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2007年工作年报



一、机构设置和聘任工作
二、科学研究方向、项目和经费
三、科学研究成果
四、学术活动
五、学科建设和人才培养
七、2007年论文摘要

一、机构设置和聘任工作
1.根据《上海高校计算科学E—研究院建设总体规划书》和《上海高校计算科学E
—研究院管理章程》等文件,制订和执行2007年度工作计划。
2.郭本瑜教授为首席研究员,并聘请下列专家为特聘研究员:
  • 张伟江 上海交通大学教授
  • 程  晋 复旦大学教授
  • 黄建国 上海交通大学教授
  • 丛玉豪 上海师范大学教授
  • 岳荣先 上海师范大学教授
  • 王元明 华东师范大学教授
  • 徐承龙 同济大学教授
  • 田红炯 上海师范大学教授
  • 王中庆 上海师范大学教授
3.由下列专家组成学术委员会:
  • 主任:石钟慈 中国科学院院士
  • 委员:林  群中国科学院院士
  • 姜礼尚 同济大学教授
  • 郭本瑜 上海师范大学教授
  • 张伟江 上海交通大学教授
  • 吴宗敏 复旦大学教授
  • 马和平 上海大学教授
  • 香港城市大学王世全教授为研究院顾问。
4.田红炯教授兼任业务秘书,王维敏同志任行政秘书。
二、科学研究方向、项目和经费
1.根据E—研究院科学研究方向,制订并资助本年度研究课题,承担国家和上海市其它科研项目,积极申请新的科研项目。
2.目前的主要研究方向:
  • 数学物理问题的高精度算法
  • 动力系统的数值研究
  • 弹性组合结构的数值方法
  • 金融随机模型的数值方法
  • 伪蒙特卡罗方法
  • 反问题的数值方法
3.本年度资助下列研究课题,共30.5万。
  • 郭本瑜 数学物理问题的高精度算法4万
  • 程  晋 数学物理方程的反问题及其算法4万
  • 黄建国组合弹性结构问题的有限元方法研究3.5万
  • 岳荣先 随机化伪蒙特卡罗方法的理论与应用研究3.5万
  • 田红炯 滞时微分系统的数值动力学分析3.5万
  • 丛玉豪 常微数值方法在求解时滞方程及Hamilton系统中的应用3万
  • 徐承龙 金融衍生物的偏微分方程定价及计算3万
  • 王元明 非线性(初)边值问题的高精度有限差分方法3万
  • 王中庆 无界区域问题和外部问题的谱方法3万
4.特聘研究员承担了15项国家、教育部和上海市科研项目,本年度到达的总经费221.8万元。
A.国家科研项目6个,本年度到达经费83.2万元。
  • 郭本瑜   国家自然科学基金项目,奇异问题及非矩形和无界区域问题的谱方法。
  • 程  晋   国家自然科学重点项目,数学物理方程的反问题及其应用,36万元。
  • 王元明   国家自然科学基金,非线性椭圆型边值问题的高精度有限差分方法,7.2万元。
  • 徐承龙   国家重点基础研究发展计划(973计划)子课题,信用风险分析和信用衍生产品定价,10万。
  • 田红炯   国家自然科学基金,滞时泛函微分动力系统的计算方法,18万元。
  • 岳荣先   国家自然科学基金,高维积分的伪蒙特卡洛算法及其在巴拿赫空间上的误差分析,12万元。
B.教育部及上海市科研项目9个,本年度到达经费138.6万元。
  • 郭本瑜   上海市科委重大科技攻关计划,高性能计算方法研究,90万元。
  • 郭本瑜   上海市科委重点项目,若干数学物理复杂问题的计算方法。
  • 郭本瑜   国家教育部博士点基金,奇异问题和无界区域问题的谱方法及其应用。
  • 程  晋   国家教育部博士点基金,偏微分方程的反问题及其数值解,3.6万元。
  • 程  晋   上海市东方论坛,5万元。
  • 田红炯   上海市科委基础研究重点项目,泛函微分动力系统的计算方法及其应用,24万元。
  • 田红炯   上海市教委曙光计划项目,泛函微分动力系统的数值模拟及其应用,6万元。
  • 岳荣先   上海市教委科研基金,随机化伪Monte Carlo积分法的易处理性研究。
  • 丛玉豪   上海市教委科研基金,非自治无穷维Hamilton系统的多辛几何算法,10万元。
5.最近申请并获准主持7个国家、教育部和上海市科研项目,总经费1002万元。这些项目将从2008年开始执行。
  • 程  晋   教育部引智基地,复旦大学现代应用数学引智基地,900万元。
  • 程  晋   上海市重点项目,飞行器设计中的数学模型及其模拟与分析,30万元。
  • 黄建国   国家自然科学基金,组合弹性结构问题的自适应有限元方法研究,24万元。
  • 黄建国   国家973计划项目,高性能科学计算研究子课题,6万元。
  • 王中庆   国家自然科学基金,外部问题的高精度算法及其应用,25万元。
  • 王中庆   国家973计划项目,高性能科学计算研究子课题,9万元。
  • 丛玉豪   国家自然科学基金,科学部主任基金项目,延时微分方程离散及连续化计算方法,8万元。
 
三、科学研究成果
通过研究院成员和国内、外学者的合作研究,本年度在奇异问题、无界区域和外部问题的高精度数值方法,数学物理反问题的数学模型和数值方法,组合弹性结构的数学模型和计算方法,随机化伪Monte-Carlo方法及其应用,动力系统的数值研究和高精度紧致差分方法等方面取得了一批具有国际先进水平的研究成果,在国内外重要学术刊物上发表了22篇论文,其中某些成果是原创性的,获得国际同行专家的高度评价。此外,出版了两本国际会议学术论文集。

1.数学物理问题的高精度算法(郭本瑜,王中庆)
  • 研究直角三角形上Dubiner正交多项式系的逼近性质,并由此构造直角三角形上的谱方法。
  • 建立一类新的广义Laguerre正交逼近及其插值的基本理论,并应用于有关退化问题及外部问题的计算。数值结果显示了这些新算法的优越性。
  • 建立混合Jacobi—球面调和正交逼近理论,并应用于三维Navier-Stokes方程的计算。数值结果显示了这些新算法的优越性。
  • 建立以Laguerre正交多项式及Laguerre正交函数系为基底的常微分方程初值问题的新配置法,为计算动力系统提供新的高精度数值方法。
有关论文:
[1]Guo Ben-yu and Wang Li-lian, Error analysis of spectral method on a triangle, Adv. in Comp. Math., 26(2007), 473-496.
[2]Ben-yu Guo and Xiao-yong Zhang, Spectral method for differential  equations of degenerate type on unbounded domains by using generalized  Laguerre functions, Appl. Numer. Math., 57(2007), 455-471.
[3]Ben-yu Guo and Wei Huang, Mixed Jacobi-spherical harmonic spectral method for Navier-Stokes equations, Appl. Numer. Math., 57(2007), 939-961.
[4]Ben-yu Guo and Zhong-qing Wang, Numerical integration based on Laguerre-Gauss interpolation, Comp. Meth. in Appl. Mech. and Engr., 196(2007), 3726-3741.

2.动力系统的数值方法(田红炯, 丛玉豪)
  • 基于奇异摄动滞时微分方程的解的渐近表达形式,构造了具有一致收敛性的非线性数值方法。
  • 研究多步法求解非线性滞时微分方程耗散性,证明了A-稳定的多步方法能够保持长时间数值计算的相应几何结构。
  • 建立了Lyapunov泛函和滞时微分方程的渐近稳定集之间的等价关系,在此基础上给出了Runge-Kutta方法求解滞时微分方程的渐近稳定集分析。
  • 给出了具有无界滞时的非线性离散Volterra方程周期解等的存在性。
  • 研究了块θ-方法求解延迟微分方程的数值稳定性。分析了用其求解多时滞微分方程系统时的渐近稳定性和L-型稳定性,给出并证明了其PLm-稳定和GPLm—稳定的充分必要条件。同时讨论了θ-方法求解多延迟系统的渐近稳定的充分必要条件。
有关论文:
[1]Y. Song and H. Tian, Periodic and Almost Periodic Solutions of Nonlinear Discrete Volterra Equations with Undounded Delay, J. Comp. Appl. Math., 205(2007), 859 –870.
[2]Y. Sun, D. Zhang and H. Tian, Numerical Methods for Singularly Perturbed Delay Differential Equations, J. Syst. Simu., 19(2007), 3943-3944,3992.
[3]Jiang Z., Fu X., Tian H., A Simple Proof of Inequalities of Integrals of Composite Functions, J. Math. Anal. Appl., 332(2007), 1307-1312.
[4]H. Tian, L. Fan and J. Xiang, Numerical Dissipativity of Multistep Methods for Delay Differential Equations, Appl. Math. Comp.,188(2007), 934-941.
[5]Cong Yuhao, Li Shundao, Tan Xiuli, GPLm-- stability of  block θ-method for delay differential equations, J. Syst. Simu., 19(2007), 3937-3939.

3.组合弹性结构问题的有限元方法(黄建国)
  • 利用控制理论中的研究技巧,通过构造恰当的非线性泛函获得求解Cauchy问题的一个有效方法。给出方法在连续情形和有限元情形的收敛性分析,并对Laplace方程和平面应力问题应用该方法进行数值模拟,取得理想计算结果。
  • 给出三维椭圆型和静态Maxwell方程interface问题的新型先验正则性估计,该估计显式地说明了介质系数对解的影响。为了得到这些结果,要建立分片调和函数的一个巧妙表达公式,并利用非光滑流形上的位势理论和形式渐近分析等技巧。法国学者S. Labrunie 在发表于MathSciNet上对该文的论文评述中,认为文中结果是“novel, elegant”。
  • 构造求解一般组合弹性结构问题的TRUNC元型有限元方法。通过建立非协调元空间上的离散Korn不等式,利用Lax-Milgram引理证明问题解的存在唯一性。同时,利用细致的误差估计技巧,获得能量范数下方法的拟最优误差估计。数值实验结果和理论结果相吻合。
有关论文:
[1]W. Han, J. Huang,  K. Kazmi, and Y. Chen, A numerical method for a Cauchy problem for elliptic partial differential equations, Inverse Problems, 23(2007),  2401-2415.
[2]J. Huang, L. Guo and Z. Shi, A finite element method for general elastic multi-structures, Computers Math. with Appl., 53(2007), 1867-1895.
[3]J. Huang and J. Zou, Uniform a priori estimates for elliptic and static Maxwell interface problems, Discrete and Continuous Dynamical Systems Ser. B, 7(2007), 145-170.

4.金融衍生物的偏微分方程定价及计算(徐承龙)
  • 研究Monte Carlo方法及其在金融中的应用,并对欧式,美式以及其它与路径有关期权的Monte Carlo计算方法进行系统的研究。
  • 对国内市场上的几个金融衍生产品进行较系统的研究。
有关论文:
[1]Xu Cheng-long, Spectral-domain decomposition method and its applications in finance, Recent Progress in Scientific Computing, edited by Wen-bin Liu, Michael Ng and Zhong-ci Shi, Science Press, Beijing, 2007, 367-381.
[2]刘畅,徐承龙,谢志华,提前还贷及其隐含期权的分析,应用数学与计算数学学报,21(2007), 1-4.
[3]徐承龙,周晶,任学敏,一类期权型外汇存款的套利分析,同济大学学报,35(2007), 994-997.
[4]徐承龙, 段为钊, 周羽宇,一种触发式汇率期权定价的数学模型,同济大学学报,35(2007), 1138-1142.

5.非线性初(边)值问题的高精度有限差分方法(王元明)
  • 对一类常系数非线性两点边值问题的紧差分格式(即 Numerov 格式)构造了外推算法,提高了数值解的精度,获得了高精度数值方法,并给出了误差估计。
  • 对一类变系数非线性四阶椭圆边值问题的标准有限差分解给出了一些数值分析, 设计了三种求解非线性差分格式的单调迭代算法, 比较了不同算法的收敛率,给出了算法几何收敛的充分条件。这些算法的一个重要优点是不需要非线性项的任何单调性,因而扩大了已有算法的应用范围,推广了已知的结果。对求解非线性四阶椭圆边值问题的有限差分解的标准单调迭代方法给出了稳定性和误差分析,比较了不同算法的收敛率,给出了算法几何收敛的充分条件。
  • 对一类非线性反应扩散方程和相应的定常问题 (即非线性椭圆边值问题)的标准有限差分解给出了一些数值分析,包括解的存在唯一性,求解的单调迭代算法和依赖于时间解的渐近收敛性。给出了一些充分条件使得解收敛于一个非负定常解,这样的结论在生物数学模型中有重要的应用, 该结论的另一个重要推论是定常解的存在唯一性。
有关论文:
[1]Yuan-Ming Wang, The extrapolation of Numerov’s scheme for nonlinear two-point boundary value problems, Appl. Numer. Math., 57(2007), 253-269.
[2]Yuan-Ming Wang, Monotone iterative technique for numerical solutions of fourth-order nonlinear elliptic boundary value problems, Appl. Numer. Math., 57(2007), 1081-1096.
[3]Yuan-Ming Wang, Error and stability of monotone method for numerical solutions of fourth-order semilinear elliptic boundary value problems, J. Comp. Appl. Math., 200(2007), 503-519.
[4]Yuan-Ming Wang, Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays, J. Math. Anal. Appl., 328(2007), 137-150.

6.随机伪蒙特卡罗方法的理论与应用(岳荣先)
  • 对有界或无界矩形区域上的高维积分,利用适当的变量变换与积分格子点序列构造新的等权求积公式,并在Banach函数空间中单位球上估计这种求积法的极端误差,还将与被积函数的周期化变换法进行比较。我们证明新的等权求积公式优于周期化变换法。
  • 对Banach函数空间上函数的高维积分,构造基于积分格子点序列的具有最优权系数的求积公式,并估计这种求积法在Banach空间中单位球上的极端误差。这里的最优是指使积分误差达到最小。
  • 利用伪Monte-Carlo方法构造多响应近似线性模型的稳健试验设计。
有关论文:
[1]R.X.Yue and F.J. Hickernell, Strong tractability of quasi-Monte Carlo quadrature using nets for certain Banach spaces, SIAM J. Numer. Anal., 44(2006), 2559-2583.

7.数学物理反问题的理论和数值方法(程晋)
  • 不适定问题的算法研究。主要研究了一般意义上的数值微分问题。提出一种简单易行的数值方法。特别值得指出的是,对于具有不连续的函数,我们的方法可以识别不连续点。这一点在许多实际问题中有比较好的应用。
  • 讨论一类从钢铁工业生产中提出的问题,从数学模型的建立和参数的辨识进行了一系列的研究。取得了比较的研究成果。部分成果已经被实际部门所采用。
  • 讨论了一类关于椭圆方程的边界系数确定问题。运用Carlman估计的方法,我们给出了条件稳定性估计。这种条件稳定性估计给出了不适定问题的Tihkonov正则化解的收敛阶。
有关论文:
[1]J. Cheng, X. Z. Jia and Y. B. Wang, Numerical differentiation and its applications, Inverse Problems in Science and Engineering, 15(2007), 339–357.
四、学术活动
遵循研究院管理章程进行日常学术活动,并举办或合办了一些国内或国际学术会议。
1.日常学术活动
  • 每月召开全体特聘研究员工作会议,相互交流科学研究工作并部署下一步研究工作。
  • 每月举办一次面向全市的学术报告会,由特聘研究员或院外专家介绍科学计算的新进展。
  • 邀请30多名国内、外专家来研究院讲学或合作研究。
  • 研究院6位成员参加了国际学术会议,共11人次,并作邀请报告或报告。多名研究员到国外或境外讲学或短期合作研究。
2.举办或合办国内、外学术会议
  • 2007年1月,参加举办第88期上海东方论坛—纳米光学数学和计算的挑战
  • 和机遇,参加者约75人。
  • 2007年6月,举办谱方法与高阶元方法讨论会,参加者约50人。
  • 2007年6月,参加举办非线性科学国际学术会议,参加者约150人。
  • 2007年7月,参加举办第二次上海地区微分方程数值解交流会, 参加者约100人。
  • 2007年11月,参加举办第三届上海市科学与工程中的计算方法研讨会,参加
  • 者约200人。
3.拟办的学术会议
  • 2008年3月, 参加举办地震科学有关的反问题研讨会。
  • 2008年6月, 参加举办生物数学中的建模与参数辨识。
  • 2008年8月,举办常微分方程数值解研讨会。
  • 2008年10月,参加举办数学物理的反问题及其应用。
  • 2008年11月,参加举办力学模型正、反问题数值解小型研讨会。
  • 2008年11月,参加举办第四届上海市科学与工程中的计算方法研讨会。
五、学科建设和人才培养
根据上海高校E—研究院的建设宗旨,加速培养上海市各高校计算数学专业的学术带头人和高水平专业人才,促进有关高校计算数学学科的建设。
1.2007年,上海师范大学被批准设立“数学”博士后流动站。
2.2007年,上海师范大学被批准设立“科学计算上海高校重点实验室”。
3.特聘研究员程晋、徐承龙、王中庆和黄建国被聘为有关973项目研究员。
4.特聘研究员程晋被选为全国工业与应用数学学会常务理事、副秘书长以及全国计算数学学会常务理事;特聘研究员岳荣先被选为中国数学会均匀设计分会常务理事;特聘研究员丛玉豪被选为全国计算数学学会理事,全国系统仿真学会理事以及全国仿真算法专业委员会副主任委员;特聘研究员黄建国被选为上海市工业与应用数学协会理事。
5.研究院成员共指导了22名博士生(其中毕业4名),和46名硕士生(其中毕业10名),指导博士后2名。
6.黄建国教授指导的一名博士生获得求是基金会研究生奖学金。

六、2007年论文摘要

Error analysis of spectral method on a triangle
Ben-yu Guo and Li-lian Wang
Advances in Comp. Math., 26(2007),473-496.
Abstract
 In this paper, the orthogonal polynomial approximation on triangle proposed by Dubiner [2], is studied. Some approximation results are established in certain non-uniformly Jacobi-weighted Sobolev space, which play important role in numerical analysis of spectral and triangle spectral element methods for differential equations on complex geometries. As an example, a model problem is considered.

Spectral method for differential equations of degenerate type on unbounded domains by using generalized Laguerre functions
Guo Ben-yu and Zhang Xiao-yong
Appl. Numer. Math., 57(2007),455-471.
Abstract
In this paper, we develop the orthogonal approximation by using generalized Laguerre functions. Some basic results on this approximation are established, which serve as the mathematical foundation of spectral methods for various differential equations on unbounded domains. As an example of applications, we propose a spectral method for a partial differential equation of degenerate type, which plays an important role in financial mathematics and other fields. The convergence of proposed scheme is proved. Numerical results show its spectral accuracy in space.

Mixed Jacobi-spherical harmonic spectral method for Navier-Stokes equations
Guo Ben-yu and Huang Wei
Appl. Numer. Math., 57(2007),939-961.
Abstract
Mixed Jacobi-spherical harmonic spectral method is proposed for the Navier-Stokes equations in a ball. Its stability and convergence are proved. Numerical results demonstrate the efficiency of this approach. Some results on the mixed Jacobi-spherical harmonic approximation are established, which play important role in numerical analysis of spectral method in spherical geometry.

Numerical integration based on Laguerre-Gauss interpolation
Guo Ben-yu and Wang Zhong-qing
Comput. Meth. Appl. Mech. Engrg., 196(2007), 3726-3741.
Abstract
In this paper, we propose two efficient numerical integrators for ordinary differential equations based on modified Laguerre-Gauss interpolations. The global convergence of proposed algorithms is proved. Numerical results demonstrate the spectral accuracy of these new schemes and agree well with the theoretical analysis.

Periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay
Yihong Song and Hongjiong Tian
J. Comp. Appl. Math. , 205(2007), 859 –870.
Abstract
The existence of periodic and almost periodic solutions of nonlinear discrete Volterra equations with unbounded delay is obtained by using stability properties of a bounded solution.

Numerical methods for singularly perturbed delay differential equations
Yeguo Sun, Dongyue Zhang and Hongjiong Tian
J. Syst. Simu., 19(2007), 3943-3944,3992.
Abstract
This paper is concerned with uniformly convergent numerical methods for singularly perturbed delay differential equations. Two uniformly convergent numerical schemes which is based on the exponential fitting technique for linear and nonlinear problems are examined for singular perturbation problems with after-effect. Numerical examples are given to testify our theoretical results.

A simple proof of inequalities of integrals of composite functions
Zhenglu Jiang, Xiaoyong Fu and Hongjiong Tian
J. Math. Anal. Appl., 332(2007), 1307-1312.
Abstract
In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continuous convex functions on a vector space Rmand vector-valued functions in a weakly compact subset of a Banach vector space generated by m  -spaces for  . Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by m  -spaces instead.

Numerical dissipativity of multistep methods for delay differential equations
Hongjiong Tian, Liqiang Fan and Jiaxiang Xiang
Appl. Math. Comp.,188(2007), 934-941.
Abstract
Dissipative differential equations have frequently appeared in the fields of physics, engineering, and biology. In this paper we investigate numerical dissipativity of linear multistep and one-leg methods applied to a class of dissipative delay differential equations. We show that for such class of dissipative systems these numerical methods are dissipative if and only if they are A-stable for ordinary differential equations. One numerical experiment is given to illustrate our result.

GPLm—stability of block Theta method for delay differential equation
Cong Yuhao, Li Shundao and Tan Xiuli
J. of System Simulation, 19(2007), 3937-3039.
Abstract
The stability behavior of numerical solution for delay differential equations with many delays was studied. The conditions of GPm-stability and GPLm-stability of block theta method for delay differential equations with many delays were discussed .By Lagrange Interpolation, it is shown that block theta method is GPm-stable if and only if it is A-stable, block theta method is GPLm—stable if and only if theta=1.

A numerical method for a Cauchy problem for elliptic partial differential equations
W. Han, J. Huang, K. Kazmi, and Y. Chen
Inverse Problems, 23(2007), 2401-2415.
Abstract
The Cauchy problem for an elliptic partial differential equation is ill-posed.  In this paper, we study a numerical method for solving the Cauchy problem. The numerical method is based on a reformulation of the Cauchy problem through an optimal control approach coupled with a regularization term which is included to treat the severe ill-conditioning of the corresponding discretized formulation.  We prove convergence of the numerical method and present theoretical results for the limiting behaviors of the numerical solution as the regularization parameter approaches zero. Results from some numerical examples are reported.

A finite element method for general elastic multi-structures
J. Huang, L. Guo and Z. Shi
Computers Math. with Appl., 53(2007), 1867-1895.
Abstract
A finite element method is proposed for the general elastic multi-structure problem, where displacements on bodies, longitudinal displacements on plates, longitudinal displacements and rotational angles on rods are discretized using conforming linear elements, transverse displacements on plates and rods are discretized respectively using TRUNC elements and Hermite elements of third order, and the discrete generalized displacement fields in individual elastic members are coupled together by some feasible interface conditions. The unique solvability of the method is verified by the Lax-Milgram lemma after deriving generalized Korn's inequalities in some nonconforming element spaces on elastic multi-structures. The quasi-optimal error estimate in the energy norm is also established. Some numerical results are presented in the end.

Uniform a priori estimates for elliptic and static Maxwell interface problems
J. Huang and J. Zou
Discrete and Continuous Dynamical Systems Ser. B, 7(2007), 145-170.
Abstract
We present some new a priori estimates of the solutions to three-dimensional elliptic interface problems and static Maxwell interface system with variable coefficients. Different from the classical a priori estimates, the physical coefficient functions of the interface problems appear in these new estimates explicitly.

Spectral-domain decomposition method and its applications in finance
Xu Chenglong
Recent Progress in Scientific Computing, Science Press, 2007,367-381.
Abstract
The modified Laguerre spectral-finite difference schemes are proposed for a class of degenerate PDEs arising from finance with discontinuous coefficient. The domain-decomposition technique is used. Error estimation of the schemes is obtained. Numerical results are given which show the efficiency and the convergence of the schemes.

The extrapolation of Numerov’s scheme for nonlinear two-point boundary value problems
Yuan-Ming Wang
Appl. Numer. Math., 57(2007), 253-269.
Abstract
This paper is concerned with the extrapolation algorithm of Numerov's scheme for semilinear and strongly nonlinear two-point boundary value problems. The asymptotic error expansion of the solution of Numerov's scheme is obtained. Based on the asymptotic error expansion, Richardson's extrapolation is constructed, and so the accuracy of the numerical solution is greatly increased.  Numerical results are presented to demonstrate the efficiency of the extrapolation algorithm.

Monotone iterative technique for numerical solutions of fourth-order nonlinear elliptic boundary value problems
Yuan-Ming Wang
Applied Numerical Mathematics, 57(2007), 1081-1096.
Abstract
This paper is concerned with finite difference solutions of a class of fourth-order nonlinear elliptic boundary value problems. The nonlinear function is not necessarily monotone. A new monotone iterative technique is developed, and three basic monotone iterative processes for the finite difference system are constructed.  Several theoretical comparison results among the various monotone sequences are given. A simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. Numerical results for a model problem with known analytical solution are given.

Error and stability of monotone method for numerical solutions of fourth-order semilinear elliptic boundary value problems
Yuan-Ming Wang
J. Comp. Appl. Math., 200(2007), 503-519.
Abstract
This paper is concerned with the error and stability analysis of the monotone method for numerical solutions of fourth-order semilinear elliptic boundary value problems. A comparison result among the various monotone sequences is given. The global error is analyzed, and some sufficient conditions are formulated to guarantee a geometric rate of convergence. The stability of the monotone method is proved. Some numerical results are presented.

Asymptotic behavior of solutions for a class of predator-prey reaction-diffusion systems with time delays
Yuan-Ming Wang
J. Math. Anal. Appl.,328(2007), 137-150.
Abstract
The aim of this paper is to investigate the asymptotic behavior of solutions for a class of three-species predator-prey reaction-diffusion systems with time delays under homogeneous Neumann boundary condition. Some simple and easily verifiable conditions are given to the rate constants of the reaction functions to ensure the convergence of the time-dependent solution to a constant steady-state solution. The conditions for the convergence are independent of diffusion coefficients and time delays, and the conclusions are directly applicable to the corresponding parabolic-ordinary differential system and to the corresponding system without time delays.

Strong Tractability of Quasi-Monte Carlo Quadrature Using Nets for Certain Banach Spaces
R. X. Yue and F. J. Hickernell
SIAM J. Numer. Anal., 44(2006), 2559-2583
Abstract
We consider multivariate integration in the weighted spaces of functions with mixed first derivatives bounded in   norms and the weighted coefficients introduced via   norms, where  . The integration domain may be bounded or unbounded. The worst-case error and randomized error are investigated for quasi-Monte Carlo quadrature rules. For the worst-case setting the quadrature rule uses deterministic  -sequences in base  , and for the randomized setting the quadrature rule uses randomly scrambled digital  -nets in base  . Sufficient conditions are found under which multivariate integration is strongly tractable in the worst-case and randomized settings, respectively. Similar results hold for the Banach spaces of finite-order weights. Results presented in this article extend and improve upon those found previously.

Numerical differentiation and its applications
J. Cheng, X. Z. Jia and Y. B. Wang
Inverse Problems in Science and Engineering, 15(2007), 339–357
Abstract
Differentiation is one of the most important concepts in calculus, which has been used almost everywhere in many fields of mathematics and applied mathematics. It is natural that numerical differentiation should be an important technique for the engineers. However, since it is ill-posed in Hadamard’s sense, which means that any small error in the measurements will be enlarged, it is very difficult for the engineers to use this technique. In this article, we propose a new simple numerical method to reconstruct the original function and its derivatives from scattered input data and show that our method is effective and can be realized easily.



发布者: eicssu admin
发布日期: 12/31/2007
浏览次数: 1327

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