关于常系数齐次线性微分方程组的解法研究

利用凯莱-哈密顿定理给出矩阵指数函数eAt的简洁计算方法;同时利用约当标准形推导出求常系数齐次线性微分方程组通解的循环公式.     

精细积分区段混合能矩阵的一种计算方法

根据结构力学与最优控制的模拟理论中阐述的各混合能矩阵的力学意义,介绍了一种利用微分方程组的状态转移矩阵计算区段混合能矩阵的方法,其计算结果与泰勒级数展开法是一致的。     

k步转移矩阵的频谱插值计算法

本文较全面研究了系统状态转移矩阵的各种计算方法，在分析这些方法在实用计算中各自特点的同时，提出了较为有效的频谱插值计算法，作为应用与比较，文中给出了几个特殊转移矩阵的计算结果．     

k步转移矩阵的频谱插值计算法

本较全面研究了系统状态转移矩阵的各种计算方法，在分析这些方法在实用计算中各自特点的同时，提出了较为有效的频谱插值计算法。作为应用与比较，中给出了几十特殊转移矩阵的计算结果。     

改进的预处理共轭斜量法及其在工程有限元分析中的应用

本文就预处理共轭斜量法(PCCG法)给出了两个具有理论和实际意义的定理,它们分别讨论了迭代解的定性性质和迭代矩阵的构造原则.作者提出了新的非M-矩阵的不完全LU分解技术和迭代矩阵的构造方法.用此改进的PCCG法,对病态问题和大型三维有限元问题进行了计算并与其他方法作了对比,分析了PCCG法在求解病态方程组时的反常现象.计算结果表明本文建议的方法是求解大型有限元方程组和病态方程组的一种十分有效的方法.     

不同元件构成系统中元件维修率分布确定

将Markov链引入SFT理论中,计算可表示环境因素影响的元件维修率分布.研究针对不同元件构成的串联、并联和混联系统中元件的维修率分布计算方法.给出了串联和并联系统中元件维修率推导过程.对状态转移概率的计算不使用Markov状态转移矩阵求解,而是根据Markov状态转移图中的状态关系求解.使用SFT中的元件故障概率分布代替Markov链中的失效率,可得到元件维修率分布.以混联系统作为实例进行分析,使用状态关系求解各状态转移概率关系,得到了3个元件在使用时间t和使用温度c影响下的维修率分布,及正常状态转移概率范围.     

同时考虑故障相关和变故障率的可修系统可靠性计算

基于Copula相关性理论,考虑可修系统零部件工作寿命、故障部件修复时间之间的正相关性,且将零件工作寿命、修复时间放宽到一般连续分布,而不局限于指数分布.提出微时间差t→t+△t内系统一步状态转移矩阵概念,进而演算出状态转移密度矩阵,经系统状态方程,分别给出了任意时刻t单部件、串联型、二不同单元和一修理工组成的并联可修系统的可用度和稳态可用度计算模型.通过算例,说明该理论方法的可行性.     

一种估算马尔柯夫状态转移概率矩阵的新方法

在现有文献研究的基础上,对马尔柯夫状态转移概率矩阵估算方法又作了进一步研究,根据马尔柯夫状态转移概率矩阵的性质和特点,提出了一种新的估算方法.方法首先构造了一个以相对误差绝对值之和最小为目标,以某一状态转移到其他状态的概率之和等于1以及状态转移概率不小于零为约束条件的优化模型.在此基础上,通过变量替换,将该模型转化为线性规划模型.由于线性规划模型不仅能够求得解析解,而且有现成的求解软件,因此不但便于问题求解,而且更加方便、可靠.最后进行了示例计算,验证了给出的马尔柯夫状态转移概率矩阵优化算法的可行性和正确性.     

基于虚拟变形法的供水管网漏损定位方法

针对供水管网的漏损探测与定位问题,采用虚拟变形法对管网各参数进行计算,通过漏损影响矩阵和流量影响矩阵来表征管网的状态,并提出了漏损探测定位的优化计算模型.由于管网漏损点数量事先难以预先获取,提出了基于贝叶斯理论的管网漏损探测、定位模型,并结合Matlab和Epanet编制了相应的优化计算程序,数值仿真结果验证了方法的有效性和可行性.     

我国长期护理保险设计创新及定价研究

随着我国人口老龄化程度不断加深,针对老年人的长期护理保险的定价方法成为保险精算方向的热点问题.本文利用中国老年健康影响因素跟踪调查(CLHLS) 2014–2018年的数据,在传统的三、四状态马尔科夫模型的基础上,进一步将老年人的健康状况划分为六种状态,采用马尔科夫模型对各状态进行数值测算,利用Robinson幂函数综合考虑了性别和年龄两种因素求解健康状态的转移强度矩阵和转移概率矩阵,随后运用双随机Lee-Carter模型、随机游走模型和预期寿命公式估算了65、75和85岁的保费年限,以此为依据给出了长期护理保险的保费计算方法,为我国的长期护理保险定价提供理论参考.     

n阶常系数非齐次线性微分方程的降阶解法

给出一阶线性非齐次微分方程的积分因子解法,避免了常数变易法带来的不便和不自然;给出,n阶常系数非齐次线性微分方程的降阶解法,可以看出,高阶常系数线性非齐次微分方程最终都可以归结为求解一阶线性微分方程,从而避免了待定系数法求非齐次方程特解的繁琐,并最终说明了一般微积分教材中只给出两种类型常系数非齐次线性微分方程的待定系数解法的原因．     

一种辨识投入产出模型中主系数的方法

目前在投入产出模型中四种常用辨识主系数的方法都是针对国民经济主系统而言,而随着我国经济的发展,一些具有相同经济特征的部门组成的国民经济子系统正在国民经济中的地位日趋显著,辨识国民经济子系统的主系数也就显得尤为重要.本文依据直接消耗系数的剔除将会使子系统蕴含受影响对象的信息量遭受损失,没有被剔除的系数集应蕴含受影响对象最少的信息量,以此达到剔除主系数损失最大的思想提出了损失量法,用以辨识国民经济子系统的主系数,并对2002年中国投入产出表的农业系统作了实证分析.同时将损失量法扩展到辨识国民经济主系统的主系数方面,并和信息量法作了比较,证明了损失量法能够比信息量法以更少的主系数达到相同的阈值,即在相同的阈值下损失量法选出的主系数个数比信息量法少.     

Optimization of technology parameters for the plane-strain component in the process of gas quenching

The paper introduces an optimization method for the technology parameters of the plane-strain component in the process of gas quenching. Distortion, residual stress, average surface hardness and standard deviation of surface hardness are regarded as the optimization objectives. A new heat transfer coefficient model is presented, five distinct heat transfer coefficients are used at various regions of the model. The five heat transfer coefficients are regarded as the design variables, and four regressive equations are established by using response surface method. The four equations, respectively represent the relations between the four optimization objectives and the design variables. A multi-objectives optimization model is established, and the multi-objectives optimization model is optimized by the non-linear method. The optimized technology parameters are used to simulate the gas quenching process by FEM software. The quenching results after optimization are compared with those before optimization. The comparison shows that the quenching quality after optimization is better than that before optimization. After optimization, the four optimization objectives are all improved.     

Slip and slide method of factoring trinomials with integer coefficients over the integers

In intermediate and college algebra courses there are a number of methods for factoring quadratic trinomials with integer coefficients over the integers. Some of these methods have been given names, such as trial and error, reversing FOIL, AC method, middle term splitting method and slip and slide method. The purpose of this article is to discuss the Slip and Slide Method and present a theoretical justification of why it works.     

Analysis of reaction-diffusion systems by the method of linear determining equations

Exact solutions to two-component systems of reaction-diffusion equations are sought by the method of linear determining equations (LDEs) generalizing the methods of the classical group analysis of differential equations. LDEs are constructed for a system of two second-order evolutionary equations. The results of solving the LDEs are presented for two-component systems of reaction-diffusion equations with polynomial nonlinearities in the diffusion coefficients. Examples of constructing noninvariant solutions are presented for the reaction-diffusion systems that possess invariant manifolds.     

A numerical approach to solve the model of an electromechanical system

In this paper, the model of an electromechanical system, which is a system of linear differential equations, is studied. Haar wavelet collocation method (HWCM) is applied for finding the approximate solution of the model. HWCM reduces the system of the model into a matrix‐vector form that contains the unknown Haar coefficients, and these coefficients are easily calculated. To demonstrate the validity and applicability of HWCM, numerical solutions of the system for different parameter values in the system are presented. The obtained results demonstrate the efficiency and accuracy of the method. All of the computations are performed via a program written in Mathematica.     

A compact multipoint flux approximation method with improved robustness

Multipoint flux approximation (MPFA) methods were introduced to solve control‐volume formulations on general grids. Although these methods are general in the sense that they may be applied to any grid, their convergence and monotonicity properties vary. We introduce a new MPFA method for quadrilateral grids termed the L‐method. This method seeks to minimize the number of entries in the flux stencils, while honoring uniform flow fields. The methodology is valid for general media. For homogeneous media and uniform grids in two dimensions, this method has four‐point flux stencils and seven‐point cell stencils, whereas the MPFA O‐methods have six‐point flux stencils and nine‐point cell stencils. The reduced stencil of the L‐method appears as a consequence of adapting the method to the closest neighboring cells, or equivalently, to the dominating principal direction of anisotropy. We have tested the convergence and monotonicity properties for this method and compared it with the O‐methods. For moderate grids, the convergence rates are the same, but for rough grids with large aspect ratios, the convergence of the O‐methods is lost, while the L‐method converges with a reduced convergence rate. Also, the L‐method has a larger monotonicity range than the O‐methods. For homogeneous media and uniform parallelogram grids, the matrix of coefficients is an M‐matrix whenever the method is monotone. For strongly nonmonotone cases, the oscillations observed for the O‐methods are almost removed for the L‐method. Instead, extrema on no‐flow boundaries are observed. These undesired solutions, which only occur for parameters not common in applications, should be avoided by requiring that the previously derived monotonicity conditions are satisfied. For local grid refinements, test runs indicate that the L‐method yields almost optimal solutions, and that the solution is considerably better than the solutions obtained by the O‐methods. The efficiency of the linear solver is in many cases better for the L‐method than for the O‐methods. This is due to lower condition number and a reduced number of entries in the matrix of coefficients. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

两类五阶解非线性方程组的迭代算法

本文首先根据Runge-Kutta方法的思想,结合Newton迭代法,提出了一类带参数的解非线性方程组F(x)=0的迭代算法,然后基于解非线性方程f(x)=0的King算法,给出第二类解非线性方程组的迭代算法,收敛性分析表明这两类算法都是五阶收敛的.其次给出了本文两类算法的效率指数,以及一些已知算法的效率指数,并且将本文算法的效率指数与其它方法进行详细的比较,通过效率比率R_(i,j)可知本文算法具有较高的计算效率.最后给出了四个数值实例,将本文两类算法与现有的几种算法进行比较,实验结果说明本文算法收敛速度快,迭代次数少,有明显的优势.     

An exponential matrix method for solving systems of linear differential equations

This paper presents an exponential matrix method for the solutions of systems of high‐order linear differential equations with variable coefficients. The problem is considered with the mixed conditions. On the basis of the method, the matrix forms of exponential functions and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown coefficients are determined and thus the approximate solutions are obtained. Also, an error estimation based on the residual functions is presented for the method. The approximate solutions are improved by using this error estimation. To demonstrate the efficiency of the method, some numerical examples are given and the comparisons are made with the results of other methods. Copyright © 2012 John Wiley & Sons, Ltd.     

辛甫生公式的推广

给出了具有3次代数精度的条件,对辛甫生求积公式进行了推广,利用待定系数方法推导出了一般性的公式,并从该推广的方法导出了三点高斯公式.