当凌绝顶一览众山——一道课本习题的溯源过程及应用

《普通高中课程标准实验教科书·数学5》(必修)P75页习题3.1A组第3题:例题 当x＞0时,求证:√1+x＜1+x/2.本题在高中的不同阶段有不同的解法,由于本题出现在必修五《不等式》的习题中,教师一般会向学生介绍综合法或分析法证明该不等式.     

球面上第二类Fredholm积分方程配置方法

球面上第二类 Fredholm积分方程经球坐标变换可化为矩形域 H0 上的问题求解 .用有限元法构造H0 上的插值函数 ,它必须满足在 H0 的左、右两边连续 ,然后用配置方程求方程的近似解     

2010年高考江苏卷第18题的别解

2010年高考江苏卷第18题一定有难度,但前两小题应该很好解决,主要涉及直线与圆锥曲线的交点、轨迹的概念和轨迹方程的求法,有难度的在第(3)小题,主要集中在计算上,有很多同学有解决问题的方法和方向,但要真正解决问题,计算是关键.2010年高考江苏卷第18题:在平面直角坐标     

一个多因子信用违约互换定价模型

考虑一个违约互换的多因子的简化模型,通过测度的转化并求解一个多变量Riccati常微分方程而得出了这个模型的解析解.此模型是Duffie-Pan-Singleton(2000)模型的特例,但是这个模型存在解析解,而Duffie-Pan-Singleton模型并不存在解析解.模型的解析解对获得直觉的判断和进行模型的计算都非常有帮助,而且模型的解析解对实证检验也是有帮助的.     

二阶Dirichlet问题的三维等参元逼近及其数值积分格式

本文研究了求解三维二阶Drichlet问题的四面体等参元逼近格式.基于对等参变换的细致估计,证明了格式的收敛性,并得到了H~1-误差的最优估计.为简化等参元的复杂计算,本文还构造了一种简单的数值积分格式,并证明了其收敛性.     

NUMERICAL LOCALIZATION OF ELECTROMAGNETIC IMPERFECTIONS FROM A PERTURBATION FORMULA IN THREE DIMENSIONS

This work deals with the numerical localization of small electromagnetic inhomogeneities. The underlying inverse problem considers, in a three-dimensional bounded domain, the time-harmonic Maxwell equations formulated in electric field. Typically, the domain contains a finite number of unknown inhomogeneities of small volume and the inverse problem attempts to localize these inhomogeneities from a finite number of boundary measurements. Our localization approach is based on a recent framework that uses an asymptotic expansion for the perturbations in the tangential boundary trace of the curl of the electric field. We present three numerical localization procedures resulting from the combination of this asymptotic expansion with each of the following inversion algorithms： the Current Projection method, the MUltiple Signal Classification （MUSIC） algorithm, and an Inverse Fourier method. We perform a numerical study of the asymptotic expansion and compare the numerical results obtained from the three localization procedures in different settings.     

A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF THE CAHN-HILLIARD EQUATION AND THE HELE-SHAW FLOW

This paper develops a posteriori error estimates of residual type for conforming and mixed finite element approximations of the fourth order Cahn-Hilliard equation ut ＋ △（ε△Au-ε^-1f（u）） = 0. It is shown that the a posteriori error bounds depends on ε^-1 only in some low polynomial order, instead of exponential order. Using these a posteriori error estimates, we construct at2 adaptive algorithm for computing the solution of the Cahn- Hilliard equation and its sharp interface limit, the Hele-Shaw flow. Numerical experiments are presented to show the robustness and effectiveness of the new error estimators and the proposed adaptive algorithm.