*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...

The object of this paper is to establish certain generalized fractional integration and differentiation involving generalized Mittag-Leffler function defined by Salim and Faraj [25]. The considered generalized fractional calculus operators contain the Appell's function $F_3$ [2, p.224] as kernel and are introduced by Saigo and Maeda [23]. The Marichev-Saigo-Maeda fractional calculus operators a...

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...

A novel method of an optimal summation is developed that allows for calculating from small-variable asymptotic expansions the characteristic amplitudes variables tending to infinity. The in two versions, as self-similar Borel–Leroy or Mittag–Leffler summations. It based on optimized iterated roots approximants applied and Mittag–Leffler- transformed series with subsequent inverse transformation...

Abstract In this paper, we introduce a new integral transform, namely Aboodh and apply the transform to investigate Hyers–Ulam stability, Hyers–Ulam–Rassias Mittag-Leffler–Hyers–Ulam Mittag-Leffler–Hyers–Ulam–Rassias stability of second order linear differential equations.

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.

We examine critically some of the existing descriptions of the envelope of a 1-parameter family of surfaces in 3-space. An old, natural, description is that the envelope is the union of the characteristic curves (defined as the “limit intersection curves” of surfaces from the family). This description fell in disrepute during the 20th century. We claim that this relegation of the limit intersec...

Abstract: Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships between q-exponential and other well-known functional forms, such as Mittag–Leffler functio...

We investigate the asymptotic behaviour of the Mittag-Leffler polynomials Gn(z) for large n and z , where z is a complex variable satisfying 0 arg z 2 π . A summary of the asymptotic properties of Gn(ix) for real values of x and an approximation for its extreme zeros as n→∞ are given. When the variables are such that z/n is finite, an expansion is obtained using the method of steepest descents ...

In this paper a new function called as K-function, which is an extension of the generalization of the Mittag-Leffler function[10,11] and its generalized form introduced by Prabhakar[20], is introduced and studied by the author in terms of some special functions and derived the relations that exists between the Kfunction and the operators of Riemann-Liouville fractional integrals and derivatives.