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A SPECTRAL METHOD FOR PANTOGRAPH-TYPE DELAY DIFFERENTIAL EQUATIONS AND ITS CONVERGENCE ANALYSIS

Journal of Computational Mathematics 2009年第2期27卷 254-265页

作者： Ishtiaq Ali hermann brunner Institute of Computational Mathematics Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China Department of Mathematics COMSA TS Institute of Information Technology IslamabadPakistan hermann brunner Department of Mathematics and Statistics Memorial University of NewfoundlandSt.John’sCanada Department of Mathematics Hong Kong Baptist UniversityHong KongChina

We propose a novel numerical approach for delay differential equations with vanishing proportional delays based on spectral methods. A Legendre-collocation method is employed to obtain highly accurate numerical approximations to the exact solution. It is proved theoretically and demonstrated numerically that the proposed method converges exponentially provided that the data in the are smooth. given pantograph delay differential equation

关键词： Spectral methods Legendre quadrature formula Pantograph-type delay differential equations Error analysis Exponential convergence

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Blow-up behavior of Hammerstein-type delay Volterra integral equations

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Frontiers of Mathematics in China 2013年第2期8卷 261-280页

作者： Zhanwen YANG hermann brunner Science Research Center Academy of Fundamental and Interdisciplinary Science Department of Mathematics Harbin Institute of Technology Harbin 150001 China Department of Mathematics Hong Kong Baptist University Hong Kong SAR China Department of Mathematics and Statistics Memorial University of NewfoundlandSt. John's NL AIC 557 Canada

We consider the blow-up behavior of Hammerstein-type delay Volterra integral equations （DVIEs）. Two types of delays, i.e., vanishing delay （pantograph delay） and non-vanishing delay （constant delay）, are considered. With the same assumptions of Volterra integral equations （VIEs）, in a similar technology to VIEs, the blow-up conditions of the two types of DVIEs are given. The blow-up behaviors of DVIEs with non-vanishing delay vary with different initial functions and the length of the lag, while DVIEs with pantograph delay own the same blow-up behavior of VIEs. Some examples and applications to delay differential equations illustrate this influence.

关键词： Delay Volterra integral equation (DVIE) non-vanishing delay vanishing delay blow-up of solution

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Numerical Soliton Solutions for a Discrete Sine-Gordon System

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Communications in Computational Physics 2009年第9期6卷 903-918页

作者： Houde Han Jiwei Zhang hermann brunner Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong KongChina Department of Mathematics and Statistics Memorial University of NewfoundlandSt.John’sNLCanada A1C 5S7

In this paper we use an analytical-numerical approach to find,in a systematic way,new 1-soliton solutions for a discrete sine-Gordon system in one spatial *** the spatial domain is unbounded,the numerical scheme employed to generate these soliton solutions is based on the artificial boundary method.A large selection of numerical examples provides much insight into the possible shapes of these new 1-solitons.

关键词： Sine-Gordon equation soliton solutions numerical single solitons artificial boundary method

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Numerical Solution of Blow-Up Problems for NonlinearWave Equations on Unbounded Domains

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Communications in Computational Physics 2013年第8期14卷 574-598页

作者： hermann brunner Hongwei Li Xiaonan Wu Department of Mathematics Hong Kong Baptist UniversityKowloon TongHong Kong Department of Mathematics and Statistics Memorial University of NewfoundlandSt.John’sNLA1C 5S7Canada College of Mathematical Sciences Shandong Normal UniversityJinan250014P.R.China.

The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is *** the unified approach,which is based on the operator splitting method,we construct the efficient nonlinear local absorbing boundary conditions for the nonlinear wave equation,and reduce the nonlinear problem on the unbounded spatial domain to an initial-boundary-value problem on a bounded *** the finite difference method is used to solve the reduced problem on the bounded computational ***,a broad range of numerical examples are given to demonstrate the effectiveness and accuracy of our method,and some interesting propagation and behaviors of the blow-up problems for nonlinear wave equations are observed.

关键词： Finite-time blow-up nonlinear wave equation absorbing boundary conditions finite differencemethod unbounded domains

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广义自卷积Volterra积分方程的配置法分析

广义自卷积Volterra积分方程的配置法分析

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作者：张然梁慧 hermann brunner 吉林大学

成果内容该成果主要研究了第二类广义自卷积Volterra积分方程解的存在性、唯一性和正则性,并用于研究这一类积分方程分片多项式配置解的全局和局部最优收敛阶以及超收敛现象。本文中还设计了多种数值实验来验证理论分析的结论。在此工...

成果内容该成果主要研究了第二类广义自卷积Volterra积分方程解的存在性、唯一性和正则性,并用于研究这一类积分方程分片多项式配置解的全局和局部最优收敛阶以及超收敛现象。本文中还设计了多种数值实验来验证理论分析的结论。在此工作基础上,我们还研究了泛函积分方程的高阶数值格式,解决了积分方程领域的一个公开问题,这将对进一步研究带消失时滞的一般积分(微-积分)方程数值算法的构造

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