The generalized Mittag-Leffler function Eα,β (z) has been studied for arbitrary complex argument z ∈ C and parameters α ∈ R+ and β ∈ R. This function plays a fundamental role in the theory of fractional differential equations and numerous applications in physics. The Mittag-Leffler function interpolates smoothly between exponential and algebraic functional behaviour. A numerical algorithm for i...

A new type of an integrable mapping is presented. This map is equipped with fractional difference and possesses an exact solution, which can be regarded as a discrete analogue of the Mittag-Leffler function.

The present paper deals with the study of a generalized Mittag-Leffler function and associated fractional operator. The operator has been discussed in the space of Lebesgue measurable functions. The composition with Riemann-Liouville fractional integration operator has been obtained.

Using inverse subordinators and Mittag-Leffler functions, we present a new definition of a fractional Poisson process parametrized by points of the Euclidean space R+. Some properties are given and, in particular, we prove a long-range dependence property.

In this paper, a mathematical model containing two-parameter Mittag-Leffler function in its definition is proposed to be used for the first time to fit the relation between unemployment rate and inflation rate, also known as the Phillips curve. The Phillips curve is in the literature often represented by an exponential-like shape. On the other hand, Phillips in his fundamental paper used a powe...

We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potent...

In this paper, we give state space realizations for the classical recursive solutions of operator-valued Carathéodory, Nevanlinna – Pick and Hermite – Fejér interpolation problems. These realizations are special in the sense that they satisfy an energy balance law; hence they are called conservative. Observability, controllability and minimality (including the property known as “simplicity”) of...

*Correspondence: [email protected]; [email protected] 1Department of Applied Mathematics, Yanshan University, Qinhuangdao, 066001, China Full list of author information is available at the end of the article Abstract This paper is concerned with the global projective synchronization issue for fractional neural networks in the Mittag-Leffler stability sense. Firstly, a fractional-order differ...

We consider some fractional extensions of the recursive differential equation governing the Poisson process, i.e. d d t pk(t) =−λ(pk(t)− pk−1(t)), k ≥ 0, t > 0 by introducing fractional time-derivatives of order ν , 2ν , ..., nν . We show that the so-called “Generalized Mittag-Leffler functions” Ek α,β(x), x ∈ R (introduced by Prabhakar [24]) arise as solutions of these equations. The correspon...

The objective of this paper is to establish some new refinements of fractional Hermite-Hadamard inequalities via a harmonically convex function with a kernel containing the generalized Mittag-Leffler function.