In this paper,we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay.This equation is discretized by the backward Euler for differential part and the composite numerica...In this paper,we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay.This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived.Based on the a priori error bound and mesh equidistribution principle,we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter.The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm.Furthermore,we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations.Numerical results are provided to demonstrate the effectiveness of our presented monitor function.Meanwhile,it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.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
In this paper, we propose a collocation method to obtain the approximate solutions of a population model and the delay linear Volterra integro-differential equations. The method is based on the shifted Legendre polyno...In this paper, we propose a collocation method to obtain the approximate solutions of a population model and the delay linear Volterra integro-differential equations. The method is based on the shifted Legendre polynomials. By using the required matrix operations and collocation points, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. The matrix equation corresponds to a system of linear algebraic equations. Also, an error estimation method for method and improve- ment of solutions is presented by using the residual function. Applications of population model and general delay integro-differential equation are given. The obtained results are compared with the known results.展开更多
基金This work is supported by the State Key Program of National Natural Science Foundation of China(11931003)National Science Foundation of China(41974133,11761015,11971410)the Natural Science Foundation of Guangxi(2020GXNSFAA159010).
文摘In this paper,we study a nonlinear first-order singularly perturbed Volterra integro-differential equation with delay.This equation is discretized by the backward Euler for differential part and the composite numerical quadrature formula for integral part for which both an a priori and an a posteriori error analysis in the maximum norm are derived.Based on the a priori error bound and mesh equidistribution principle,we prove that there exists a mesh gives optimal first order convergence which is robust with respect to the perturbation parameter.The a posteriori error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm.Furthermore,we extend our presented adaptive grid algorithm to a class of second-order nonlinear singularly perturbed delay differential equations.Numerical results are provided to demonstrate the effectiveness of our presented monitor function.Meanwhile,it is shown that the standard arc-length monitor function is unsuitable for this type of singularly perturbed delay differential equations with a turning point.
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.
文摘In this paper, we propose a collocation method to obtain the approximate solutions of a population model and the delay linear Volterra integro-differential equations. The method is based on the shifted Legendre polynomials. By using the required matrix operations and collocation points, the delay linear Fredholm integro-differential equation is transformed into a matrix equation. The matrix equation corresponds to a system of linear algebraic equations. Also, an error estimation method for method and improve- ment of solutions is presented by using the residual function. Applications of population model and general delay integro-differential equation are given. The obtained results are compared with the known results.
