In this work, we recall some definitions on fractional calculus with discrete-time. Then, introduce a discrete-time Hopfield neural network (D.T.H.N.N) non-commensurate variable-order (V.O) for three neurons. After that, phase-plot portraits, bifurcation and Lyapunov exponents diagrams are employed to verify that the proposed discrete time variable order has chaotic behavior. Furthermore, use 0...

31 www.erpublication.org  Abstract—In order to overcome the nonlinear characteristics of permanent magnet synchronous motor (PMSM), fractional order (FO) fuzzy proportional integral (PI) controller was designed based on the fractional order calculus and fuzzy technology for PMSM with the nonlinearity and time-varying parameters at work. Fuzzy control rules are designed in the light of running ...

In this article, we propose the definition of one parameter matrix Mittag-Leffler functions of fractional nabla calculus and present three different algorithms to construct them. Examples are provided to illustrate the applicability of suggested algorithms.

In recent years, it has been found that derivatives of non-integer order are very effective for the description of many physical phenomena such as rheology, damping laws, and diffusion processes. These findings invoked the growing interest on studies of the fractal calculus in various fields such as physics, chemistry, and engineering [1 – 4]. In general, there exists no method that yields an e...

We consider a stochastic volatility model where the volatility process is a fractional Brownian motion. We estimate the memory parameter of the volatility from discrete observations of the price process. We use criteria based on Malliavin calculus in order to characterize the asymptotic normality of the estimators. 2000 AMS Classification Numbers: 60F05, 60H05, 60G18.

The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the ...

and Applied Analysis 3 2. Preliminaries on Time Scales A time scale T is an arbitrary nonempty closed subset of the real numbers R. The most wellknown examples are T R, T Z, and T q : {qn : n ∈ Z}⋃{0}, where q > 1. The forward and backward jump operators are defined by σ t : inf{s ∈ T : s > t}, ρ t : sup{s ∈ T : s < t}, 2.1 respectively, where inf ∅ : supT and sup ∅ : inf T. A point t ∈ T is sa...

Fractional calculus is the field of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order. The purpose of this work is to use Hadamard fractional integral to establish some new integral inequalities of Gruss type by using one or two parameters which ensues four main results . Furthermore, other integral inequalities of reverse ...

In this work, a non-integer order Airy equation involving Liouville differential operator is considered. Proposing an undetermined integral solution to the left fractional Airy differential equation, we utilize some basic fractional calculus tools to clarify the closed form. A similar suggestion to the right FADE, converts it into an equation in the Laplace domain. An illustration t...