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Mittag-Leffler Functions, Related Topics and Applications

verfasst von: Prof. Rudolf Gorenflo, Prof. Anatoly A. Kilbas, Prof. Francesco Mainardi, Prof. Sergei Rogosin

Verlag: Springer Berlin Heidelberg

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The 2nd edition of this book is essentially an extended version of the 1st and provides a very sound overview of the most important special functions of Fractional Calculus. It has been updated with material from many recent papers and includes several surveys of important results known before the publication of the 1st edition, but not covered there.

As a result of researchers’ and scientists’ increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have caught the interest of the scientific community. Focusing on the theory of Mittag-Leffler functions, this volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular, the Mittag-Leffler functions make it possible to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and related special functions.

The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, control theory and several other related areas.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

This book is devoted to an extended description of the properties of the Mittag-Leffler function, its numerous generalizations and their applications in different areas of modern science.

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Chapter 2. Historical Overview of the Mittag-Leffler Functions

Abstract

Gösta Magnus Mittag-Leffler was born on March 16, 1846, in Stockholm, Sweden. His father, John Olof Leffler, was a school teacher, and was also elected as a member of the Swedish Parliament.

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Chapter 3. The Classical Mittag-Leffler Function

Abstract

In this chapter we present the basic properties of the classical Mittag-Leffler function \(E_\alpha (z)\) (see (1.0.1)). The material can be formally divided into two parts. Starting from the basic definition of the Mittag-Leffler function in terms of a power series, we discover that for parameter \(\alpha \) with positive real part the function \(E_\alpha (z)\) is an entire function of the complex variable z. Therefore we discuss in the first part the (analytic) properties of the Mittag-Leffler function as an entire function. Namely, we calculate its order and type, present a number of formulas relating it to elementary and special functions as well as recurrence relations and differential formulas, introduce some useful integral representations and discuss the asymptotics and distribution of zeros of the classical Mittag-Leffler function.

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Chapter 4. The Two-Parametric Mittag-Leffler Function

Abstract

In this chapter we present the basic properties of the two-parametric Mittag-Leffler function \(E_{\alpha ,\beta } (z)\) (see (1.0.3)), which is the most straightforward generalization of the classical Mittag-Leffler function \(E_{\alpha } (z)\) (see (3.1.1)). As in the previous chapter, the material can be formally divided into two parts. Starting from the basic definition of the Mittag-Leffler function as a power series, we discover that, for the first parameter \(\alpha \) with positive real part and any complex value of the second parameter \(\beta \), the function \(E_{\alpha ,\beta } (z)\) is an entire function of the complex variable z.

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Chapter 5. Mittag-Leffler Functions with Three Parameters

Abstract

The Prabhakar generalized Mittag-Leffler function [Pra71] is defined as where \((\gamma )_n = \gamma (\gamma +1)\ldots (\gamma +n-1)\) (see formula (A.17) in Appendix A).

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Chapter 6. Multi-index and Multi-variable Mittag-Leffler Functions

Abstract

Consider the function defined for \(\alpha _1,\ \alpha _2\in {\mathbb R}\) \((\alpha _1^2+\alpha _2^2\ne 0)\) and \(\beta _1,\beta _2 \in {\mathbb C}\) by the series

$$\begin{aligned} E_{\alpha _1,\beta _1;\alpha _2,\beta _2}(z)\equiv \sum ^{\infty }_{k=0}\frac{z^k}{\varGamma (\alpha _1k+\beta _1) \varGamma (\alpha _2k+\beta _2)}\ \ (z\in {\mathbb C}). \end{aligned}$$

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Chapter 7. The Classical Wright Function

Abstract

This chapter deals with the classical Wright function. Like the functions of Mittag-Leffler type, the functions of Wright type are known to play fundamental roles in various applications of the fractional calculus. This is mainly due to the fact that they are interrelated with the Mittag-Leffler functions through Laplace and Fourier transformations.

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Chapter 8. Applications to Fractional Order Equations

Abstract

In this chapter we consider a number of integral equations and differential equations (mainly of fractional order). In representations of their solution, the Mittag-Leffler function, its generalizations and some closely related functions are used.

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Chapter 9. Applications to Deterministic Models

Abstract

Here we present material illuminating the role of the Mittag-Leffler function and its generalizations in the study of deterministic models. It has already been mentioned that the Mittag-Leffler function is closely related to the Fractional Calculus (being called ‘The Queen Function of the Fractional Calculus’). This is why we focus our attention here on fractional (deterministic) models. We start with a technical Sect. 9.1 in which the fractional differential equations, related to the fractional relaxation and oscillation phenomena, are discussed in full detail.

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Chapter 10. Applications to Stochastic Models

Abstract

This chapter is devoted to the application of the Mittag-Leffler function and related special functions in the study of certain stochastic processes. As this topic is so wide, we restrict our attention to some basic ideas. For more complete presentations of the discussed phenomena we refer to some recent books and original papers which are mentioned in Sect. 10.6.

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei Rogosin

Backmatter

Titel: Mittag-Leffler Functions, Related Topics and Applications
verfasst von: Prof. Rudolf Gorenflo
Prof. Anatoly A. Kilbas
Prof. Francesco Mainardi
Prof. Sergei Rogosin
Copyright-Jahr: 2020
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-61550-8
Print ISBN: 978-3-662-61549-2
DOI: https://doi.org/10.1007/978-3-662-61550-8

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