Mittag-Leffler Functions and Their Applications

By Hans J. Haubold, Arak M. Mathai, R. K. Saxena. Journal of Applied Mathematics, 2009

1,010 citations recorded by the source corpus.

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 G. M. 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 Mittag-Leffler, due to the vast potential of its applications in solving the problems of physical, biological, engineering, and earth sciences, and so forth. 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.

Subfields: Modeling and Simulation, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/448c58a4cbd342f923632bdd8767d42d592c0cbf

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Citations in the corpus

A Note on Fractional Integral Operator Associated with Multiindex Mittag-Leffler Functions (2016, background): “1) can be found in the review article [6] (see also [5]).”
Comparing Fréchet and positive stable laws (2013, background): “Since its introduction in [16] it has been the study of many studies - see Chapter XVIII in [9] for classical properties and [11] for a recent account with updated references.”
Multi-parametric mittag-leffler functions and their extension (2013, methodology): “Solutions of fractional differential equations by the operational methods are also obtained in terms of certain multivariate Mittag-Leffler functions of n complex variables z1, . . . , zn, defined as (see, e.g., Luchko [37], KilbasSrivastava-Trujillo [30], Haubold-Mathai-Saxena [18], etc.): Ea1,...,”
Hermite‐Hadamard inequalities in fractional calculus defined using Mittag‐Leffler kernels (2020, methodology): “Using the Riemann-Liouville fractional integrals (17) and the series formula (20), it follows that”
Some New Fractional-Calculus Connections between Mittag–Leffler Functions (2019, background): “1), [12]) but without reference to fractional calculus.”
The Mittag-Leffler function (2020, background): “The Mittag-Leffler function Ea,b (z) naturally arises in fractional calculus and in the solution of fractional order integral or differential equations, as illustrated in [22] and [16]; for example, in the fractional generalization of the heat equation, random walks, Lévy flights, superdiffusive tra”
Fractional Skellam processes with applications to finance (2014, background): “where (γ)r = Γ(γ + r)/Γ(γ) whenever the Gamma function Γ is defined, and (γ)0 = 1 for γ = 0, see [13].”
The characteristic function of rough Heston models (2016, positive): “The following properties of fα,λ and Fα,λ can be found in [21, 34, 36].”
Models of dielectric relaxation based on completely monotone functions (2016, background): “Important results on the asymptotic behaviour of the standard (one or two parameter) ML function are largely available in the literature (see, for instance, [35, 47, 79, 80, 96]).”
Global Padé Approximations of the Generalized Mittag-Leffler Function and its Inverse (2013, background): “Later on, some studies [5, 4, 12, 17, 13, 22, 10] contributed to several properties and applications of the (generalized) Mittag-Leffler functions.”
Perfect hedging in rough Heston models (2017, neutral): “The following properties of fα,λ and Fα,λ can be found in [14, 17, 18].”
Fractional differential equations: a novel study of local and global solutions in Banach spaces (2013, neutral): “For more information, see the literature [12, 36, 37, 54].”