler functions. It also states fractional-integral and derivative relations, Laplace-transform formulas, and multivariate extensions.

Mellin-Barnes representations are obtained by contours separating the poles of Γ(s) and Γ(1 − s), yielding analytic continuation formulas.
Relations with left- and right-sided Riemann-Liouville fractional operators are stated through theorems for integrals and derivatives under explicit restrictions on α and β.
The Prabhakar generalization is introduced through a series representation and is connected to Laplace-transform formulas under positivity and convergence conditions.
Operational methods use generalized Mittag-Leffler functions to obtain closed-form solutions for certain fractional differential and integral equations.
The generalized functions have Mellin-Barnes and Laplace-transform representations subject to contour, pole-separation, and real-part conditions.
Further generalizations include multivariate Mittag-Leffler functions and the Kilbas-Saigo function, with special cases reducing to Mittag-Leffler forms.

12. Laplace and Fourier transforms, fractional calculus operators

This section defines Laplace and Fourier transforms and fractional-calculus operators used to derive solutions of fractional equations. It also introduces transform identities and operator conventions needed for applications to physical problems.

Transforms: Laplace and Fourier transforms are defined for functions N(x,t) and their inverses to support derivations of fractional differential-equation solutions.The section specifies transforms with respect to time and space, alongside inverse formulas.
Transform identities: The section states Laplace-transform formulas for H-functions under explicit real-part and positivity conditions.These identities are used with H-function representations in subsequent derivations.
Fractional calculus operators: Riemann-Liouville fractional integrals and derivatives are defined, including their action on power functions under stated restrictions.For f(t)=t^ρ, the resulting expressions involve Gamma functions and shifted powers of t.
Fractional calculus operators: Caputo and Weyl fractional operators are introduced with corresponding Laplace- or Fourier-transform relations.The Riemann-Liouville derivative of unity is explicitly noted to be nonzero, while a modified Fourier-space convention suppresses the imaginary unit.
Applications: These definitions prepare the analysis of Mittag-Leffler applications in kinetic equations and related physical problems.The following section derives solutions of two kinetic equations.

13. Application in kinetic equations

This section applies operational and transform methods to fractional kinetic integral and differential equations. The resulting solutions are expressed using Mittag-Leffler and generalized Mittag-Leffler functions, including a general formula for integrable forcing.

Fractional kinetic equations: Theorem 13.1 gives a solution of a fractional integral equation under the condition ℜ(ν)>0.Its proof uses operator manipulations, summation over integer orders, and rewriting the resulting series as a Mittag-Leffler function.
Fractional kinetic equations: The solution of the associated fractional diffusion equation is also given by the Mittag-Leffler expression in (13.6).The equation is obtained by applying a fractional operator to the integral equation.
Related methods: The section situates these results within prior operational methods for closed-form solutions of fractional differential and integral equations.Earlier work obtained such solutions in terms of generalized Mittag-Leffler functions, while alternate derivations are cited for the presented theorems.
Generalized Mittag-Leffler solutions: Theorem 13.2 gives a solution of another integral equation when min{ℜ(ν),ℜ(µ)}>0, expressed through the generalized Mittag-Leffler function.Laplace-transform arguments and an alternate operator derivation both lead to the stated result.
General integral-equation result: Theorem 13.3 establishes a formula for an integral equation with c>0, ℜ(ν)>0, and an integrable function f(t) on [0,b].The derivation uses Laplace transforms, H-function representations, and the convolution property.

14. Application to time-fractional diffusion

This section derives the fundamental solution of a time-fractional diffusion equation using joint Laplace-Fourier transforms. The solution is represented through the Mittag-Leffler function and reduces to the Gaussian density when α=1.

Derivation: The fundamental solution is derived by applying the Laplace transform in time and Fourier transform in space.The transformed equation is s^α Ñ*(k,s)−s^(α−1)=−Dk^2 Ñ*(k,s).
Solution: The inverse Laplace transform introduces E_α, and subsequent Fourier inversion yields the required closed-form solution.The derivation uses an integral identity to complete the Fourier inversion.
Classical limit: For α=1, the resulting expression reduces to the Gaussian density.This identifies the classical diffusion case within the fractional model.

15. Application to fractional-space diffusion

This section treats fractional-space diffusion and reaction-diffusion models with delta initial conditions. Their fundamental or closed-form solutions are expressed using the paper’s fractional operators and special-function framework.

Fractional-space diffusion: Theorem 15.1 states that the fractional-space diffusion model has a fundamental solution under the specified initial and boundary conditions.The proof follows lines similar to the preceding time-fractional diffusion theorem.
Reaction-diffusion model: Theorem 16.1 gives a formula for the solution of the fractional reaction-diffusion model under these conditions.A corresponding corollary provides a formula for a fractional reaction-diffusion equation with delta initial condition.
Classical and stable-law limits: The closed-form representation in (16.4) is a Lévy stable law and approaches the classical Gaussian solution as α→2.The paper explicitly identifies this limiting behavior for the fractional reaction-diffusion solution.

17. Application to nonlinear waves

This section derives a solution for a nonlinear fractional reaction-diffusion equation using inverse Laplace and Fourier transforms, with generalized Mittag-Leffler functions appearing in the resulting expression.

Theorem 17.1: The section establishes a theorem giving a solution formula for a fractional reaction-diffusion equation with nonlinear reaction term φ(x,t).The equation is posed for x ∈ R and t > 0, with fractional orders 0 ≤ α, β ≤ 1 and specified initial and boundary conditions.
Transform method: The solution framework uses Laplace transforms in time and Fourier transforms in space to transform the governing equation.The proof applies the Laplace transform first, then the Fourier transform, before performing the inverse operations.
Transform method: The transformed equation contains fractional powers of the Laplace parameter and spatial-frequency terms involving diffusion and nonlinear coefficients.The transformed relation includes s^α, as^β, −ν^2|k|^γ, and ζ^2 terms.
Inverse transforms: The generalized Mittag-Leffler function enters the solution through inversion of the transformed algebraic expression.The parameter b is defined as b = ν^2|k|^γ − ζ^2, and the generalized function is identified as the one defined earlier in the paper.
Inverse transforms: The final inverse Fourier transform produces the desired solution of the nonlinear-wave problem.The Laplace inversion uses a previously derived result and requires the stated parameter conditions and convergence of the series.

18. Generalized Mittag-Leffler type functions

The section surveys multiindex and other generalized Mittag-Leffler functions, emphasizing their entire-function properties, fractional-calculus relations, and connections with established special functions.

Multiindex functions: The multiindex Mittag-Leffler function is defined by a power series with multiple index pairs and arbitrary real parameters under stated positivity conditions.For m > 1, the parameters ρ1, ..., ρm and μ1, ..., μm are arbitrary real parameters.
Multiindex functions: The multiindex function is an entire function whose order and asymptotic estimate are given for arbitrary admissible index sets.The asymptotic estimate holds for sufficiently large |z| and every positive ε.
Special cases: For m = 2, the multiindex function reduces to Dzherbashyan’s generalized Mittag-Leffler function.The resulting two-index function is denoted φρ1,ρ2(z; μ1, μ2) and is also entire.
Connections and applications: Relations with H-functions, generalized Wright functions, Riemann-Liouville operators, and other special functions yield broader fractional-calculus results.The cited results generate known and previously unknown relations in generalized Mittag-Leffler-function theory.
Fractional-calculus relations: Left- and right-sided Riemann-Liouville fractional integrals satisfy relations involving the multiindex Mittag-Leffler function.These relations are stated in Theorems 18.2 and 18.3 under positive parameter conditions.
Other generalizations: The M-series is presented as another generalization, but it is identified as a special case of the generalized Wright function rather than a new special function.Fractional integration and differentiation results for the M-series are also reproduced.

19. Mittag-Leffler Distributions and Processes

This section develops Mittag-Leffler distributions through Laplace transforms, product representations, Mellin-Barnes integrals, and extensions of gamma and exponential distributions.

Statistical distribution: A product of independent random variables involving a Lévy variable has a Mittag-Leffler distribution characterized by its Laplace transform.The theorem states that if y is Lévy and x is independent, then u = yx^α has the Mittag-Leffler distribution.
Statistical distribution: The distributional result extends beyond continuous variables because it requires only existence of the relevant expected values.The result applies to continuous, discrete, or mixed random variables.
Statistical distribution: Statistical independence is sufficient but not necessary when the conditional Laplace transform has the required form and the marginal transform is ψ(t).The paper explicitly states that independence is not a basic requirement for the result.
Corrections: Several cited representations and logarithmic-moment formulas are identified as needing correction.The paper specifically flags an interchange of exponential and Lévy variables in earlier representations.
Density representations: Mellin-Barnes representations provide moments and connect Mittag-Leffler densities with H-functions and gamma-type densities.The generalized Mittag-Leffler density extends a gamma density, reducing to the exponential density when η = 1.
Density representations: The Mittag-Leffler density is non-negative and normalized as a density, although normalization is not directly evident from its series representation.The paper establishes the density property through its integral representation rather than the series alone.
Structural representations: A generalized Mittag-Leffler variable with Laplace transform (1+δt^α)^−β is represented through independent Lévy and gamma variables.The paper also describes a pathway from the generalized Mittag-Leffler density to a positive Lévy density by varying the relevant parameter.

20. Mittag-Leffler Stochastic Processes

The section introduces Mittag-Leffler stochastic processes through Laplace transforms and relates them to autoregressive processes and applications in fractional physical systems.

Process definition: A stochastic process with stationary independent increments and a Mittag-Leffler-distributed value at t = 1 is called a Mittag-Leffler stochastic process.Its value at t = 1 has the Laplace transform specified earlier in the section.
Process properties: The Laplace transform, density, and distribution function of x(t) are derived for the process.The section presents these quantities as consequences of the process definition.
Autoregressive connections: The process construction is connected to sequences of independent and identically distributed innovations through their Laplace transforms.The representation leads to autoregressive settings based on class L distributions.
Autoregressive connections: The resulting autoregressive cases include the first-order exponential autoregressive process and the first-order Mittag-Leffler autoregressive process.These are denoted EAR(1) and MLAR(1), respectively.
Applications: Mittag-Leffler functions are presented as useful for modeling deviations from exponential behavior in physical phenomena.The paper highlights applications in stochastic systems, dynamical systems, disordered systems, reaction, diffusion, and reaction-diffusion.