We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

Fractional Calculus: Integral and Differential Equations of Fractional Order

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].

Multi-index Mittag-Leffler Functions

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.

https://downloads.hindawi.com/journals/jam/2011/298628.pdf

Mittag-Leffler Functions and Their Applications

On Mittag-Leffler-type functions in fractional evolution processes

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.

The paper is devoted to the study of the function E γ ρ,μ(z) defined for complex ρ, μ, γ (Re(ρ) > 0) by which is a generalization of the classical Mittag-Leffler function E ρ,μ(z) and the Kummer confluent hypergeometric function Φ(γ, μ; z). The properties of E γ ρ,μ(z) including usual differentiation and integration, and fractional ones are proved. Further the integral operator with such a function kernel is studied in the space L(a, b). Compositions of the Riemann–Liouville fractional integration and differentiation operators with E γ ρ,μ,ω;a+ are established. An analogy of the semigroup property for the composition of two such operators with different indices is proved, and the results obtained are applied to construct the left inversion operator to the operator E γ ρ,μ,ω;a+. Since, for γ = 0, E 0 ρ,μ,ω;a+ coincides with the Riemann–Liouville fractional integral of order μ, the above operator and its inversion can be considered as generalized fractional calculus operators involving the generalized Mitta...

Generalized mittag-leer function and generalized fractional calculus operators

Some inclusion properties of a certain family of integral operators

A class of distortion theorems involving certain operators of fractional calculus

This paper is devoted to the study of the H -function as defined by the 
Mellin-Barnes integral
 H p , q m , n ( z ) = 1 2 π i ∫ ℒ ℋ p , q m , n ( s ) z − s d s ,
where the function ℋ p , q m , n ( s ) is a certain ratio of products of the Gamma-functions with the argument s and the contour ℒ specially chosen. The 
conditions for the existence of H p , q m , n ( z ) are discussed and explicit power 
and power-logarithmic series expansions of H p , q m , n ( z ) near zero and infinity 
are given. The obtained results define more precisely the known results.

On the H-function

The special entire function of the form with is introduced, where α>0, m>0 and α(im+1)+1≠ 0,−1, −2,.....for i=0,1,2,....For m = 1, Eα1,l(z) coincides with the Mittag-Leffler function Eα,α+1 ,with exactness to the constant multiplier γ(αl+1)The connections of Eα,m,l(z) with the Riemann-Liouville fractional integrals and derivatives are investigated and their applications to solving the linear Abel-Volterra integral equations are given.

Megumi Saigo

Papers

Generalized mittag-leer function and generalized fractional calculus operators

Some inclusion properties of a certain family of integral operators

A class of distortion theorems involving certain operators of fractional calculus

On the H-function

On mittag-leffler type function, fractional calculas operators and solutions of integral equations