Mittag-Leffler Functions and Their Applications

Abstract

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.

Short note on the emergence of fractional kinetics

“…In the framework of fractional kinetics, when the survival probability Ψ(t) is of the Mittag-Leffler type (4), the equation Ψ(s) = s β−1 /(s β + 1) holds and from (12) [19].…”

Design of longitudinal finned tubes adjusted with the Cesàro curve

“…The analysis was carried out by finite elements, where it was necessary to enter and adjust the internal surface at a temperature of 200 °C, assigning to the copper tube a coefficient of thermal conductivity of 400 W/(M * °C), then a room temperature (22 °C) was assigned with a thermal convection coefficient of 5 W/(M 2 * °C). The development of CAE analysis tools, from logistical models, can be estimated with the so-called fractional or fractional model [25], of the type MittagLeffler generalized, developed from the equation (2), where .…”

The Role of the Mittag-Leffler Function in Fractional Modeling

“…Different aspects of the distinguished role of this function in fractional theory and its applications have been described in several monographs and surveys on fractional calculus (see, e.g., [2][3][4][5][6][7][8][9][10]) and fractional modeling (see, e.g., [11][12][13][14][15][16][17][18]). The paper is organized as follows.…”