This paper introduces a numerical technique for the estimation of stochastic response of the Duffing oscillator with fractional or variable order damping and driven by white noise excitation. The Wiener-Hermite expansion is integrated with the Grunwald-Letnikov approximation in case of fractional order damping and with Coimbra approximation in case of variableorder damping. The numerical solver...

This paper presents a novel adaptive fuzzy logic controller (FLC) equipped with an adaptive algorithm to achieve synchronization performance for fractional chaotic systems. By introducing the fuzzy control design and robustness tracking approach, a desired synchronization error can be attenuated to a prescribed level, even in the presence of the high level of uncertainties and noisy training da...

In this paper, we consider some aspects of the numerical analysis mathematical model fractional Duffing with a derivative variable order Riemann-Liouville type. Using methods: an explicit finite-difference scheme based on Grunwald-Letnikov and Adams-Bashford-Moulton approximations (predictor-corrector), proposed is found. These methods have been verified test case. It shown that predictor-corre...

In this paper, we further discuss the properties of three kinds of fractional derivatives: the Grünwald–Letnikov derivative, the Riemann–Liouville derivative and the Caputo derivative. Especially, we compare the Riemann–Liouville derivative with the Caputo derivative. And sequential property of the Caputo derivative is also derived, which is helpful in translating the higher fractional-order di...

The theory of derivatives and integrals of non-integer order goes back to Leibniz, Liouville, Grunwald, Letnikov and Riemann. The history of fractional vector calculus (FVC) has only 10 years. The main approaches to formulate a FVC, which are used in the physics during the past few years, will be briefly described in this paper. We solve some problems of consistent formulations of FVC by using ...

and Applied Analysis 3 Definition 2. The Grünwald-Letnikov fractional derivative with order α of function x(t) is defined below:

Two approaches for defining fractional derivatives of periodic distributions are presented. The first is a distributional version of the Weyl fractional derivative in which a derivative of arbitrary order of a periodic distribution is defined via Fourier series. The second is based on the Grünwald-Letnikov formula for defining a fractional derivative as a limit of a fractional difference quotie...

• We propose a discrete fractional order model to analyze the behaviour of an epidemic process with indirect transmission. Our main contribution is study influence and memory steps on solution. The stability around equilibrium points our analyzed. obtain some bounds tan introduction derivhe population size assure nonnegativity solution model. A proposed This based version Grunwald-Letnikov deri...

System identification is important in science and engineering. This contribution proposes a simple yet elegant scheme for identifying a fractional order dynamic system based on its observed response to a standard excitation. Our approach consists in obtaining and solving a set of simultaneous linear equations connecting the parameters to yield the desired estimates. To minimise the effect of no...