北京科学与工程计算研究院学术报告之二十四

报告人/Speaker:王晚生/Wansheng Wang

报告题目/Title:非线性中立型泛函微分方程理论及数值分析/Theory and numerical analysis of nonlinear neutral functional differential equations

时间/Date & Time: 1013日(星期四)上午10-11/13rd Oct. 10am-11am

地点/Location: 理科楼M842

报告摘要/Abstract

这个报告主要回顾了我们十几年来在非线性中立型泛函微分方程(包括Volterra泛函微分方程)理论和数值解法方面的一些结果。

1)基于单边Lipschitz条件深入研究了非线性NFDEs的稳定性,获得了一系列关于收缩性、渐近稳定性及指数稳定性的结果,统一了此前相关结论,为这类方程的数值稳定性分析奠定理论基础;首次证明了求解非线性中立型变延迟微分方程数值方法的保稳定性;系统给出了几类非线性NFDEs数值方法的稳定性准则,为数值方法稳定性提供了重要判据;

2)突破经典Lipschitz条件的限制,基于单边Lipschitz条件系统研究了NFDEs数值方法的收敛性,获得了更精确和逼真的误差估计;

3)采用直接方法或利用所推广的Halanay不等式获得了几类NFDEs的耗散性结果,以此为基础研究了几类保耗散的数值算法,为这些方程的长时间计算提供了理论指南。

This report presents a review of recent progress in the field of numerical solutions of a class of nonlinear neutral functional differential equations (NFDEs) which includes as an important special case the Volterra functional differential equations (VFDEs) .

1. A series of stability, contractivity, asymptotic stability and exponential asymptotic stability results of the theoretical solutions to nonlinear NFDEs in Banach spaces were obtained. From a numerical point of view, it is important to study the potential of numerical methods in preserving the qualitative behaviour of the underlying system. Consequently, these results presented in this report provide the theoretical foundation for analyzing the stability of the numerical methods when they are applied to these systems. The contractivity and asymptotic stability properties of the implicit Euler method for nonlinear functional differential equations (FDEs) are discussed. One of main result we established is that the implicit Euler method with linear interpolation can completely preserve these stability properties of the theoretical solution to such FDEs. We also review the results on one-leg methods and Runge-Kutta methods for solving NDDEs.

2. On bases of one-sided Lipschitz condition, we studied the convergence of numerical methods for NFDEs, and obtained the optimal error estimates.

3. We established the dissipativity theory for several classes of FDEs by means of direct approach or Halanay inequality. Based on these analytical dissipativity results, we investigated the dissipativity-preserving properties of numerical methods for such FDEs.

报告人简介/About the speaker:

王晚生:长沙理工大学教授。20086月博士毕业于湘潭大学,20106月从华中科技大学数学博士后流动站出站。一直从事微分方程数值解方面的研究工作,发表学术论文50余篇(SCI收录40篇),其中以第一作者在《Numer. Math.》、《SIAM J. Sci. Comput.》等计算数学权威期刊上发表学术论文40余篇,研究成果获湖南省自然科学奖二等奖2项(1项排名第一,1项排名第6)、霍英东青年教师奖三等奖、中国计算数学学会2009年优秀青年论文竞赛二等奖等。主持国家自然科学基金面上项目1项、湖南省自然科学基金杰出青年基金项目1项、湖南省教育厅重点项目1项,主持完成了国家自然科学基金青年基金项目、中国博士后基金特别资助和面上资助等科研项目。20109-20116月访问北京大学数学院,201210-20131月获AMS(美国数学会)的Ky and Yu-Fen Fan基金资助访问美国加州大学尔湾分校,20138-20147月年由国家留学基金委资助访问剑桥大学应用数学与理论物理系。系湖南省新世纪“121人才工程第二层次人选、湖南省普通高校学科带头人培养对象、长沙理工大学青年英才支持计划人选。现为中国系统仿真学会仿真算法专业委员会委员、湖南省数学会常务理事,长沙理工大学数学与计算科学学院副院长、学术委员会委员。

Wansheng Wang: Professor in Numerical Analysis of Differential Equations, Changsha University of Science & Technology.

Education  
 M.S., Xiangtan University, 2004
 Ph.D., Xiangtan University, 2008, Supervisor:Prof. Shoufu Li
Professional Experience    
 Lecturer, Changsha University of Science and Technology, 2006.10-2008.11
 Associate Professor, Changsha University of Science and Technology, 2008.11-2013.12
 Postdoctoral Fellow, Huazhong University of Science and Technology, 2008.7- 2010.6
 Visiting member, Peking University, 2010.9-2011.6
 Visiting member, University of California at Irvine, 2012.10-2013.1
 Postdoctoral Fellow, University of Cambridge, 2013.8-2014.7
Awards, Honors    
 2004  President Award, Xiangtan University,
 2006  Excellent Thesis Award for Master’s Degree of Hunan Province, Hunan Province
 2009  Science Prize of Hunan Province, Hunan Province
 2009  Developing Member of the Core of the Young in the Colleges and Universities of
Hunan Province, Hunan Province
 2009  Second Prize winners in the 4th Paper Competition for Young Computational
Mathematicians, Chinese Society for Computational Mathematics
 2010  Excellent Postdoctoral fellow, Huazhong University of Science and Technology
 2010  The elected of the second level of Hunan New Century 121 Talent Project, Hunan
Province
 2011  The elected of the Young-Support program of Changsha University of Science and
Technology, Changsha University of Science and Technology
 2012  Fok Ying Tung Award for young teachers, Ministry of Education of China, Fok Ying
Tung Education Foundation
 2012  Ky and Yu-Fen Fan Fund Travel Grant from the AMS, AMS
 2015  Science Prize of Hunan Province, Hunan Province
Research Grants  
 Natural Science Foundation of China (NSFC): High order dissipative methods and their
a posteriori error estimates for nonlinear functional differential equations  (No. 11371074,
2014.1-2017.12, RMB 620,000);
 Natural Science Foundation of Hunan Province for Outstanding Youth: High efficient
algorithms and numerical simulation for nonlinear infinite dimensional dissipative systems
(No. 13JJ1020, 2013.1-2015.12, RMB200,000);
 Natural Science Foundation of China (NSFC): High order preserving stability methods for
nonlinear neutral functional differential equations and their applications  (No. 11001033,
2011.1-2013.12, RMB 180,000);
 China Postdoctoral Science Foundation: Granted Special Grade of the Financial Support
(No. 200902437, RMB100,000);
Publications    
 Published papers more than 50, including 40 papers published in SCI journals
Selected publications:
[1] Wansheng Wang and Chengjian Zhang. Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space. Numer. Math. 2010, 115(3): 451-474.
[2] Wansheng Wang and Shoufu Li. Stability analysis of Θ-methods for nonlinear neutral functional differential equations, SIAM J. Sci. Comput. 2008, 30(4): 2181-2205. May.
[3] Wangsheng Wang, Long Chen and Jie Zhou, Postprocessing mixed finite element methods for solving Cahn–Hilliard equation: methods and error analysis, J. Sci. Comput. 2016, 67(2), 724-746
[4] Wansheng Wang and Chengjian Zhang. Analytical and numerical dissipativity for nonlinear generalized pantograph equations. Discrete Contin. Dyn. Syst. 2011, 29(3): 1245-1260.
[5] Wansheng Wang, Yuan Zhang and Shoufu Li. Nonlinear stability of one-leg methods for delay differential equations of neutral type. App. Numer. Math., 2008, 58(2): 122-130.