Numerical Analysis of the Fractional-Order Nonlinear System of Volterra Integro-Differential Equations

This paper presents the nonlinear systems of Volterra-type fractional integro-differential equation solutions through a Chebyshev pseudospectral method. The proposed method is based on the Caputo fractional derivative. The results that we get show the accuracy and reliability of the present method. Different nonlinear systems have been solved; the solutions that we get are compared with other methods and the exact solution. Also, from the presented figures, it is easy to conclude that the CPM error converges quickly as compared to other methods. Comparing the exact solution and other techniques reveals that the Chebyshev pseudospectral method has a higher degree of accuracy and converges quickly towards the exact solution. Moreover, it is easy to implement the suggested method for solving fractional-order linear and nonlinear physical problems related to science and engineering.

Iterative Learning Control for Fractional order Nonlinear System with Initial Shift

In this study, a closed-loop Dα -type iterative learning control(ILC) with a proportional D-type iterative learning updating law for the initial shift is applied to nonlinear conformable fractional system. First, the system with the initial shift is introduced. Then, fractional-order ILC (FOILC) frameworks that experience the initial shift problem for the path-tracking of nonlinear conformable fractional order systems are addressed. Moreover, the sufficient condition for the convergence of tracking errors is obtained in the time domain by introducing λ -norm and Hölder’s inequality. Lastly, numerical examples are provided to illustrate the effectiveness of the proposed methods.

Random Response and Crossing Rate of Fractional Order Nonlinear System with Impact

The random response and mean crossing rate of the fractional order nonlinear system with impact are investigated through the equivalent nonlinearization technique. The random additive excitation is Gaussian white noise, while the impact is described by a phenomenological model, which is developed from the actual impact process experiments. Based on the equivalent nonlinearization technique, one class of random nonlinear system with exact probability density function (PDF) solution of response is selected. The criterion of the appropriate equivalent nonlinear system is the similarity with the original system on the damping, stiffness, and inertia. The more similar, the higher the precision. The optimal unknown parameters of the equivalent random nonlinear system in the damping and stiffness terms are determined by the rule of smallest mean-square difference. In the view of equivalent nonlinearization technique, the response of the original system is the same as that of the equivalent system with the optimal unknown parameters in analytical solution manner. Then, the mean crossing rate is derived from stationary PDF. The consistence between the results from proposed technique and Monte Carlo simulation reveals the accuracy of the proposed analytical procedure.

The New Semianalytical Technique for the Solution of Fractional-Order Navier-Stokes Equation

In this paper, we introduce a modified method which is constructed by mixing the residual power series method and the Elzaki transformation. Precisely, we provide the details of implementing the suggested technique to investigate the fractional-order nonlinear models. Second, we test the efficiency and the validity of the technique on the fractional-order Navier-Stokes models. Then, we apply this new method to analyze the fractional-order nonlinear system of Navier-Stokes models. Finally, we provide 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.

Multi-model estimation using neural network and fault detection in unknown time continuous fractional order nonlinear systems

In this paper, multi-model estimation and fault detection using neural network is proposed for an unknown time continuous fractional order nonlinear system. Fractional differentiation is considered based on Caputo concept and the fractional order is considered to be between 0 and 1. In order to estimate a time continuous fractional order nonlinear system with unknown term in its dynamic, single-layer and double-layer RBF neural network is used. First, a parallel-series neural network observer is designed for state estimation. Weights of the neural network are updated adaptively and updating laws are presented in fractional order form. Using Lyapunov method, it is proved that state estimation error and weight estimation error of the neural network are bounded. Parameters of the neural estimator converge to ideal parameters which satisfy excitation condition stability. Then, multi-model estimation structure of fractional order nonlinear systems is presented and its application in fault detection is investigated. Finally, simulation results are presented to show efficiency of the proposed method.