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2014 | Buch

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Mittag-Leffler Functions, Related Topics and Applications

Theory and Applications

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

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 recently caught the interest of the scientific community. Focusing on the theory of the Mittag-Leffler functions, the present 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 the applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular the Mittag-Leffler functions allow us to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and its successors. 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, theory of control and several other related areas.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

The 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 V. 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. His mother, Gustava Vilhelmina Mittag, was a daughter of a pastor, who was a person of great scientific abilities. At his birth Gösta was given the name Leffler and later (when he was a student) he added his mother’s name “Mittag” as a tribute to this family, which was very important in Sweden in the nineteenth century. Both sides of his family were of German origin.

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

Chapter 3. The Classical Mittag-Leffler Function

Abstract

In this chapter we present the basic properties of the classical Mittag-Leffler function E _α(z) (see (1.0.1)). The material can be formally divided into two parts.

Rudolf Gorenflo, Anatoly A. Kilbas, Francesco Mainardi, Sergei V. 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 _α, β(z) (see (1.0.3)), which is the most straightforward generalization of the classical Mittag-Leffler function E _α(z) (see (3.1.1)).

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

Chapter 5. Mittag-Leffler Functions with Three Parameters

Abstract

The Prabhakar generalized Mittag-Leffler function [Pra71] is defined as

$$\displaystyle{ E_{\alpha,\beta }^{\gamma }(z):=\sum _{ n=0}^{\infty } \frac{(\gamma )_{n}} {n!\varGamma (\alpha n+\beta )}\,z^{n}\,,\quad Re\,(\alpha ) > 0,\,Re\,(\beta ) > 0,\,\gamma > 0, }$$

(5.1.1)

where (γ)_n = γ(γ + 1)…(γ + n − 1) (see formula (A.1.17)).

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

Chapter 6. Multi-index Mittag-Leffler Functions

Abstract

Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α ₁ ² +α ₂ ² ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α ₁ > 0, α ₂ > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

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

Chapter 7. 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 V. Rogosin

Chapter 8. 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 to fractional (deterministic) models. We start with a technical Sect. 8.1 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 V. Rogosin

Chapter 9. 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. 9.6.

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

Backmatter

Titel: Mittag-Leffler Functions, Related Topics and Applications
verfasst von: Rudolf Gorenflo
Anatoly A. Kilbas
Francesco Mainardi
Sergei V. Rogosin
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-662-43930-2
Print ISBN: 978-3-662-43929-6
DOI: https://doi.org/10.1007/978-3-662-43930-2