极小解法求解最大-最小型关系方程的系统研究

最大-最小型关系方程已有许多学者进行过研究并给出一些解法,这些解法在应用中仍显得过于复杂.本文在文献[1]的基础上对最大-最小型关系方程的极小解的性质和特点进行了系统的研究,并对文献[1]提出的求解这类方程的图法和分枝法在理论上进行了补充和完善.研究表明,图法和分枝法是求解最大-最小型关系方程的有效方法.     

一类非自共轭非线性Schro(?)dinger方程组的有限差分法

近年来,在很多物理问题中都遇到了非线性Schrodinger方程。由于它具有孤立子解和类似于KDV方程的许多性质,因此对其解的适定性研究和数值解法也越来越引起人们的重视,这方面的工作可见[1—5]。然而上述工作中所考虑的方程都是自共轭的,但在一维晶体和α-螺旋生物分子所产生的激子中出现了一类非自共轭的非线性Schrodinger方程,[6]研究了这类非线性Schrodinger方程组的适定性。由于在方程组中出现了非自     

关于二次域理论求解一类Diophantus方程的整数解

本文研究了两个典型Diophantus方程在实二次域中整数解的问题.利用二次域中的理论和二次代数整数环中算术基本定理,得到了该类方程的一般解法和在实二次域中的所有整数解的相关结论,推广了文献[1]和[2]的结果.     

一个方程的几种解法

问题　同学们 ,你会解方程ｘ =2 2 ｘ吗 ?请动笔一试 .解法 1(平方法 )　这是一个无理方程 ,早在读初中的时候 ,同学们就知道无理方程可以通过两边平方将原方程转化为多项式方程 ,从而得 :(ｘ2 - 2 ) 2 - 2 -ｘ =0解这个四次方程 ,可求得ｘ1=- 1- 52 ,ｘ2 =- 1,ｘ3=- 1 52 ,ｘ4 =2 .经检验 ,原方程的根为ｘ =2 .本解法很自然 ,但有一个明显的缺点就是转化后所得的多项式方程次数太高 ,不利于求解 ,也于解法的推广不利 .还有别的解法吗 ?进高中学了不等式性质和熟悉反证法后 ,我们想到 :解法 2 (反证法 )　直接观察就知ｘ =2是原方程的一个…     

任意体上的矩阵方程[XnnAns,XnnBnt]=[Ans,0]

本研究了任意体上的矩阵方程[XnnAns,XnnBnt]=[Ans,0]，（1）?A给出了(1)相容的充要条件、通解的表达式、解的性质及其实用解法。     

奇异积分方程的数值解法(Ⅱ)

本文接[1],继续讨论奇异积分方程的完全方程的数值解法。以下所使用的记号,术语未加指明的均同[1]。     

带Hilbert核的奇异积分方程的数值解法

作者在[1—4]中巳经系统讨论了带Cauchy核的奇异积分方程的数值解法.本文考虑带Hilbert核的奇异积分方程     

也谈慎用图象解法

文 [1 ]告诫人们 :代数问题应慎用图象解法 .下面再举三个例子说明为什么要“慎用图象解法” .例 1　求函数ｙ =2ｓｉｎｘ-13ｃｏｓｘ +2 的最大值和最小值 (文 [2 ]例 2 ) .关于例 1 ,文 [2 ]的解答是解法 1　引进参数ｕ =3ｃｏｓｘ,ｖ =2·ｓｉｎｘ,则点 (ｕ ,Ｖ)在椭圆ｕ23 +ｖ22 =1上 ,易知 (-2 ,1 )到椭圆的切线斜率是ｙ的最大值与最小值 ,设切线方程ｖ =ｋ(ｕ+2 ) +1 ,将其代入椭圆方程并化简为(3ｋ2 +2 )ｕ2 +(1 2ｋ2 +6ｋ)ｕ +(1 2ｋ2 +1 2ｋ -3 ) =0 ,由Δ =0得 ,ｋ2 +4ｋ -1 =0 ,所以ｋ =-2 ± 5 ,所以ｙｍａｘ =-2 +5 ,ｙｍｉｎ …     

离散椭圆型方程的矩阵方程表示及其直接解法

在[1]中讨论了解离散椭圆型方程的直接法,[2]、[3]中在椭圆型方程的类型和区域上得到了进一步的推广.它们的优点是计算起来快速且仅需要极小的存贮量.本文把求解的线性方程组写成矩阵方程形式,进一步发展矩阵分解法.首先,为了简单,如[1]那样考虑方程组     

根式方程的简捷解法

贵刊1984第8期刊出《根式方程解法集锦》文给出要式方程几种简捷解法读后很受启发。根据自己教学实践其给出如下二种解法作为对该文的补充。一、辅助等式法: 先看下面例子: 铡1:解方程 (2x~2-7x+1)~(1/2)-(2x~2-9x+4)~(1/2)=1 解:为解这个方程,引入恒等式; (2x~2-7x+1)~2)~(1/2)-(2x~2-9x~+4)~(1/2)=2x-3与原方程联立成方程组:     

An efficient hybrid numerical method based on an additive scheme for solving coupled systems of singularly perturbed linear parabolic problems

We construct an efficient hybrid numerical method for solving coupled systems of singularly perturbed linear parabolic problems of reaction-diffusion type. The discretization of the coupled system is based on the use of an additive or splitting scheme on a uniform mesh in time and a hybrid scheme on a layer-adapted mesh in space. It is proven that the developed numerical method is uniformly convergent of first order in time and third order in space. The purpose of the additive scheme is to decouple the components of the vector approximate solution at each time step and thus make the computation more efficient. The numerical results confirm the theoretical convergence result and illustrate the efficiency of the proposed strategy.     

一类抛物－椭圆耦合方程组混合初边值问题的二阶收敛差分格式Ⅰ

一类抛物－椭圆耦合方程组混合初边值问题的二阶收敛差分格式Ⅰ孙志忠（东南大学数学力学系）ＡＳＥＣＯＮＤ－ＯＲＤＥＲＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＴＨＥＭＩＸＥＤＩＮＩＴＩＡＬ－ＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＯＦＡＣＬＡＳＳＯＦＰＡＲＡ...     

A note on phase synchronization in coupled chaotic fractional order systems

The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.     

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

This paper is devoted to developing an Il'in‐Allen‐Southwell (IAS) parameter‐uniform difference scheme on uniform meshes for solving strongly coupled systems of singularly perturbed convection‐diffusion equations whose solutions may display boundary and/or interior layers, where strong coupling means that the solution components in the system are coupled together mainly through their first derivatives. By decomposing the coefficient matrix of convection term into the Jordan canonical form, we first construct an IAS scheme for 1D systems and then extend the scheme to 2D systems by employing an alternating direction technique. The robustness of the developed IAS scheme is illustrated through a series of numerical examples, including the magnetohydrodynamic duct flow problem with a high Hartmann number. Numerical evidence indicates that the IAS scheme appears to be formally second‐order accurate in the sense that it is second‐order convergent when the perturbation parameter ϵ is not too small and when ϵ is sufficiently small, the scheme is first‐order convergent in the discrete maximum norm uniformly in ϵ.     

地球科学中的一类耦合抛物方程组的数值模拟

地质流体的性质和动力学行为是当前地球科学研究的前沿领域.铜陵冬瓜山层控夕卡岩型铜矿床成矿作用中矿质输运-化学反应耦合过程的数学模型是一个非局部的耦合抛物方程组初边值问题.本文考虑这一数学模型的数值模拟,用降阶法对此耦合方程组建立了一个具有二阶精度的差分格式.用能量方法给出了差分方程问题解的先验估计式,并证明了差分格式的可解性、稳定性和收敛性,其收敛阶在L2范数下关于时间步长和空间步长均是二阶的.最后给出了数值例子,数值结果和理论分析结果是吻合的.     

Stability analysis and a numerical scheme for fractional Klein‐Gordon equations

Fractional order nonlinear Klein‐Gordon equations (KGEs) have been widely studied in the fields like; nonlinear optics, solid state physics, and quantum field theory. In this article, with help of the Sumudu decomposition method (SDM), a numerical scheme is developed for the solution of fractional order nonlinear KGEs involving the Caputo's fractional derivative. The coupled method provides us very efficient numerical scheme in terms of convergent series. The iterative scheme is applied to illustrative examples for the demonstration and applications.     

Numerical Solution of the Kiessl Model

The Kiessl model of moisture and heat transfer in generally nonhomogeneous porous materials is analyzed. A weak formulation of the problem of propagation of the state parameters of this model, which are so-called moisture potential and temperature, is derived. An application of the method of discretization in time leads to a system of boundary-value problems for coupled pairs of nonlinear second order ODE's. Some existence and regularity results for these problems are proved and an efficient numerical approach based on a certain special linearization scheme and the Petrov-Galerkin method is suggested.     

A nonlinear iteration method for solving a two-dimensional nonlinear coupled system of parabolic and hyperbolic equations

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.     

一类抛物-椭圆耦合方程组混合初边值问题的二阶收敛差分格式(Ⅱ)

2.差分格式的可能性和收敛性我们证明[1]中建立的差分格式(6.1-6.17)是唯一可解且二阶收敛的.记(3.1-3.10)的解为{ψ ,p,q},(4.1-4.13)的解为{ψ,p,q;u_1,u_2,v_1,v_2,w_1,w_2}.假设(3)的系数满足如下条件:当|ε_l|≤ε_0,1≤l≤4时,使得     

Efficient finite-difference multi-scheme approach to the simulation of seismic waves in anisotropic media

This paper presents an original multi-scheme approach to the numerical simulation of seismic wave propagation in models with anisotropic formations. To simulate wave propagation in the anisotropic parts of the model, the Lebedev scheme is used. This scheme is rather universal, but highly expensive in terms of computational efficiency. In the main part of the model, a highly efficient standard staggered grid scheme is proposed. The two schemes are coupled to ensure convergence of the reflection/propagation coefficients with a prescribed order. The algorithm combines the universality of the Lebedev scheme and the efficiency of the standard staggered grid scheme.