The main object of this paper is to investigate the solution of the following free electron laser equations associated with generalized MittagLeffler function and a confluent hypergeometric function in two variables: Dα τ h(τ) = ω ∫ τ 0 t β−1 h(τ − t)E σ,β(iνt σ)dt+ δf(τ), where γ, δ, ω ∈ C; ν ∈ R,α > 0, σ > 0, β > 0; E σ,β(z) is the generalized Mittag-Leffler function and Dα τ h(τ) = λ Γ(γ) ∫ ...

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.

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.

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.

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.

This paper studies a prototype of inverse obstacle scattering problems whose governing equation is the Helmholtz equation in two dimensions. An explicit method to extract information about the location and shape of unknown obstacles from the far field operator with a fixed wave number is given. The method is based on: an explicit construction of a modification of Mittag-Leffler’s function via t...

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...

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...