评“二阶线性微分方程可解的条件”一文

指出该文定理不是新的,例子中的解法也很繁杂,介绍了作者所得到的关于Riccati方程和二阶线性微分方程的一些新的可积类型.     

几类变系数二阶线性微分方程的通解

研究了几类变系数二阶线性微分方程,利用变量代换法将其化为可积方程,从而得到二阶线性微分方程的通解.     

二阶线性微分算子的分解及其应用

本文给出由二阶线性微分算子的分解式求解二阶线性微分方程和二维线性微分方程组的方法,并由此得到它们的一些可积类型与可积的充要条件.     

二阶变系数线性微分方程的几个可积类型

利用变量代换把二阶变系数线性微分方程降阶为一阶线性微分方程,讨论了二阶变系数线性微分方程可积4个充分条件及通解公式.     

关于常微分方程解法的一点评注

著名的 Riccati 方程和二阶线性齐次微分方程,一般说来是不可积的.本文首先评述了前人的某些结果是非实质性的,然后对这两类方程统一地引入了不变式、预解方程和预解常数的概念,得到了这两类方程的一个新的、实用的可积充分条件,导出了一些新的可积类型.     

二阶线性微分方程的可积性判据

文章研究二阶线性微分方程y″+p(x)y′+q(x)y=0可积性.通过寻找p(x),q(x)满足的关系式得到方程可积的充分条件.     

Euler方程的降阶解法

给出了二阶 Euler方程的降阶解法 ,这种解法与传统的解法——通过换元化为常系数线性微分方程相比较有着显著的优点 .对一般的 f(x)易写出通解 ,且该方法易于推广至三阶甚至更高阶的 Euler方程上去 .     

关于几类高阶变系数线性方程的求解

线性常微分方程有着广泛的应用。常系数线性方程及其代数解法已是力学、电学及工程技术中的重要解析工具。在一般的系数激励振动、波导传输理论以及其它许多系统中,人们还常会遇到高阶变系数线性方程或可化为这种方程的一阶线性方程组。如果能求出它们的解析解,将对有关问题的归纳、分析与应用大有帮助。 在文献[4]中,我们利用Riccati方程的几类新的可积类型给出了二阶变系数线性方程的几类新的可积类型,积出了     

模空间上的多线性乘子估计

给出了模空间上多线性乘子新的估计,改进了已知结论,并把所得结果应用于微分方程非线性项的估计.     

具有特殊解结构的二阶线性微分方程的可积性判据

针对二阶线性微分方程y″+p(x)y′+q(x)y=0具有某种特殊解结构的情况下,进行可积性判据研究,利用降阶的思想,得到p(x),q(x)满足的关系式,找到了方程可积的充分条件.     

On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlev′e Equation: an Operator-Splitting Approach

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevé transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang’s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows: (i) The first Painlevé ordinary differential equation, (ii) a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather “violent” phenomenon.     

一类二阶变系数线性微分方程的积分因子解法

通过寻求积分因子f1(x),f2(x),求解一类二阶线性微分方程,包括二阶常系数线性微分方程和二阶Euler方程.     

用“升阶法”求解一类一阶线性微分方程——兼谈逆向思维能力的培养

"升阶法"能够把一类特殊的一阶线性微分方程化为二阶常系数齐次线性微分方程求解,而一般的一阶线性微分方程的求解问题可以转化为二元函数全微分的求积问题.利用"升阶法"和"全微分法"对学生进行逆向思维训练,培养学生的创新思维能力.     

二阶常系数非齐次线性微分方程的通解公式

根据二阶常系数齐次线性微分方程的特征根,利用降阶法,可给出求解一般二阶常系数非齐次线性微分方程的通解公式.     

一个求解约束非线性优化问题的微分方程方法

本文构造的求解非线性优化问题的微分方程方法包括两个微分方程系统,第一个系统基于问题函数的一阶信息,第二个系统基于二阶信息．这两个系统具有性质:非线性优化问题的局部最优解是它们的渐近稳定的平衡点,并且初始点是可行点时,解轨迹都落于可行域中．我们证明了两个微分方程系统的离散迭代格式的收敛性定理和基于第二个系统的离散迭代格式的局部二次收敛性质．还给出了基于两个系统的离散迭代方法的数值算例,数值结果表明基于二阶信息的微分方程方法速度更快．     

The Singular Function Boundary Integral Method for singular Laplacian problems over circular sections

The Singular Function Boundary Integral Method (SFBIM) for solving two-dimensional elliptic problems with boundary singularities is revisited. In this method the solution is approximated by the leading terms of the asymptotic expansion of the local solution, which are also used to weight the governing partial differential equation. The singular coefficients, i.e., the coefficients of the local asymptotic expansion, are thus primary unknowns. By means of the divergence theorem, the discretized equations are reduced to boundary integrals and integration is needed only far from the singularity. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers, the discrete values of which are additional unknowns. In the case of two-dimensional Laplacian problems, the SFBIM converges exponentially with respect to the numbers of singular functions and Lagrange multipliers. In the present work the method is applied to Laplacian test problems over circular sectors, the analytical solution of which is known. The convergence of the method is studied for various values of the order p of the polynomial approximation of the Lagrange multipliers (i.e., constant, linear, quadratic, and cubic), and the exact approximation errors are calculated. These are compared to the theoretical results provided in the literature and their agreement is demonstrated.     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations

In this paper, we propose a new three-level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions, based on cubic spline approximation in x-direction and finite difference approximation in t-direction for the numerical solution of one-space dimensional second order non-linear hyperbolic partial differential equations. We describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method.     

解模守恒微分方程的显式平方守恒格式

对具有模守恒的微分方程,经典的显式Runge—Kutta方法和线性多步方法不能保微分方程的模守恒特性．我们利用李群算法和Cayley变换构造了高阶显式平方守恒格式,应用到模守恒的微分方程如Euler方程,Landau—Lifshitz方程,并且与相同阶的显式Runge—Kutta方法在保模守恒和精度方面进行了比较,数值结果表明用李群算法构造的新的显式平方守恒格式能保微分方程模守恒的特性且它和相应Runge—Kutta方法有相同的精度．     

Linear B-spline finite element method for the improved Boussinesq equation

In this paper, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.