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 approx...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展开更多
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 dimension.Since the spatial domain is unbounded,the numerical ...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 dimension.Since 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.展开更多
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 conside...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.展开更多
The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered.Applying the unified approach,which is based on the operator splitting method,we construct the efficie...The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered.Applying 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 domain.Then the finite difference method is used to solve the reduced problem on the bounded computational domain.Finally,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.展开更多
基金The research of HB was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada and by the Research Grants Council of Hong KongThe research of TT was supported by Hong Kong Baptist University,the Research Grants Council of Hong Kong and he was supported in part by the Chinese Academy of Sciences while visiting its Institute of Computational Mathematics.
文摘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
基金The research was supported in part by the Natural Sciences and Engineering Research Council(NSERC)of Canada,by Hong Kong Research Grants Council and FRG of Hong Kong Baptist University.
文摘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 dimension.Since 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.
基金Acknowledgements The authors thank the anonymous referees for the constructive criticism and the many valuable suggestions that led to a significant improvement in the presentation of the main results. This work was supported by the National Natural Science Foundation of China （Grant No. 11071050）, the Fundamental Research Funds for the Central Universities （Grant No. HIT.NSRIF.2010051）, the Hong Kong Research Grants Council （RGC Project No. 200210）, and the Natural Sciences and Engineering Research Council of Canada （NSERC Discovery Grant A9406）.
文摘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.
基金supported by FRG of Hong Kong Baptist University,and RGC of Hong Kong.
文摘The numerical solution of blow-up problems for nonlinear wave equations on unbounded spatial domains is considered.Applying 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 domain.Then the finite difference method is used to solve the reduced problem on the bounded computational domain.Finally,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.
