New oscillation criteria of second-order nonlinear differential equations

By employing a class of new functions Φ=Φ(t,s,l) and a generalized Riccati technique, some new oscillation and interval oscillation criteria are established for the second-order nonlinear differential equation

(r(t)y^′(t))^′+Q(t,y(t),y^′(t))=0.     

Symbolic computation of exact solutions expressible in rational formal hyperbolic and elliptic functions for nonlinear partial differential equations

With the aid of symbolic computation, some algorithms are presented for the rational expansion methods, which lead to closed-form solutions of nonlinear partial differential equations (PDEs). The new algorithms are given to find exact rational formal polynomial solutions of PDEs in terms of Jacobi elliptic functions, solutions of the Riccati equation and solutions of the generalized Riccati equation. They can be implemented in symbolic computation system Maple. As applications of the methods, we choose some nonlinear PDEs to illustrate the methods. As a result, we not only can successfully obtain the solutions found by most existing Jacobi elliptic function methods and Tanh-methods, but also find other new and more general solutions at the same time.     

Asymptotic behavior of a class of nonlinear evolution equations

In this paper, using a new method (or technology), we prove the existence and upper semi-continuity of the global attractors A_ω for a class of nonlinear evolution equations in D(A)×D(A), where the nonlinear term f satisfies a critical exponential growth condition.     

Gain-scheduled fault detection on stochastic nonlinear systems with partially known transition jump rates

In this paper, the problem of continuous gain-scheduled fault detection (FD) is studied for a class of stochastic nonlinear systems which possesses partially known jump rates. Initially, by using gradient linearization approach, the nonlinear stochastic system is described by a series of linear jump models at some selected working points. Subsequently, observer-based residual generator is constructed for each jump linear system. Then, a new observer-design method is proposed for each re-constructed system to design H_∞ observers that minimize the influences of the disturbances, and to formulate a new performance index that increase the sensitivity to faults. Finally, continuous gain-scheduled approach is employed to design continuous FD observers on the whole nonlinear stochastic system. Simulation example is given to show the effectiveness and potential of the developed techniques.     

Solving nonconvex nonlinear programming problems via a new aggregate constraint homotopy method

In this paper, by introducing C² mappings ξ_i(x,z_i),i=1,…,m and using the idea of the aggregate function method, a new aggregate constraint homotopy method is proposed to solve the Karush-Kuhn-Tucker (KKT) point of nonconvex nonlinear programming problems. Compared with the previous results, the choice scope of initial points is greatly enlarged, so use of the new aggregate constraint homotopy method may improve the computational efficiency of reduced predictor-corrector algorithms.     

Symbolic computation and a uniform direct ansätze method for constructing travelling wave solutions of nonlinear evolution equations

On the basis of the computer symbolic system Maple and the tanh method, the Riccati equation method as well as all kinds of improved versions of these methods, we present a further uniform direct ansätze method for constructing travelling wave solutions of nonlinear evolution equations. Compared with the existing methods, the presented method can be used to construct more new general solutions. And we give some examples to illustrate the key step of our method.     

Structure of multiple solutions for nonlinear differential equations

Based on the eigensystem {λ_j,φ_j} of -Δ, the multiple solutions for nonlinear problem Δu + f(u) = 0 in Ω,u = 0 on ?Ω are approximated. A new search-extension method (SEM), which consists of three steps in three level subspaces, is proposed. Numerical simulations for several typical nonlinear cases, i.e. f(u) = u ³, u ², (u - p), u ²(u ² - p), are completed and some conjectures are presented.     

Coupled nonlinear stochastic differential equations

Systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co-workers. Examples include two coupled Riccati equations, coupled linear equations, stochastic coupled equations with product terms, and n coupled stochastic differential equations.     

Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems

In this paper, we present an algorithm to solve nonlinear semi-infinite programming (NSIP) problems. To deal with the nonlinear constraint, Floudas and Stein (SIAM J. Optim. 18:1187?C1208, 2007) suggest an adaptive convexification relaxation to approximate the nonlinear constraint function. The ??BB method, used widely in global optimization, is applied to construct the convexification relaxation. We then combine the idea of the cutting plane method with the convexification relaxation to propose a new algorithm to solve NSIP problems. With some given tolerances, our algorithm terminates in a finite number of iterations and obtains an approximate stationary point of the NSIP problems. In addition, some NSIP application examples are implemented by the method proposed in this paper, such as the proportional-integral-derivative controller design problem and the nonlinear finite impulse response filter design problem. Based on our numerical experience, we demonstrate that our algorithm enhances the computational speed for solving NSIP problems.     

Bi-directional evolutionary topology optimization of geometrically nonlinear continuum structures with stress constraints

This paper proposes a design method to maximize the stiffness of geometrically nonlinear continuum structures subject to volume fraction and maximum von Mises stress constraints. An extended bi-directional evolutionary structural optimization (BESO) method is adopted in this paper. BESO method based on discrete variables can effectively avoid the well-known singularity problem in density-based methods with low density elements. The maximum von Mises stress is approximated by the p-norm global stress. By introducing one Lagrange multiplier, the objective of the traditional stiffness design is augmented with p-norm stress. The stiffness and p-norm stress are considered simultaneously by the Lagrange multiplier method. A heuristic method for determining the Lagrange multiplier is proposed in order to effectively constrain the structural maximum von Mises stress. The sensitivity information for designing variable updates is derived in detail by adjoint method. As for the highly nonlinear stress behavior, the updated scheme takes advantages from two filters respectively of the sensitivity and topology variables to improve convergence. Moreover, the filtered sensitivity numbers are combined with their historical sensitivity information to further stabilize the optimization process. The effectiveness of the proposed method is demonstrated by several benchmark design problems.     

Solving nonlinear equations by a new derivative free iterative method

We develop a new simple iteration formula, which does not require any derivatives of f(x), for solving a nonlinear equation f(x) = 0. It is proved that the convergence order of the new method is quadratic. Furthermore, the new method can approximate complex roots. By several numerical examples we show that the presented method will give desirable approximation to the root without a particularly good initial approximation and be efficient for all cases, regardless of the behavior of f(x).     

Modified method of simplest equation and its application to nonlinear PDEs

We search for traveling-wave solutions of the class of PDEswhere A_p(Q),B_r(Q),C_s(Q),D_u(Q) and F(Q) are polynomials of Q. The basis of the investigation is a modification of the method of simplest equation. The equations of Bernoulli, Riccati and the extended tanh-function equation are used as simplest equations. The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto-Sivashinsky equation, reaction-diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.     

Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations

This paper aims to design full-order and reduced-order observers for one-sided Lipschitz nonlinear systems. The system under consideration is an extension of its known Lipschitz counterpart and possesses inherent advantages with respect to conservativeness. For such system, we first develop a novel Riccati equation approach to design a full-order observer, for which rigorous mathematical analysis is performed. Consequently, we show that the conditions under which a full-order observer exists also guarantee the existence of a reduced-order observer. A design method for the reduced-order observer that is dependent on the solution of the Riccati equation is then presented. The proposed conditions are easily and numerically tractable via standard numerical software. Furthermore, it is theoretically proven that the obtained conditions are less conservative than some existing ones in recent literature. The effectiveness of the proposed observers is illustrated via a simulative example.     

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.     

Sparse time-frequency representation of nonlinear and nonstationary data

Adaptive data analysis provides an important tool in extracting hidden physical information from multiscale data that arise from various applications. In this paper, we review two data-driven time-frequency analysis methods that we introduced recently to study trend and instantaneous frequency of nonlinear and nonstationary data. These methods are inspired by the empirical mode decomposition method (EMD) and the recently developed compressed (compressive) sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form {a(t) cos(θ(t))}, where a is assumed to be less oscillatory than cos(θ(t)) and θ′ ? 0. This problem can be formulated as a nonlinear l ⁰ optimization problem. We have proposed two methods to solve this nonlinear optimization problem. The first one is based on nonlinear basis pursuit and the second one is based on nonlinear matching pursuit. Convergence analysis has been carried out for the nonlinear matching pursuit method. Some numerical experiments are given to demonstrate the effectiveness of the proposed methods.     

Modeling and nonlinear control of a flexible-link manipulator

The problem of modeling and controlling the tip position of a one-link flexible manipulator is considered. The proposed model has been used to investigate the effect of the open-loop control torque profile, and the payload. The control strategy is based on the nonlinear State Dependent Riccati Equation (SDRE) design method in the context of application to robotics and manufacturing systems. In this paper, an experimental test-bed was developed to demonstrate the concept of end-point position feedback on a single-link elastic manipulator, and the control strategy for a single-link flexible manipulator. The controller is designed based on the nonlinear SDRE developed by the authors and applied to a flexible manipulator. The experimental results are compared with conventional PD controller strategy. The results reveal that the nonlinear SDRE controller is near optimal and robustly; and its performance is improved comparing to the PD control scheme.     

The extended homogeneous balance method and its applications for a class of nonlinear evolution equations

The extended homogeneous balance method with the aid of computer algebraic system Maple, is proposed for seeking the travelling wave solutions for a class of nonlinear evolution equations, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear evolution equations, respectively. Many new exact travelling wave solutions are successfully obtained. The method is straightforward and concise, and it can be also applied to other nonlinear evolution equations.     

New generalized and improved (G′/G)-expansion method for nonlinear evolution equations in mathematical physics

In this article, new extension of the generalized and improved (G′/G)-expansion method is proposed for constructing more general and a rich class of new exact traveling wave solutions of nonlinear evolution equations. To demonstrate the novelty and motivation of the proposed method, we implement it to the Korteweg-de Vries (KdV) equation. The new method is oriented toward the ease of utilize and capability of computer algebraic system and provides a more systematic, convenient handling of the solution process of nonlinear equations. Further, obtained solutions disclose a wider range of applicability for handling a large variety of nonlinear partial differential equations.     

Filled function method for nonlinear equations

Systems of nonlinear equations are ubiquitous in engineering, physics and mechanics, and have myriad applications. Generally, they are very difficult to solve. In this paper, we will present a filled function method to solve nonlinear systems. We will first convert the nonlinear systems into equivalent global optimization problems with the property: x^∗ is a global minimizer if and only if its function value is zero. A filled function method is proposed to solve the converted global optimization problem. Numerical examples are presented to illustrate our new techniques.     

An effective iterative method for computing real and complex roots of systems of nonlinear equations

A simple and flexible iterative method is proposed to determine the real or complex roots of any system of nonlinear equations F(x)=0. The idea is based on passing defined functions G_j(x_j),j=1,…,n tangent to F_i(x_j),i,j=1,…,n at an arbitrary starting point. Choosing G_j(x_j) in the form of or or any other reversible function compatible to F_i(x_j), where k is obtained for the best correlation with the function F_i(x_j), gives an added freedom, which in contrast with all existing methods, accelerates the convergence.The method that was first proposed for computing the roots of any single function is now adopted for a system of nonlinear equations. This method is compared to some classical and famous methods such as Newton’s method and Newton-Simpson’s method. The results show the effectiveness and robustness of this new method.