In this research a new fractional-order derivative is defined and applied to the fractional perturbed Chen-Lee-Liu nonlinear equation. Analytical soliton solutions are obtained by the modified
$ \exp ...(-\Omega (\xi )) $
exp
(
−
Ω
(
ξ
)
)
-expansion function method (MEFM). Three new groups of bright soliton solutions, dark soliton solutions, periodic tan-type singular soliton solutions and rational singular soliton solutions are constructed with specific contains over the parameters of the nonlinear fractional perturbed Chen-Lee-Liu equation. Solutions are illustrated in 3D graphs, contour plots and 2D plots. Next, we compare our results with the results obtained previously for the nonlinear perturbed CCL. It is noticed that the solutions here introduced present some complex propagation features that appears for different time values in the 2D space graphs, these complex propagation phenomena are not observed in the integer-order nonlinear perturbed CCL equation or in the fractional case for other type of fractional derivatives previously introduced in the literature The fractional-order derivative here introduced will be very useful for the description of some nonlinear dissipation, nonlinear dispersion and other complex effects in optical wave phenomena.
Abstract In this paper, we present a derivative-free computational method for solving fuzzy nonlinear equation. Derivative-free technique avoids computing the derivative by generating an estimate to ...the derivative. This is made possible by inserting the estimate of in Levenberg-Marquardt’s method. Numerical experiments are carried out which shows that, the method is efficient.
Nonlinear equation systems are ubiquitous in a variety of fields, and how to tackle them has drawn much attention, especially dynamic ones. As a particular class of recurrent neural network, the ...zeroing neural network (ZNN) takes time-derivative information into consideration, and thus, is a competent approach to dealing with dynamic problems. Hitherto, two kinds of ZNN models have been developed for solving systems of dynamic nonlinear equations. One of them is explicit, involving the computation of a pseudoinverse matrix, and the other is of implicit dynamics essentially. To address these two issues at once, a low-computational-complexity ZNN (LCCZNN) model is proposed. It does not need to compute any pseudoinverse matrix, and is in the form of explicit dynamics. In addition, a novel activation function is presented to endow the LCCZNN model with finite-time convergence and certain robustness, which is proved rigorously by Lyapunov theory. Numerical experiments are conducted to validate the results of theoretical analyses, including the competence and robustness of the LCCZNN model. Finally, a pseudoinverse-free controller derived from the LCCZNN model is designed for a UR5 manipulator to online accomplish a trajectory-following task.
This paper is dedicated to studying the nonlinear Schrödinger equations of the form ... where a ∈ C1(RN, 0, ∞)) is nonsymmetric and satisfies a(x) < (...) ... and f ∈ C(R, R). By using concentration ...compactness arguments and projections on a general Pohozaev type manifold, we prove that the above problem has a nontrivial solution under weaker assumptions on a and f than the existing results in literature.
Nonlinear equation is an important cornerstone of nonlinear science, and many practical problems in scientific and engineering fields can be described by nonlinear equation in mathematics. In this ...paper, improved zeroing neural network (IZNN) models are presented and investigated for finding the solutions of the time-invariant nonlinear equation (TINE) and time-varying nonlinear equation (TVNE) in predictable and finite time. Compared with the exponential convergence zeroing neural network (ZNN), the convergence time of the IZNN models is finite and able to be estimated; in addition, the IZNN model is more stable and reliable for solving high-order TVNE. Both of the theoretical and numerical simulation results of the ZNN and IZNN for finding the solutions of the TINE and TVNE are presented to demonstrate the superiority and effectiveness of the IZNN model.
The high-precision parameters in distribution networks are difficult to obtain, which brings difficulties to the model-based methods and analysis. With the widespread deployment of high-precision ...measurement units, data-driven methods have greater advantages in practice. In addition, with massive integration of distributed renewable generation and fast electric vehicle chargers, the fluctuations of net load increase significantly. The data-driven power flow calculation method based on the linear model becomes difficult to obtain accurate results for the low nonlinear adaptability. To improve the data-driven power flow calculation accuracy under high penetration of renewable distribution generation, this paper proposed an approach with high adaptability to the nonlinearity of power flow. Based on the thought of Koopman operator theory, the nonlinear relationship in power flow calculation is converted into a linear mapping in a higher dimension state space, which can significantly improve the calculation accuracy. Case studies on different IEEE cases have demonstrated that the proposed method can realize higher accuracy in power flow calculation with large-scale power fluctuations, compared to the existing data-driven method, in both mesh and radial networks. Finally, measurement data of a practical 10kV distribution network has been further used to verify the effectiveness of the proposed method in practical applications.
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