and Applied Analysis 3 where the first term refers to the probability mass concentrated in the origin, δ y denotes the Dirac delta function, and fYβ denotes the density of the absolutely continuous component. The function gYβ given in 1.5 satisfies the following fractional master equation, that is, ∂ ∂tβ gYβ ( y, t ) −λgYβ ( y, t ) λ ∫ ∞ −∞ gYβ ( y − x, t ) fX x dx, 1.6 where ∂/∂t is the Caputo...

In this paper, we derive a number of interesting results concerning subordination and superordination relations for certain analytic functions associated with an extension the Mittag–Leffler function.

In the paper, the authors generalize the notion “k-Mittag-Leffler function”, establish some integral transforms of the generalized k-Mittag-Leffler function, and derive several special and known conclusions in terms of the generalized Wright function and the generalized k-Wright function. 1. Preliminaries Throughout this paper, let C, R, R0 , R, Z − 0 , and N denote respectively the sets of com...

In this paper, we consider an anti-periodic Boundary Value Problem for Volterra integro-differential equation of fractional order 1 < α ≤ 2, with generalized Mittag-Leffler function in the kernel. Some existence and uniqueness results are obtained by using some well known fixed point theorems. We give some examples to exhibit our results. c ©2016 All rights reserved.

Inspired by a number of recent investigations, we introduce the new analogues of the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials, the Apostol-Genocchi polynomials based on Mittag-Leffler function. Making use of the Caputo-fractional derivative, we derive some new interesting identities of these polynomials. It turns out that some known results are derived as special cases.

In this paper, we study the controllability of linear and nonlinear fractional damped dynamical systems, which involve fractional Caputo derivatives, with different order in finite dimensional spaces using the Mittag-Leffler matrix function and the iterative technique. A numerical example is provided to illustrate the theory.

This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by Rudolf Gorenflo, Anatoly Kilbas, Francesco Mainardi and Sergei Rogosin.

The main objective of this paper is to introduce k-fractional derivative operator by using the definition of k-beta function. We establish some results related to the newly defined fractional operator such as Mellin transform and relations to khypergeometric and k-Appell’s functions. Also, we investigate the k-fractional derivative of k-Mittag-Leffler and Wright hypergeometric functions.