完整系统三阶Lagrange方程的一种推导与讨论

从牛顿运动方程出发,推导了完整系统关于广义加速度的Lagrange方程.讨论了该方程与传统分析力学中的Lagrange方程的相容性问题.结果显示,三阶Lagrange方程可以通过对Lagrange方程求一阶时间导数得到,表明它们是相容的.因此三阶Lagrange方程提供了一种不同于传统Lagrange方程方法的求解物体运动方程的途径.     

完整系统三阶Lagrange方程的一种推导与讨论

从牛顿运动方程出发，推导了完整系统关于广义加速度的Lagrange方程.讨论了该方程与传统分析力学中的Lagrange方程的相容性问题.结果显示，三阶Lagrange方程可以通过对Lagrange方程求一阶时间导数得到，表明它们是相容的.因此三阶Lagrange方程提供了一种不同于传统Lagrange方程方法的求解物体运动方程的途径.     

短波模型的Novikov方程族与Sawada-Kotera方程族的刘维尔相关性

主要研究短波模型的Novikov方程族与Sawada-Kotera方程族的对应关系.通过短波模型的Novikov方程与Sawada-Kotera方程等谱问题之间的刘维尔变换联系两个方程族的递推算子,从而建立两个方程族中每一对可积方程之间以及每一对哈密顿守恒律之间的一一对应关系.     

几种广义非线性发展方程的新解

本文研究了构造了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解的问题.利用三种辅助方程及其新解,获得了广义Kd V方程和广义KP-Burgers方程等几种广义非线性发展方程的新解.这些解由双曲余割函数、双曲正切函数、双曲正割函数、双曲余切函数和余割函数组成.     

耦合Gerdjikov-Ivanov方程的多孤子解和无穷守恒律

借用Hirota方法找到耦合Gerdjikov-Ivanov方程的多孤子解.描述了单孤子解和双孤子解的动力特征.耦合Gerdjikov-Ivanov方程可约化至Gerdjikov-Ivanov方程,并且得出Gerdj ikov-Ivanov方程的解.还给出了耦合Gerdj ikov-Ivanov方程的无穷多守恒律.     

FitzHugh-Nagumo方程的分歧北大核心

运用非线性分歧理论,研究FitzHugh-Nagumo方程的定态分歧和Hopf分歧.证明了FitzHugh-Nagumo方程在适当条件下有定态分歧发生,此时FitzHugh-Nagumo方程的定态方程有非平凡解存在.另外还证明了FitzHugh-Nagumo方程在适当的条件下有Hopf分歧发生,此时该方程从平凡解分歧出非平凡的周期解.最后分析得出影响FitzHugh-Nagumo方程分歧发生的主要因素是离子电压门控通道打开与关闭的延迟反应的快慢.理论分析所得结果与实验现象是相一致的.     

带有爆破现象的反应扩散方程的移动网格方法

研究了带有指数非线性项的反应扩散方程的数值解.针对方程在有限时间内会变得非常奇异,提出了移动网格方法和维数分析方法来解该方程.数值结果验证了当移动网格方程具有等分布占优这个性质的时候,移动网格方法求解方程非常有效.另外,数值结果同样显示了等分布占优不是一个必要条件.     

扩展形式的修正的Kadomtsev-Petviashvili方程的多重峰波

研究修正的Kadomtsev-Petviashvili(mKP)方程的一个扩展形式.使用由Hereman和Nuseir提出的、一个可以信赖的、Hirota双线性法的简化形式.由该方程(这里称为mKP方程)直接导出多重峰波解.研究还表明,扩展项并不会破坏mKP方程的可积性.     

具源项的波动方程的非古典对称

给出了具源项的波动方程的非古典对称的完全分类和相应源项的所有可能的具体表达式.除了古典对称对应的巳知源项外,获得了允许非古典对称的新源项,其中包括著名的演化方程,如线性(齐次和非齐次)波动方程,双曲Liouville方程和Klein-Gordon方程等.这些结果解答了Clarkson在2001年中提出的关于波方程非古典对称的公开问题.同时,用分类中得到的对称,通过求不变解构造了以上演化方程的一些新的精确解.     

各项异性椭圆方程基本解的存在性

证明了右端可测的各项异性椭圆方程基本解的存在性,其中应用了各项异性Sobolev空间和Lebesgue空间.首先得到近似方程的解,然后通过对这些解的子列取极限,得到原方程的解.关键是要有一个近似函数空间以及近似方程的先验估计.最后运用Vitali定理证明了原方程基本解的存在性,推广和改进了已有方程.     

Justification of the collocation method for the integral equation for a mixed boundary value problem for the Helmholtz equation

The surface integral equation for a spatial mixed boundary value problem for the Helmholtz equation is considered. At a set of chosen points, the equation is replaced with a system of algebraic equations, and the existence and uniqueness of the solution of this system is established. The convergence of the solutions of this system to the exact solution of the integral equation is proven, and the convergence rate of the method is determined.     

The King-Werner method for solving nonsymmetric algebraic Riccati equation

For the nonsymmetric algebraic Riccati equation arising from transport theory, we concern about solving its minimal positive solution. In [1], Lu transferred the equation into a vector form and pointed out that the minimal positive solution of the matrix equation could be obtained via computing that of the vector equation. In this paper, we use the King-Werner method to solve the minimal positive solution of the vector equation and give the convergence and error analysis of the method. Numerical tests show that the King-Werner method is feasible to determine the minimal positive solution of the vector equation.     

Universal boundary value problem for equations of mathematical physics

A new statement of a boundary value problem for partial differential equations is discussed. An arbitrary solution to a linear elliptic, hyperbolic, or parabolic second-order differential equation is considered in a given domain of Euclidean space without any constraints imposed on the boundary values of the solution or its derivatives. The following question is studied: What conditions should hold for the boundary values of a function and its normal derivative if this function is a solution to the linear differential equation under consideration? A linear integral equation is defined for the boundary values of a solution and its normal derivative; this equation is called a universal boundary value equation. A universal boundary value problem is a linear differential equation together with a universal boundary value equation. In this paper, the universal boundary value problem is studied for equations of mathematical physics such as the Laplace equation, wave equation, and heat equation. Applications of the analysis of the universal boundary value problem to problems of cosmology and quantum mechanics are pointed out.     

二维连续信号的近似采样定理

利用加细方程的面具,给出该方程的一个近似解,并根据这个近似解构造出二维连续信号的近似采样定理.其近似采样函数是所求加细方程的近似解,它是由加细方程的面具唯一确定的逐段线性函数,且有显示的计算公式.因此可以根据需要选择加细方程的面具,从而达到控制近似采样函数的衰减速度.     

Camassa-Holm方程的拟周期解及其渐近行为

近20年来,浅水波模型Camassa-Holm（CH）方程受到诸多研究者关注。在之前的工作中,通过Hirota双线性方法得到了CH方程的单周期解.基于此,该文将对N=2时CH方程的拟周期解及其渐近行为进行研究.首先,回顾了坐标变换,扩展的双线性形式和Riemann(黎曼)θ-函数等内容,并在此基础上利用Hirota双线性方法构造了在N=2时CH方程的含有多个参数的拟周期解,并且此拟周期解是由Riemannθ-函数表示的。其次,发现了此拟周期解渐近行为的一个特点,即CH方程的此拟周期解可以退化为其二孤子解.     

New travelling wave solutions for a nonlinearly dispersive wave equation of Camassa-Holm equation type

In this paper, the integral bifurcation method is used to study a nonlinearly dispersive wave equation of Camassa-Holm equation type. Loop soliton solution and periodic loop soliton solution, solitary wave solution and solitary cusp wave solution, smooth periodic wave solution and non-smooth periodic wave solution of this equation are obtained, their dynamic characters are discussed. Some solutions have an interesting phenomenon that one solution admits multi-waves when parameters vary.     

Solitary wave solution for the generalized Kawahara equation

The travelling wave ansatz is used to find the solitary wave solution of the generalized Kawahara equation. The ansatz is obtained from the structure of the soliton solution of the Kawahara equation and the modified Kawahara equation. The first two integrals of motion of the generalized Kawahara equation are also computed in this work.     

Solution of the matrix Riccati equation for the linear quadratic control problems

This paper is concerned with the solution of the matrix Riccati differential equation with a terminal boundary condition. The solution of the matrix Riccati equation is given by using the solution of the algebraic form of the Riccati equation. An illustrative example for the proposed method is given.     

An Auxiliary Equation for the Bellman Equation in a One-Dimensional Ergodic Control

In this paper we consider the Bellman equation in a one-dimensional ergodic control. Our aim is to show the existence and the uniqueness of its solution under general assumptions. For this purpose we introduce an auxiliary equation whose solution gives the invariant measure of the diffusion corresponding to an optimal control. Using this solution, we construct a solution to the Bellman equation. Our method of using this auxiliary equation has two advantages in the one-dimensional case. First, we can solve the Bellman equation under general assumptions. Second, this auxiliary equation gives an optimal Markov control explicitly in many examples. \keywords{Bellman equation, Auxiliary equation, Ergodic control.} \amsclass{49L20, 35G20, 93E20.} Accepted 11 September 2000. Online publication 16 January 2001.     

An efficient method for quadratic Riccati differential equation

In this paper, a new form of homotopy perturbation method (NHPM) has been adopted for solving the quadratic Riccati differential equation. In this technique, the solution is considered as a Taylor series expansion converges rapidly to the exact solution of the nonlinear equation. Having found the exact solution of the Riccati equation, the capability and the simplicity of the proposed technique is clarified.