曲线搜索的有关理论与数值方法

统一表达了无约束优化问题下降曲线的常微分方程组．证明了两个常见的方程组实质上是参数不同的同一曲线．指出并证明了一种方程组是有利于数值计算的．本文还提出了两个算法一基于积分的搜索法和附加插值法．研究表明曲线寻优与累积迭代信息的策略可以提高优化算法的效率和稳定性．借助于对偶规划本方法对约束优化问题也获得了效率．     

无约束最优化的微分方程方法

一、引言 无约束最优化问题(UO) min f(x) (1.1)就是要寻找目标函数f(x)的极小点x~*. 如果有一从初始点出发的曲线,其每点的切线方向是该点的一个下降方向,就可望能沿该曲线找到x~*,而该种曲线可用常微分方程初值问题加以描述.     

轴对称压差系统的自相似解

考察了二维压差系统的轴对称活塞均匀膨胀而产生的自相似流动.在轴对称和自相似假设下,该问题可以简化为一个自治的非线性常微分方程组的自由边值问题.通过对常微分方程组的积分曲线性质的详细分析,建立该自由边值问题正光滑解的整体存在性.     

关于无约束规划的一个ODE算法的收敛性质

1.引言 所谓无约束规划的ODE算法,就是沿着一个常微分方程组初值问题的解曲线寻找光滑函数f(x)(x∈R~n)的极值点。这类方法近来很受重视,许多人对之进行了研究,见[1]中所列文献。[1]对现有的各种ODE方法进行了总结,并通过大量的数值试验对多种ODE方法以及两个公认的好的传统方法(一个是拟牛顿法,另一个是修正牛顿法)进     

无约束最优化中的一阶曲线法

本文对无约束最优化问题提出一阶曲线法，给出其全局收敛结果．对由微分方程定义的一阶曲线，提出渐近收敛指数概念，并计算出几个常用曲线模型的渐近收敛指数．     

常系数线性分数阶微分方程组的解

本文研究了常系数线性分数阶微分方程组的求解问题.利用逆Laplace变换,Jordan标准矩阵和最小多项式,得到矩阵变量Mittag-Leffler函数的三种不同的计算方法,包含了常系数线性一阶微分方程组的解.     

常微分方程中的反问题

研究某函数或函数组是什么常微分方程的通解或特解,这可以称为常微分方程中的反问题.这类问题,可以用"微分法"来解决.研究这类问题的意义在于通过利用"微分法"及"逆向思维方法"解决反问题的过程来加强对常微分方程理论内涵的深刻理解.     

一类三维神经元模型的分支研究

考虑一类三维神经元模型的分支问题.利用常微分方程的定性与分支理论的知识,讨论了模型的平衡点个数及其稳定性,主要分析了平衡点的Hopf分支和Bogdanov-Takens分支,并得到了相应的鞍结点分支曲线,Hopf分支曲线与同宿分支曲线.     

二阶椭圆型偏微分方程奇异摄动问题的差分解法

近十多年来出现了一系列奇异摄动问题的数值解法,例如C.E.Pearson,F.W.Door,H.O.Kreiss,A.M.,K.B.等人的工作,他们大都对常微分方程和常微分方程组奇异摄动问题来讨论的。 苏煜城、吴启光在1980年用差分方法讨论了椭圆——抛物偏微分方程的奇异摄动问     

人卫精密定轨中的算法问题

对当今人卫精密定轨问题 ,由于力学模型复杂 ,精密星历和状态转移矩阵的计算均采用数值方法 ,这就需要积分两组常微分方程 .现给出一个方法可避免数值求解两组常微分方程 ,并以实际算例证实了算法的有效性 .     

Complete group classification of systems of two linear second-order ordinary differential equations

We give a complete group classification of the general case of linear systems of two second-order ordinary differential equations excluding the case of systems which are studied in the literature. This paper gives the initial step in the study of nonlinear systems of two second-order ordinary differential equations. It can also be extended to systems of equations with more than two equations. Furthermore the complete group classification of a system of two linear second-order ordinary differential equations is done. Four cases of linear systems of equations with inconstant coefficients are obtained.     

Complex Lie symmetries for scalar second-order ordinary differential equations

A scalar complex ordinary differential equation can be considered as two coupled real partial differential equations, along with the constraint of the Cauchy–Riemann equations, which constitute a system of four equations for two unknown real functions of two real variables. It is shown that the resulting system possesses those real Lie symmetries that are obtained by splitting each complex Lie symmetry of the given complex ordinary differential equation. Further, if we restrict the complex function to be of a single real variable, then the complex ordinary differential equation yields a coupled system of two ordinary differential equations and their invariance can be obtained in a non-trivial way from the invariance of the restricted complex differential equation. Also, the use of a complex Lie symmetry reduces the order of the complex ordinary differential equation (restricted complex ordinary differential equation) by one, which in turn yields a reduction in the order by one of the system of partial differential equations (system of ordinary differential equations). In this paper, for simplicity, we investigate the case of scalar second-order ordinary differential equations. As a consequence, we obtain an extension of the Lie table for second-order equations with two symmetries.     

Numerical Solution of Systems of Loaded Ordinary Differential Equations with Multipoint Conditions

A system of loaded ordinary differential equations with multipoint conditions is considered. The problem under study is reduced to an equivalent boundary value problem for a system of ordinary differential equations with parameters. A system of linear algebraic equations for the parameters is constructed using the matrices of the loaded terms and the multipoint condition. The conditions for the unique solvability and well-posedness of the original problem are established in terms of the matrix made up of the coefficients of the system of linear algebraic equations. The coefficients and the righthand side of the constructed system are determined by solving Cauchy problems for linear ordinary differential equations. The solutions of the system are found in terms of the values of the desired function at the initial points of subintervals. The parametrization method is numerically implemented using the fourth-order accurate Runge–Kutta method as applied to the Cauchy problems for ordinary differential equations. The performance of the constructed numerical algorithms is illustrated by examples.     

New origins of hidden symmetries for partial differential equations

Hidden symmetries of differential equations are point symmetries that arise unexpectedly in the increase (equivalently decrease) of order, in the case of ordinary differential equations, and variables, in the case of partial differential equations. The origins of Type II hidden symmetries (obtained via reduction) for ordinary differential equations are understood to be either contact or nonlocal symmetries of the original equation while the origin for Type I hidden symmetries (obtained via increase of order) is understood to be nonlocal symmetries of the original equation. Thus far, it has been shown that the origin of hidden symmetries for partial differential equations is point symmetries of another partial differential equation of the same order as the original equation. Here we show that hidden symmetries can arise from contact and nonlocal/potential symmetries of the original equation, similar to the situation for ordinary differential equations.     

Meromorphic solutions of nonlinear ordinary differential equations

Exact solutions of some popular nonlinear ordinary differential equations are analyzed taking their Laurent series into account. Using the Laurent series for solutions of nonlinear ordinary differential equations we discuss the nature of many methods for finding exact solutions. We show that most of these methods are conceptually identical to one another and they allow us to have only the same solutions of nonlinear ordinary differential equations.     

Criteria for fourth-order ordinary differential equations to be linearizable by contact transformations

Solution of linearization problem of fourth-order ordinary differential equations via contact transformations is presented in the paper. We show that all fourth-order ordinary differential equations that are linearizable by contact transformations are contained in the class of equations which is at most quadratic in the third-order derivative. We provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation. Moreover, we obtain the general form of ordinary differential equations of order greater than four linearizable via contact transformations.     

Application of group analysis to classification of systems of three second‐order ordinary differential equations

Here, we give a complete group classification of the general case of linear systems of three second‐order ordinary differential equations excluding the case of systems which are studied in the literature. This is given as the initial step in the study of nonlinear systems of three second‐order ordinary differential equations. In addition, the complete group classification of a system of three linear second‐order ordinary differential equations is carried out. Four cases of linear systems of equations are obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Degasperis-Procesi方程的类孤子新解

利用辅助方程与函数变换相结合的方法,构造了Degasperis-Procesi(D-P)方程的无穷序列类孤子新解.首先,通过两种函数变换,把D-P方程化为常微分方程组.然后,利用常微分方程组的首次积分,把D-P方程的求解问题化为几种常微分方程的求解问题.最后,利用几种常微分方程的Bcklund变换等相关结论,构造了D-P方程的无穷序列类孤子新解.这里包括由Riemannθ函数、Jacobi椭圆函数、双曲函数、三角函数和有理函数组成的无穷序列光滑孤立子解、尖峰孤立子解和紧孤立子解.     

Group properties of the extended Green–Naghdi equations

The group analysis method is applied to the extended Green–Naghdi equations. The equations are studied in the Eulerian and Lagrangian coordinates. The complete group classification of the equations is provided. The derived Lie symmetries are used to reduce the equations to ordinary differential equations. For solving the ordinary differential equations the Runge–Kutta methods were applied. Comparisons between solutions of the Green–Naghdi equations and the extended Green–Naghdi equations are given.     

Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations

In this work, the asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses is investigated. By impulsive differential inequality and Riccati transformation, sufficient conditions of asymptotic behavior of all solutions of second-order nonlinear ordinary differential equations with impulses are obtained. An example is also inserted to illustrate the impulsive effect.