Grunwald-Letnikov Difference Research Articles

Discrete fracmemristor-based chaotic map by Grunwald–Letnikov difference and its circuit implementation

The topic of discrete memristor has attracted more and more attention in recent years, but there is not enough discussion about discrete memristor under the fractional-order difference definition, especially for the system model defined by Grunwald–Letnikov. In this paper, a discrete fractional-order memristor model based on Grunwald–Letnikov definition is derived, and two new chaotic maps are designed based on this fractional-order model. Dynamic behaviors are investigated by the volt–ampere curve, bifurcation diagram, Lyapunov exponent spectrum, attractor phase diagram and fuzzy entropy complexity algorithm. Simulation results show that the two fractional-order systems can produce rich dynamic behaviors with the change of fractional order value, and both of them can present hyperchaos. Finally, the proposed systems are implemented in FPGA digital circuit and analog circuit respectively, and the phenomena in the numerical simulation are verified.

Special Fractional-Order Map and Its Realization

Recent works have focused the analysis of chaotic phenomena in fractional discrete memristor. However, most of the papers have been related to simulated results on the system dynamics rather than on their hardware implementations. This work reports the implementation of a new chaotic fractional memristor map with “hidden attractors”. The fractional memristor map is developed based on a memristive map by using the Grunwald–Letnikov difference operator. The fractional memristor map has flexible fixed points depending on a system’s parameters. We study system dynamics for different values of the fractional orders by using bifurcation diagrams, phase portraits, Lyapunov exponents, and the 0–1 test. We see that the fractional map generates rich dynamical behavior, including coexisting hidden dynamics and initial offset boosting.

Numerical simulation for nonlinear space-fractional reaction convection-diffusion equation with its application

In this research, we adopt a fully implicit approach with a weighted shifted Grunwald–Letnikov difference operator to find the numerical solution of the one and two-dimensional nonlinear space fractional convection–diffusion-reaction equation over a bounded domain. We employ the Peacemann-Rachford alternative direction implicit technique for 2D problems, which is computationally efficient and is used to reduce a 2D problem to a 1D problem by alternately applying x and y directions. This type of problem allows us to successfully predict and investigate pollutant distribution in groundwater, which is based on two processes: convection and diffusion. Because computing the numerical solution of the nonlinear system of equations at each time step using the fixed-point iteration technique is computationally expensive, we have linearized our problem. The numerical scheme’s unconditional stability and second-order convergence in both space and time are theoretically investigated. Finally, numerical experiments with several variations of the nonlinear reaction term are performed, confirming the theoretical analysis and demonstrating the effectiveness of the suggested strategy.

Hyperchaotic fractional Grassi–Miller map and its hardware implementation

Most of the papers published so far on fractional discrete systems are related to theoretical results on their chaotic behaviors or to applications based on the mathematical modeling of the chaotic phenomena. No paper has been published to date regarding the hardware implementation of hyperchaotic fractional maps. This manuscript makes a contribution to the topic by presenting the first example of hardware implementation of hyperchaotic fractional maps. In particular, by exploiting the Grunwald–Letnikov difference operator, the paper introduces a new version of the fractional Grassi–Miller map. The system dynamics are analyzed via bifurcation diagrams and Lyapunov exponents, showing that the conceived map is hyperchaotic when the fractional order μ belongs to the interval [0.966,1]. Finally, a hardware implementation of the fractional map is illustrated, with the aim to concretely highlight the presence of hyperchaos in physical systems described by fractional difference equations. Arduino, an open-source platform, has been used to illustrate the simplicity as well as the feasibility of the implementation.

Tobit Kalman filtering for fractional‐order systems with stochastic nonlinearities under Round‐Robin protocol

AbstractIn this article, the Tobit Kalman filtering problem is investigated for a class of discrete time‐varying fractional‐order systems in the presence of measurement censoring and stochastic nonlinearities under the Round‐Robin protocol (RRP). The fractional‐order dynamic model is described by the Grunwald–Letnikov difference equation, and the statistical means are utilized to characterize the stochastic nonlinearities that include state‐dependent stochastic disturbances as a special case. The RRP is employed to decide the transmission sequence of sensors so as to alleviate undesirable data collisions. Under the RRP scheduling, only one sensor is permitted to transmit its measurement over the network at each time instant. In light of the renowned orthogonality projection principle, a protocol‐based fractional Tobit Kalman filter is devised with the fractional dynamics and stochastic nonlinearities elaborately addressed. In the pursuit of the filter design, a couple of new terms appear which are in relation to the RRP, fractional dynamics and stochastic nonlinearities arise, and these terms are adequately handled recursively or off‐line. Simulation results are provided to demonstrate the usefulness of the proposed method.

An improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term

In this paper, we propose an improved finite difference/finite element method for the fractional Rayleigh–Stokes problem with a nonlinear source term. The second-order backward differentiation formula (BDF2) and weighted and shifted Grunwald-Letnikov difference (WSGD) formula are employed to discretize first-order time derivative and the time fractional-order derivative, respectively. Moreover, a linearized difference scheme is proposed to approximate the nonlinear source term. Together with the Galerkin finite element method in the space direction, we present a fully discrete scheme for the fractional Rayleigh–Stokes problem with a nonlinear source term. Based on a novel analytical technique, the stability and the convergence accuracy in $$L^{2}$$ -norm with $$O(\tau ^{2}+h^{k+1})$$ are derived in detail, and this convergence order is higher than the previous work. Finally, some numerical examples are presented to validate our theoretical results.

Extended Kalman Filters for Continuous-time Nonlinear Fractional-order Systems Involving Correlated and Uncorrelated Process and Measurement Noises

In order to improve the estimation accuracy of the state information and save the computing time for fractional-order systems containing correlated and uncorrelated process and measurement noises, this paper investigates fractional-order extended Kalman filters for continuous-time nonlinear fractional-order systems using the method of fractional-order average derivative. Compared with Grunwald-Letnikov difference, the estimation accuracy is improved via the fractional-order average derivative method. Meanwhile, the computing time in the state estimation is saved. To deal with the correlated and uncorrelated process and measurement noises, two kinds of extended Kalman filters for nonlinear fractional-order systems are given. Finally, the effectiveness of the proposed fractional-order extended Kalman filters based on fractional-order average derivative is validated by two examples.

State Estimation of Continuous-Time Linear Fractional-Order Systems Disturbed by Correlated Colored Noises via Tustin Generating Function

This paper proposes Kalman filters to investigate the state estimation of continuous-time linear fractional-order systems disturbed by correlated colored noises. A difference equation is obtained by discretizing the fractional-order differential equation via Tustin generating function. Besides, an augmented vector is determined by the state vector and the colored noise vector to deal with the problems on the colored process noise or the colored measurement noise with fractional-orders. In fact, a more accurate state estimation can be achieved using the discretization method via Tustin generation function for the investigated fractional-order systems, compared with Grunwald-Letnikov difference. Finally, two illustrative examples are provided to validate the effectiveness of the proposed algorithms in this paper.

A conservative numerical method for the fractional nonlinear Schrödinger equation in two dimensions

This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schrodinger (NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grunwald-Letnikov difference (WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schrodinger (FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations (ODEs) in matrices formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson (CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor (cIIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.

Long and Short Memory in Economics: Fractional-Order Difference and Differentiation

<div><p><em>Long and short memory in economic processes is usually described by the so-called discrete fractional differencing and fractional integration. We prove that the discrete fractional differencing and integration are the </em><em>Grunwald-Letnikov fractional differences of non-integer order d. Equations of ARIMA(p,d,q) and ARFIMA(p,d,q) models are the fractional-order difference equations with the Grunwald-Letnikov differences of order d. We prove that the long and short memory with power law should be described by the exact fractional-order differences, for which </em><em>the Fourier transform </em><em>demonstrates the power law exactly. The fractional differencing and the Grunwald-Letnikov fractional differences cannot give exact results for the long and short memory with power law, since </em><em>the Fourier transform</em><em> of these discrete operators satisfy the </em><em>power law in the neighborhood of zero only. We prove that the economic processes with t</em><em>he continuous time long and short memory, which is characterized by the power law, should be described by the fractional differential equations.</em></p></div>

Interval Shannon Wavelet Collocation Method for Fractional Fokker-Planck Equation

Metzler et al. introduced a fractional Fokker-Planck equation (FFPE) describing a subdiffusive behavior of a particle under the combined influence of external nonlinear force field and a Boltzmann thermal heat bath. In this paper, we present an interval Shannon wavelet numerical method for the FFPE. In this method, a new concept named “dynamic interval wavelet” is proposed to solve the problem that the numerical solution of the fractional PDE is usually sensitive to boundary conditions. Comparing with the traditional wavelet defined in the interval, the Newton interpolator is employed instead of the Lagrange interpolation operator, so, the extrapolation points in the interval wavelet can be chosen dynamically to restrict the boundary effect without increase of the calculation amount. In order to avoid unlimited increasing of the extrapolation points, both the error tolerance and the condition number are taken as indicators for the dynamic choice of the extrapolation points. Then, combining with the finite difference technology, a new numerical method for the time fractional partial differential equation is constructed. A simple Fokker-Planck equation is taken as an example to illustrate the effectiveness by comparing with the Grunwald-Letnikov central difference approximation (GL-CDA).