Mittag-Leffler Function: Generalizations and Applications

Dear Colleagues,

Since the seminal works of the Swedish mathematician Gösta Magnus Mittag-Leffler (1902–1905), where he introduced his famous function as a power series, many generalizations and applications of the Mittag-Leffler function have been published in the literature. This function provides the simplest non-trivial generalization of the exponential function. However, many other special functions have been introduced in the literature as generalizations of the Mittag-Leffler function.

Among the most important generalizations, we found the two-parametric Mittag-Leffler function (introduced and studied by Agarval and Humbert in 1953, and independently by Djrbashyan in 1954), the three parametric Mittag-Leffler function (introduced by Prabhakar in 1971), and other generalized Mittag-Leffler functions (introduced more recently by Kilbas and Saigo in 1995).

Among the most important applications, it is worth mentioning the solution of different types of integral equations in terms of the Mittag-Leffler function. Nevertheless, the most relevant applications come from the special role of the Mittag-Leffler function in Fractional Calculus (known as “The Queen Function of the Fractional Calculus”). Therefore, we found the Mittag-Leffler function in physical fractional models such as linear viscoelasticity, fractional Newton equations, fractional Ohm law, and fractional equations for heat transfer.

The main scope of this Special Issue is to publish recent research papers about new generalizations and applications of the Mittag-Leffler function, Also, papers describing the new analytic and asymptotic properties of this function, as well as its connection to other special functions in order to solve applied problems in Mathematics and Physical Sciences, are welcome. 

Prof. Dr. Juan Luis González-Santander
Guest Editor

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• the Mittag-Leffler function in fractional calculus
• the analytical properties of the Mittag-Leffler function
• the mittag-leffler function in physical models
• generalizations and extensions of the Mittag-Leffler function
• the mittag-leffler function’s connection to other special functions
• applications of the Mittag-Leffler function in engineering problems
• applications of the Mittag-Leffler function in mathematical problems

Review - The Mittag-Leffler Functions - the Special Functions Unexpectedly Modeling Complex Scientific and Technological Systems

Author(s) Prof. Dr. Juan Luis González-Santander

Abstract
At the beginning of the last century, the Swedish Mathematician G. M. Mittag-Leffler introduced a new power series to generalize the exponential function. During the last four decades, this special function, called after him Mittag-Leffler function, was generalized to a variety of special functions that found considerable attention in mathematical analysis, especially in the solution of integral equations and fractional differential equations. In parallel with the large number of mathematical investigations dedicated to the properties and generalizations of Mittag-Leffler functions, it is surprising to observe that these functions can adequately represent a wide spectrum of scientific and technical situations. In this study, the ability of the Mittag-Leffler function to describe various physical, chemical, medical, biological, technological and social processes is discussed. In theoretical and applied physics, applications of the Mittag-Leffler function include nonlinear mass and heat transfer phenomena in porous media; the relaxation behavior of mechanical, dielectric and magnetic systems, as well as viscoelastic materials; the flow of radiant gas; and the motion and hydrodynamic fluctuations of non-Newtonian fluids and ferrofluids. Likewise, anomalous and ultra-slow diffusion, chaos phenomena, statistical distribution functions and Brownian motion are usually represented by Mittag-Leffler functions. On the other hand, the use of these functions is illustrated in various cases, such as solar cells, transmission lines, energy storage systems, electrical sensors, low-pass filters, dynamic performance, traffic control and pricing problems. In chemistry, they are used to describe oil extraction processes, water purification and contamination, electrochemical reactions, as well as polymers, hydrogels, and synthetic fiber production. What is even more surprising is that Mittag-Leffler functions are applied for the simulation of various diseases and treatments (pregnancy, brain tumors, cancer spread, tuberculosis, kidney and dental diseases), as well as in fractional release kinetics of drugs and in epidemic models (Covid-19, HIV/AIDS and related vaccines). Further, they are applied in perfusion and blood flow, in magnetic resonance imaging (MRI), as well as in electrical impedance tomography (EIT). Mathematical modeling extends to the survival and growth of microbial cultures, such as viruses and bacterial infections. Finally, some less known and recently reported properties of the inverse Mittag-Leffler functions are also discussed in the Appendix.