Kupershmidt浅水波方程的线性化,相似约化和孤波解

利用未知函数变换方法，找到了Ｋｕｐｅｒｓｈｍｉｄｔ方程到Ｂｕｒｇｅｒｓ方程及热传导方程间的Ｂａｃｋｌｕｎｄ变换并借此给出了Ｋｕｐｅｒｘｈｍｉｄｔ方程四种杨似约化和一组孤波解。     

渗流方程自适应非均匀网格Dagan粗化算法

在粗网格内先统计渗透率在粗网格中的概率分布，利用Dagan渗透率粗化积分方程通过渗透率概率分布计算粗化网格的等效渗透率，并由等效渗透率计算了粗化网格的压强分布，计算压强时还将渗透率自适应网格技术应用于三维渗流方程的网格粗化算法中，在渗透率或孔隙度变化异常区域自动采用精细网格，用直接解法求解渗透率或孔隙度变化异常区域的压强分布。整个求解区采用不均匀网格粗化，在流体流速高的区域采用精细网格。利用本文方法计算了三维渗流方程的压强分布，结果表明这种算法的解在渗透率或孔隙度异常区的压强分布规律非常逼近精细网格的解，在其他区域压强分布规律非常逼近粗化算法的解，计算速度比采用精细网格提高了约100倍。     

层合板分叉方程的Lyapunov—Schmidt约化分析

采用广义傅立叶级数法建立了具有弹性约束的复合材料矩形层板在面内载荷作用下的非线性稳定性控制方程,并简化为矩阵形式。利用分叉理论和泛函知识,对有限维的该分叉方程进行了Lyapunov-Schmidt约化,获得了三种典型的分叉图形式,同时指出当非齐次项等于零时必然发生分叉。数值计算结果表明了三种分叉图分别所对应的典型的力学模型,主要因素在于边界条件、铺层方式及初始缺陷三方面。     

梁中非线性弯曲波传播特性的研究

研究了梁中的非线性弯曲波的传播特性，同时考虑了梁的大挠度引起的几何非线性效应和 梁的转动惯性导致的弥散效应，利用Hamilton变分法建立了梁中非线性弯曲波的波动方程. 对该方程进行了定性分析，在不同的条件下，该方程在相平面上存在同宿轨道或异宿轨道， 分别对应于方程的孤波解或冲击波解. 利用Jacobi椭圆函数展开法，对该非线性方程进行 求解，得到了非线性波动方程的准确周期解及相对应的孤波解和冲击波解，讨论了这些解存 在的必要条件，这与定性分析的结果完全相同. 利用约化摄动法从非线性弯曲波动方程中导 出了非线性Schr\"{o}dinger方程，从理论上证明了考虑梁的大挠度和转动惯性时梁中存在 包络孤立波.     

网络RTK参考站间模糊度动态解算中病态方程的一种解算方法

在网络RTK参考站间的模糊度估计中,若误差方程严重病态,将导致模糊度解与其准确值偏差较大或整周模糊度无法固定,因此提出了一种适于网络RTK模糊度动态解算的新方案：1）法方程病态性的判断;2）Tikhonov正则化解算病态方程;3）LAMBDA方法搜索固定整周模糊度。同时,深入研究了Tikhonov正则化矩阵的构造方法和正则化参数的选取准则。最后以实例验证了采用此方案解算病态方程是可行的,通过选取合适的正则化参数可以解得准确的整周模糊度;详细讨论了选择不同的正则化参数对模糊度解算结果的影响。     

解两点边值问题的精细循环约化方法

将精细积分技术与循环约化方法相结合,提出两点边值问题的一种高精度、高效率求解方法。将求解域均匀离散,利用相邻两点间的传递关系式建立区段代数方程,将各区段的代数方程集成代数方程组,并利用循环约化方法对其求解。由于离散过程中几乎没有引入离散误差,并且在循环约化过程中采用了大量、小量分离技术,因此本方法具有极高的精度;同时循环约化过程充分利用2N算法的特点,使计算效率高、存储量小。研究结果表明,相对于已有的求解两点边值问题的精细积分法,本文方法适用范围更广,效率更高。例如对两端固支、受均布横向荷载作用下梁的非齐次方程计算,本文方法的精度可达到小数点后十三位,已经非常精确。     

Fokker-Planck方程有限解析/Monte Carlo数值模拟方法

对白噪声驱动随机系统的Fokker-Planck方程进行约化，求得约化方程的解析解，使 用局部解析解和Monte Carlo结合方法求解常系数Fokker-Planck方程，并与常系数Fokker-Planck方程的精确解 进行对比，之后求解了变驱动力系统的行为. 数值模拟结果表明，有限解析/Monte Carlo结合的方法，能成功求解一维Fokker-Planck方程，求解粒子数为10$^{5}$个，能获得 十分光滑的PDF分布曲线，计算颗粒在300个时，就能获得较好的均值. 其研究为两相 湍流PDF模型新计算方法研究提供基础.     

Banach空间非线性脉冲微分积分方程的整体解

本文考虑Ｂａｎａｃｈ空间中混合单调脉冲微分积分方程,利用混合单调迭代方法及Ｍｏｅｎｃｈ不动点定理,给出所论方程解和耦合最小最大解的存在性定量及单调迭代方法,改进和推广文（１－４）中的相应结果。     

摄动有限差分方法研究进展

振动有限差分（ＰＦＤ）方法,既离散徽商项也离散非微商项（包括微商系数）,在微商用直接差分近似的前提下提高差分格式的精度和分辨率．ＰＦＤ方法包括局部线化微分方程的摄动精确数值解（ＰＥＮＳ）方法和摄动数值解（ＰＮＳ）方法以及考虑非线性近似的摄动高精度差分（ＰＨＤ）方法。论述了这些方法的基本思想、具体技巧、若干方程（对流扩散方程、对流扩散反应方程、双曲方程、抛物方程和ＫｄＶ方程）的ＰＥＮＳ、ＰＮＳ和ＰＨＤ格式,它们的性质及数值实验．并与有关的数值方法作了必要的比较．最后提出值得进一步研究的一些课题．      

关于Lindelf方程——分析力学札记之十四

研究Lindelof方程的历史作用以及现实意义.失败是成功之母,科学错误也是有功劳的.     

一种求解局部非线性结构稳态响应的方法

为了降低求解局部非线性结构稳态响应的计算量,基于子结构和阻抗缩聚提出了一种用于求解局部非线性结构稳态响应的计算方法.将局部非线性结构分解为线性子结构和非线性子结构,利用谐波平衡构造各个子结构的阻抗方程,对线性子结构进行缩聚,将局部非线性动力学方程转化为求解一组非线性代数方程组问题,通过迭代求解非线性代数方程组,求解系统的稳态响应.     

Higher Conditional Symmetry and Reduction of Initial Value Problems

We give the exposition of a generalized symmetry approach toreduction of initial value problems for nonlinear evolutionequations in one spatial variable. Using this approach we classifythe initial value problems for third-order evolution equationsthat admit reduction to Cauchy problems for systems of twoordinary differential equations. These reductions are shown tocorrespond to higher conditional symmetries admitted by thecorresponding nonlinear evolution equations.     

New Answers on the Interaction Between Polymers and Vortices in Turbulent Flows

Numerical data of polymer drag reduced flows is interpreted in terms of modification of near-wall coherent structures. The originality of the method is based on numerical experiments in which boundary conditions or the governing equations are modified in a controlled manner to isolate certain features of the interaction between polymers and turbulence. As a result, polymers are shown to reduce drag by damping near-wall vortices and sustain turbulence by injecting energy onto the streamwise velocity component in the very near-wall region.     

行进绳索在横向流体激励下的运动

给出横向流体对行进绳索的作用力描述,建立了绳索的动力学方程。由于该方程具有零刚度特征,引入Pilipchuk变换,以自平衡状态起度量的径向振动和回转运动来描述绳的运动,获得非零刚度系统。然后,用两变量参数摄动法求得绳索关于扩张振动和回转运动的约化型,使得绳索运动可近似由一个二维动力系统来描述。最后,用数值方法讨论了行进速度和重力对绳索运动形态的影响。     

Hybrid (numerical-experimental) modeling of complex structures with linear and nonlinear components

The present work investigates the performance of two systematic methodologies leading to hybrid modeling of complex mechanical systems. This is done by applying numerical methods in determining the equations of motion of some of the substructures of large order mechanical systems, while the dynamic characteristics of the remaining components are determined through the application of appropriate experimental procedures. In their simplest version, the models examined are assumed to possess linear characteristics. For such systems, it is possible to apply several hybrid methodologies. Here, the first of the methods selected is performed in the frequency domain, while the second method has its roots and foundation in time domain analysis. Originally, the accuracy and effectiveness of these methodologies is illustrated by numerical results obtained for two complex mechanical models, where the equations of motion of each substructure are first set up by applying the finite element method. Then, the equations of motion of the complete system are derived and their dimension is reduced substantially, so that the new model is sufficiently accurate up to a prespecified level of forcing frequencies. The formulation is developed in a general way, so that application of other methods, including experimental techniques, is equally valid. This is actually performed in the final part of this study, where experimental results are employed in conjunction with numerical results in order to predict the dynamic response of a mechanical structure possessing a linear substructure with high modal density, supported on four substructures with strongly nonlinear characteristics.     

Numerical viscosity reduction in the resolution of the shallow water equations with turbulent term

A first‐order finite volume model for the resolution of the 2D shallow water equations with turbulent term is presented. An upwind discretization of the equations that include the turbulent term is carried out. A method to reduce the excess of numerical viscosity (or diffusion) produced by the upwinding of the flux term is proposed. Two different discretizations of the turbulent term are compared, and results for uniform distributions of the viscosity are presented. Finally, two discretizations of the time derivative which are more efficient than Euler's are proposed and compared. Copyright © 2008 John Wiley & Sons, Ltd.     

Gram determinant solutions to nonlocal integrable discrete nonlinear Schrödinger equations via the pair reduction

In this paper, Gram determinant solutions of local and nonlocal integrable discrete nonlinear Schrödinger (IDNLS) equations are studied via the pair reduction. A generalized IDNLS equation is firstly introduced which possesses the single Casorati determinant solution. Two kinds of Gram determinant solutions are presented from Casorati determinant ones due to the gauge freedom. The different pair constraint conditions for wave numbers are imposed and then solutions of local and nonlocal IDNLS equations are derived in terms of Gram determinant.     

Dimensional Reduction for Nonlinear Time-Delayed Systems Composed of Stiff and Soft Substructures

This paper presents a new approach, based on the center manifoldtheorem, to reducing the dimension of nonlinear time-delay systemscomposed of both stiff and soft substructures. To complete the reductionprocess, the dynamic equation of a delayed system is first formulated asa set of singular perturbed equations that exhibit dynamic behaviorevolving in two different time scales. In terms of the fast time scale,the dynamic equation of system can be converted into the standard formof a functional differential equation in critical cases, namely, to aform that can be treated by means of the center manifold theorem. Then,the approximated center manifold is determined by solving a series ofboundary-value problems. The center manifold theorem ensures that thedominant dynamics of the system is described by a set of ordinarydifferential equations of low order, the dimension of which is identicalto that of the phase space of slowly variable states. As an applicationof the proposed approach, a detailed stability analysis is made for aquarter car model equipped with an active suspension with a time delaycaused by a hydraulic actuator. The analysis shows that the dimensionalreduction is surprisingly effective within a wide range of the systemparameters.     

Integration of Ordinary Differential Equations via Nonlocal Symmetries

We present a nonlocal symmetry method to reduce scalar first- and second-orderordinary differential equations (ODEs) to quadratures. It is shown that a second-orderODE admitting a non-Abelian two-dimensional Lie algebra of point symmetriesis reducible to quadratures via a nonideal of the algebra. We also providea direct method of integration for a first-order ODE admitting an exponential nonlocal symmetry which satisfies an additional property.Moreover, we give examples, two on double reduction and several on Abel equations of the second kind, that illustrate ourapproaches.