Here we define a Caputo like discrete nabla fractional difference and we produce discrete nabla fractional Taylor formulae for the first time. We estimate their remaiders. Then we derive related discrete nabla fractional Opial, Ostrowski, Poincaré and Sobolev type inequalities .

A matrix form representation of discrete analogues of various forms of fractional differentiation and fractional integration is suggested. The approach, which is described in this paper, unifies the numerical differentiation of integer order and the n-fold integration, using the so-called triangular strip matrices. Applied to numerical solution of differential equations, it also unifies the sol...

This research investigates the synchronization of distributed delayed discrete-time fractional-order complex-valued neural networks. The necessary conditions have been established for stability proposed networks using theory discrete fractional calculus, Laplace transform, and Mittag–Leffler functions. In order to guarantee global asymptotic stability, adequate criteria are determined Lyapunov’...

Given the recent advances regarding studies of discrete fractional calculus, and fact that dynamics discrete-time neural networks in variable-order cases have not been sufficiently documented, herein, we consider a novel class fractional-order using nabla operator variable-order. An adequate criterion for existence solution addition to its uniqueness such systems is provided with use Banach fix...

Differences are introduced as outputs of linear systems called differencers, being considered two classes: shift and scale-invariant. Several types presented, namely: nabla delta, bilateral, tempered, bilinear, stretching, shrinking. Both continuous discrete-time differences described. ARMA-type based on differencers exemplified. In passing, the incorrectness usual delta difference is shown.

The aim of this paper is to deduce a discrete version of the fractional Laplacian in matrix form defined on the 1D periodic (cyclically closed) linear chain of finite length. We obtain explicit expressions for this fractional Laplacian matrix and deduce also its periodic continuum limit kernel. The continuum limit kernel gives an exact expression for the fractional Laplacian (Riesz fractional d...

Fractional dynamics is a field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power-law nonlocality, power-law long-term memory or fractal properties by using integrations and differentiation of noninteger orders, i.e., by methods in the fractional calculus. This paper is a review of physical models that look very promising for futu...

The aim of this paper is to establish the existence of solutions of boundary value problems of nonlinear fractional integro-differential equations involving Caputo fractional derivative by using the techniques such as fractional calculus, H"{o}lder inequality, Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type. Examples are exhibited to illustrate the main resu...

in this paper, we consider the second-kind chebyshev polynomials (skcps) for the numerical solution of the fractional optimal control problems (focps). firstly, an introduction of the fractional calculus and properties of the shifted skcps are given and then operational matrix of fractional integration is introduced. next, these properties are used together with the legendre-gauss quadrature fo...

The first power weighted version of Hardy's inequality can be rewritten as [Formula: see text] where the constant [Formula: see text] is sharp. This inequality holds in the reversed direction when [Formula: see text]. In this paper we prove and discuss some discrete analogues of Hardy-type inequalities in fractional h-discrete calculus. Moreover, we prove that the corresponding constants are sh...