Theory and numerical analysis of nonlinear neutral functional differential equations

Speaker: Wansheng Wang

Title: Theory and numerical analysis of nonlinear neutral functional differential equations

Date & Time: Oct.13, 2016, 10am-11am

Location: M842, Science Research Building

Abstract

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.

About the speaker:

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.