MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CONCAVE-CONVEX NONLINEARITIES AND MULTIPLE HARDY-TYPE TERMS

In this paper, we deal with the existence and multiplicity of positive solutions for the quasilinear elliptic problem -△pu-∑i=1^kμi｜u｜^p-2/｜x-ai｜p^u=｜u｜^p^＊-2u＋λ｜u｜^q-2u,x∈Ω,where Ω belong to R^N（N ≥ 3） is a smooth bounded domain such that the different points ai∈Ω,i= 1,2,...,k,0≤μi〈μ^-=（N-p/p）^p,λ〉0,1≤q〈p,and p^＊=p^N/N-p.The results depend crucially cn the parameters λ,q and μi for i=1,2,...,k.

EXISTENCE OF MULTIPLE SOLUTIONS FOR SINGULAR QUASILINEAR ELLIPTIC SYSTEM WITH CRITICAL SOBOLEV-HARDY EXPONENTS AND CONCAVE-CONVEX TERMS

The main purpose of this paper is to establish the existence of multiple solutions for singular elliptic system involving the critical Sobolev-Hardy exponents and concave-convex nonlinearities. It is shown, by means of variational methods, that under certain conditions, the system has at least two positive solutions.

MULTIPLICITY OF SOLUTIONS AND SOURCE TERMS IN A FOURTH ORDER NONLINEAR ELLIPTIC EQUATION

The authors investigatc relations between multiplicity of solutions and sourceterms of the fourth order nonlinear elliptic boundary value problem under Dirichlet boundary condition △2u＋c△u = bu+＋f inΩ, wherc Ω is a bounded open set in Rn with smoothbonndary and the nonlinearity bu+ crosses eigenvalues of △2 ＋c△. They investigate therelatiolls when the source term is constant and when it is generated by two eigenfuntions.

CONTROLLABILITY FOR A PARABOLIC EQUATION WITH A NONLINEAR TERM INVOLVING THE STATE AND THE GRADIENT

In this article, we consider the controllability of a quasi-linear heat equation involving gradient terms with Dirichlet boundary conditions in a bounded domain of RN.The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem.

Travelling solitary wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order

In this paper, the travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved.

A new mixed scheme based on variation of constants for Sobolev equation with nonlinear convection term

A new mixed scheme which combines the variation of constants and the H1-Galerkin mixed finite element method is constructed for nonlinear Sobolev equation with nonlinear con- vection term. Optimal error estimates are derived for both semidiscrete and fully discrete schemes. Finally, some numerical results are given to confirm the theoretical analysis of the proposed method.

Oscillation of Second Order Nonlinear Neutral Differential Equations with Mixed Neutral Term

In this paper, we obtained some sufficient conditions for the oscillation of all solutions of the second order neutral differential equation of the form where , and . Examples are provided to illustrate the main results.

Blow-Up Result for a Semi-Linear Wave Equation with a Nonlinear Memory Term of Derivative Type

In this paper,we study the blow-up of solutions to a semi-linear wave equation with a nonlinear memory term of derivative type.By using methods of an iteration argument and di erential inequalities,we obtain the blow-up result for the semi-linear wave equation when the exponent of p is under certain conditions.Meanwhile,we derive an upper bound of the lifespan of solutions to the Cauchy problem for the semi-linear wave equation.

Oscillation of Second Order Delay Differential Equations with Nonlinear Neutral Term

由使用平均技术，我们为第二顺序延期获得新摆动标准有非线性的中立术语的微分方程。这些结果关于中立延期概括并且改善一些已知的结果第二份订单的微分方程。

Use of nonlinear variable structure control with equivalent term in design of coordinated controls for turbo generators

Presents the coordinated controls of nonlinear variable structure with an equivalent control term designed for turbo generators following the theory of nonlinear variable structure control, which integrates the valve control with the excitation control and reduce the problem of high frequency shiver with conventional variable structure control in addition to a simplified design, and concludes from simulation results that the use of this control can improve the transient of the system.

An Improved Numerical Simulation Mode of Nonlinear Wave with Consideration of High Order Terms

SUN Da-peng BAO Wei-bin, WU Hao and LI Yu-cheng （ In this paper the 0-1 combined BEM is adopted to subdivide the computational domain boundary, and to discretize the Green＇s integral expression based on Laplace equation. The FEM is used to subdivide the wave surface and deduce the surface equation which satisfies the nonlinear boundary conditions on the surface. The equations with potential function and wave surface height as an unknown quantity by application of Taylor expansion approach can be solved by iteration within the time step. In m-time iteration within the computational process of time step （n-1）At to nat, the results of the previous iteration are taken as the initial value of the two-order unknown terms in the present iteration. Thus, an improved tracking mode of nonlinear wave surface is estabIished, and numerical results of wave tank test indicate that this mode is improved obviously and is more precise than the previous numerical model which ignored the two-order unknown terms of wave surface location and velocity potential function in comparison with the theoretical values.

Classification of Single Traveling Wave Solutions to the Generalized Strong Nonlinear Boussinesq Equation without Dissipation Terms in <i>P</i>= 1

By the complete discrimination system for polynomial method, we obtained the classification of single traveling wave solutions to the generalized strong nonlinear Boussinesq equation without dissipation terms in p=1.

Multiple Solutions for a Class of Variable-Order Fractional Laplacian Equations with Concave-Convex Nonlinearity

This paper is concerned with the following variable-order fractional Laplacian equations , where N ≥ 1 and N > 2s(x,y) for (x,y) ∈ Ω × Ω, Ω is a bounded domain in RN, s(⋅) ∈ C (RN × RN, (0,1)), (-Δ)s(⋅) is the variable-order fractional Laplacian operator, λ, μ > 0 are two parameters, V: Ω → [0, ∞) is a continuous function, f ∈ C(Ω × R) and q ∈ C(Ω). Under some suitable conditions on f, we obtain two solutions for this problem by employing the mountain pass theorem and Ekeland’s variational principle. Our result generalizes the related ones in the literature.

ERROR ESTIMATE FOR INFLUENCE OF MODEL REDUCTION OF NONLINEAR DISSIPATIVE AUTONOMOUS DYNAMICAL SYSTEM ON LONG-TERM BEHAVIOURS

From viewpoint of nonlinear dynamics, the model reduction and its influence on the long-term behaviours of a class of nonlinear dissipative autonomous dynamical system with higher dimension are investigated theoretically under some assumptions. The system is analyzed in the state space with an introduction of a distance definition which can be used to describe the distance between the full system and the reduced system, and the solution of the full system is then projected onto the complete space spanned by the eigenvectors of the linear operator of the governing equations. As a result, the influence of mode series truncation on the long-term behaviours and the error estimate are derived, showing that the error is dependent on the first products of frequencies and damping ratios in the subspace spanned by the eigenvectors with higher modal damping. Furthermore, the fundamental understanding for the topological change of the solution due to the application of different model reduction is interpreted in a mathematically precise way, using the qualitative theory of nonlinear dynamics.

Nonlinear Dynamic System Identification of ARX Model for Speech Signal Identification

System Identification becomes very crucial in the field of nonlinear and dynamic systems or practical systems.As most practical systems don’t have prior information about the system behaviour thus,mathematical modelling is required.The authors have proposed a stacked Bidirectional Long-Short Term Memory(Bi-LSTM)model to handle the problem of nonlinear dynamic system identification in this paper.The proposed model has the ability of faster learning and accurate modelling as it can be trained in both forward and backward directions.The main advantage of Bi-LSTM over other algorithms is that it processes inputs in two ways:one from the past to the future,and the other from the future to the past.In this proposed model a backward-running Long-Short Term Memory(LSTM)can store information from the future along with application of two hidden states together allows for storing information from the past and future at any moment in time.The proposed model is tested with a recorded speech signal to prove its superiority with the performance being evaluated through Mean Square Error(MSE)and Root Means Square Error(RMSE).The RMSE and MSE performances obtained by the proposed model are found to be 0.0218 and 0.0162 respectively for 500 Epochs.The comparison of results and further analysis illustrates that the proposed model achieves better performance over other models and can obtain higher prediction accuracy along with faster convergence speed.

Sub-ODE's New Solutions and Their Applications to Two Nonlinear Partial Differential Equations with Higher-Order Nonlinear Terms

In the present paper,with the aid of symbolic computation,families of new nontrivial solutions of thefirst-order sub-ODE F′~2=AF^2+BF^(2+p)+CF^(2+2p)(where F'=dF/dξ,p>0)are obtained.To our best knowledge,these nontrivial solutions have not been found in[X.Z.Li and M.L.Wang,Phys.Lett.A 361(2007)115]and[S.Zhang,W.Wang,and J.L.Tong,Phys.Lett.A 372(2008)3808]and other existent papers until now.Using thesenontrivial solutions,the sub-ODE method is described to construct several kinds of exact travelling wave solutions for thegeneralized KdV-mKdV equation with higher-order nonlinear terms and the generalized ZK equation with higher-ordernonlinear terms.By means of this method,many other physically important nonlinear partial differential equations withnonlinear terms of any order can be investigated and new nontrivial solutions can be explicitly obtained with the helpof symbolic computation system Maple or Mathematica.

Nonlinear Differential Equation of Macroeconomic Dynamics for Long-Term Forecasting of Economic Development

In this article we derive a general differential equation that describes long-term economic growth in terms of cyclical and trend components. Equation is based on the model of non-linear accelerator of induced investment. A scheme is proposed for obtaining approximate solutions of nonlinear differential equation by splitting solution into the rapidly oscillating business cycles and slowly varying trend using Krylov-Bogoliubov-Mitropolsky averaging. Simplest modes of the economic system are described. Characteristics of the bifurcation point are found and bifurcation phenomenon is interpreted as loss of stability making the economic system available to structural change and accepting innovations. System being in a nonequilibrium state has a dynamics with self-sustained undamped oscillations. The model is verified with economic development of the US during the fifth Kondratieff cycle (1982-2010). Model adequately describes real process of economic growth in both quantitative and qualitative aspects. It is one of major results that the model gives a rough estimation of critical points of system stability loss and falling into a crisis recession. The model is used to forecast the macroeconomic dynamics of the US during the sixth Kondratieff cycle (2018-2050). For this forecast we use fixed production capital functional dependence on a long-term Kondratieff cycle and medium-term Juglar and Kuznets cycles. More accurate estimations of the time of crisis and recession are based on the model of accelerating log-periodic oscillations. The explosive growth of the prices of highly liquid commodities such as gold and oil is taken as real predictors of the global financial crisis. The second wave of crisis is expected to come in June 2011.

基于层叠式残差LSTM网络的桥梁非线性地震响应预测

提出了一种基于层叠式残差长短时记忆神经网络(residual long short-term memory neural network,ResLSTM)的数据驱动建模方法,实现桥梁非线性地震响应预测。该方法利用长短时记忆(long short-term memory, LSTM)网络在长序列回归中的优势,并采用残差连接结构降低深度神经网络中的梯度回传难度,提高了有限数据下的深度网络预测性能。同时,通过采用层叠式序列结构,降低深度神经网络隐藏层节点数目,进一步提升深度神经网络的预测精度。随后,通过两跨预应力混凝土连续梁桥与组合梁斜拉桥的数值算例对该方法进行验证。神经网络的训练样本与测试样本均源自桥梁有限元模型的增量动力分析结果。此外,采用该方法成功预测了美国Meloland Overpass桥的地震响应,并与历史监测数据进行对比验证。结果表明:ResLSTM网络是一种鲁棒性良好、计算效率高的非线性地震响应预测方法,能够利用少量数据快速准确地预测桥梁结构在地震作用下的动力响应,在桥梁抗震性能评价中具有重要的应用潜力。

具变指数非线性项的强阻尼波动方程的爆破

考虑一类具变指数非线性项的强阻尼波动方程的有限时刻爆破.借助凹方法和适当选取的参数,给出该问题的一个新的爆破准则,并给出爆破时间的上下界估计.结果表明,该爆破准则特别蕴含对任意高的初始能量,该问题存在有限时刻爆破解.