Higher Order Nonlinear Problems Research Articles

Dynamic Description of soliton solutions of nonlinear Schrödinger equation with higher order dispersion and nonlinear terms

Abstract The article presents an analytical solution for the higher-order nonlinear Schrödinger equation (NLSE), which describes the propagation of short light pulses in monomode optical fibers. Various traveling wave solutions are obtained using the generalized exponential rational function method, a technique with substantial applications in physics and mathematics. Additionally, the parameters leading to the occurrence of optical bright and multipeak solitons in this medium are provided along with their formation conditions. The derived solutions are graphically displayed to enhance the understanding of the model’s physical phenomena. This approach is credible, potent, and successful in solving a wide variety of different models of this kind that arise in the applied sciences. Its robustness, strength, and efficiency make it suitable for addressing various higher-order nonlinear problems in current research fields, extending beyond the models encountered in the applied sciences.

Improved nonlinear MPC for aircraft gas turbine engine based on semi-alternative optimization strategy

The development of advanced control systems is motivated to achieve the complicated aircraft engine control, so a novel semi-alternative optimization strategy based model predictive control is proposed. In this proposed controller, a nonlinear state-space model is generated as the predictive model on the basis of a baseline engine model that estimates the current operating state of engine with an extended Kalman filter every sampling instant, and a quadratic constrained problem rather than a higher order nonlinear problem is constructed based on this model. Moreover, a novel control sequence to be optimized based on semi-alternative optimization strategy is proposed, which includes two parts. The first part represents that all the control variables of the engine are optimized for the current sampling instant while the second part represents that different control variables are optimized alternatively for different future sampling instants over the control horizon. The proposed model predictive control method is implemented to a twin-spool turbofan engine. Simulation results demonstrate not only that the proposed method can achieve a smaller control error than the standard model predictive control algorithm does, but also that the proposed controller is more time-saving than the standard model predictive control, which can save up to about 61% optimization time when control horizon increases to nine.

A novel kinematic parameters calibration method for industrial robot based on Levenberg-Marquardt and Differential Evolution hybrid algorithm

• A new method is proposed to calibrate the kinematic parameters of robot, solving the high order nonlinear problem of Levenberg-Marquardt algorithm and the issue of slow convergence speed of Differential Evolution algorithm. • This method is validated on a FANUC M710ic/50 robot. • In consideration of the effect of measurement noise, such as measurement error of laser tracker, the coordinate system transformation error between the robot and laser tracker, the absolute positioning accuracy of robot has increased by almost 73.6% after calibration with hybrid algorithm. • A kind of joint space error compensation method is proposed to compensate for robot kinematic parameter deviations, overcoming the problem of authority limitation of modifying the D-H parameters of robot control system. The poor absolute positioning accuracy of industrial robots is the main obstacle for its further application in precision grinding of complex surfaces, such as blisk, blade, etc. Based on the established kinematic error model of a typical industrial robot FANUC M710ic/50, a novel kinematic parameters calibration method is proposed in this paper to improve the absolute positioning accuracy of robot. The pre-identification of the kinematic parameter deviations of robot was achieved by using the Levenberg-Marquardt algorithm. Subsequently, these identified suboptimal values of parameter deviations were defined as central values of the components of initial individuals to complete accurate identification by using Differential Evolution algorithm. The above two steps, which were regarded as the core of this Levenberg-Marquardt and Differential Evolution hybrid algorithm, were used to obtain the preferable values for kinematic parameters of the robot. On this basis, the experimental investigations of kinematic parameters calibration were conducted by using a laser tracker and numerical simulation method. The results revealed that the robot positioning error decreased from 0.994 mm, initial positioning error measured by laser tracker, to 0.262 mm after calibration with this proposed hybrid algorithm. The absolute positioning accuracy has increased by 40.86% than that of the Levenberg-Marquardt algorithm, increased by 40.31% than that of the Differential Evolution algorithm, and increased by 25.14% than that of the Simulated Annealing algorithm. This work shows that the proposed kinematic parameters calibration method has a significant improvement on the absolute positioning accuracy of industrial robot.

An approximate analytical solution of the Navier\u2013Stokes equations within Caputo operator and Elzaki transform decomposition method

In this article, a hybrid technique of Elzaki transformation and decomposition method is used to solve the Navier–Stokes equations with a Caputo fractional derivative. The numerical simulations and examples are presented to show the validity of the suggested method. The solutions are determined for the problems of both fractional and integer orders by a simple and straightforward procedure. The obtained results are shown and explained through graphs and tables. It is observed that the derived results are very close to the actual solutions of the problems. The fractional solutions are of special interest and have a strong relation with the solution at the integer order of the problems. The numerical examples in this paper are nonlinear and thus handle its solutions in a sophisticated manner. It is believed that this work will make it easy to study the nonlinear dynamics, arising in different areas of research and innovation. Therefore, the current method can be extended for the solution of other higher-order nonlinear problems.

The calculation of the reliability index in nonlinear problems: A new method

Abstract The accurate estimation of the health (reliability) index is important to estimate probability of failure in the reliability assessment of structures. The conventional first-order reliability methods including the Hasofer-Lind, Rackwitz-Fiessler, and Monte Carlo could lead to unstable, fluctuating, and distorted solutions to nonlinear problems, featuring complicated structural performance functions. The present study aimed to propose a new method by combining the particle swarm optimization and differential evolution algorithms in order to calculate the reliability index. The performance of the proposed method was evaluated by 10 examples from different studies, and the convergence results were compared to the results of studies such as Hasofer-Lind, Rackwitz-Fiessler, Monte Carlo and some other methods. To verify the accuracy of the proposed method, to verify the accuracy of the proposed method, a reliability index chart was applied. The comparisons indicated the high accuracy and speed of the proposed method. Accordingly, in higher order nonlinear problems, the proposed method successfully calculated the reliability index while some failed to solve these problems.

Implementation of Dynamics Neural Network in Solving Inverse Function

Back-Propagation Neural Network (BPNN) has been widely used in solving nonlinear problems. However, there are some limitations in using conventional BPNN especially for high order nonlinear problems. Dynamic Back-Propagation Neural Network (DBPNN) is proposed in this paper to improve the performance of conventional BPNN. Its adaptive learning ability is closer to human being learning behavior in comparing to conventional BPNN. Few simulations have been run to test the robustness of DBPNN and the results are compared to the conventional BPNN.

Single-Term Walsh Series Direct Method for the Solution of Nonlinear Problems in the Calculus of Variations

A direct method for solving variational problems using single-term Walsh series is presented. Two nonlinear examples are considered. In the first example the classical brachistochrone problem is examined. and in the second example a higher-order nonlinear problem is considered. The properties of single-term Walsh series are given and are utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement and yields accurate results.

Legendre wavelets method for the solution of nonlinear problems in the calculus of variations

A numerical technique for solving the nonlinear problems of the calculus of variations is presented. Two nonlinear examples are considered. In the first example, the brachistochrone problem is formulated as a nonlinear optimal control problem, and in the second example, a higher-order nonlinear problem is given. An operational matrix of integration is introduced and is utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement, and yields very accurate results.