In this article, we study the Volterra integral equations with two kinds of delay that are proportional delay and nonproportional delay. We mainly use Chebyshev spectral collocation method to analyze them. First, we u...In this article, we study the Volterra integral equations with two kinds of delay that are proportional delay and nonproportional delay. We mainly use Chebyshev spectral collocation method to analyze them. First, we use variable transformation to transform the equation into an new equation which is defined in [-1,1]. Then, with the help of Gronwall inequality and some other lemmas, we provide a rigorous error analysis for the proposed method, which shows that the numerical error decay exponentially in L~∞ and L_(ω~c)~2-norm. In the end, we give numerical test to confirm the conclusion.展开更多
In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transforma...In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.展开更多
In this paper,a Legendre-collocation spectral method is developed for the second order Volterra integro-differential equation with pantograph delay.We provide a rigorous error analysis for the proposed method.The spec...In this paper,a Legendre-collocation spectral method is developed for the second order Volterra integro-differential equation with pantograph delay.We provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both L^(2)-norm and L^(∞)-norm.展开更多
In this paper,we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems.The state and co-state are approximated by the lowest order Raviart-...In this paper,we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems.The state and co-state are approximated by the lowest order Raviart-Thomas mixed fi-nite element spaces and the control variable is approximated by piecewise constant functions.We derive L^(2) and L^(∞)-error estimates for the control variable.Moreover,using a recovery operator,we also derive some superconvergence results for the control variable.Finally,a numerical example is given to demonstrate the theoretical results.展开更多
基金supported by National Science Foundation of China(11671157,11626074)Hanshan Normal Uninversity projects(LF201404,Z16027)
文摘In this article, we study the Volterra integral equations with two kinds of delay that are proportional delay and nonproportional delay. We mainly use Chebyshev spectral collocation method to analyze them. First, we use variable transformation to transform the equation into an new equation which is defined in [-1,1]. Then, with the help of Gronwall inequality and some other lemmas, we provide a rigorous error analysis for the proposed method, which shows that the numerical error decay exponentially in L~∞ and L_(ω~c)~2-norm. In the end, we give numerical test to confirm the conclusion.
基金supported by the State Key Program of National Natural Science Foundation of China(11931003)the National Natural Science Foundation of China(41974133,11671157)。
文摘In this paper,a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay,which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1,1],so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation,the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end,we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L^(∞)-norm and the weighted L^(2)-norm.
基金This work is supported by National Science Foundation of China(11271145)Foundation for Talent Introduction of Guangdong Provincial University,Specialized Research Fund for the Doctoral Program of Higher Education(20114407110009)the Project of Department of Education of Guangdong Province(2012KJCX0036).
文摘In this paper,a Legendre-collocation spectral method is developed for the second order Volterra integro-differential equation with pantograph delay.We provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both L^(2)-norm and L^(∞)-norm.
基金This work is supported by National Science Foundation of China,Foundation for Talent Introduction of Guangdong Provincial University,Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2008)Specialized Research Fund for the Doctoral Program of Higher Education(20114407110009).
文摘In this paper,we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems.The state and co-state are approximated by the lowest order Raviart-Thomas mixed fi-nite element spaces and the control variable is approximated by piecewise constant functions.We derive L^(2) and L^(∞)-error estimates for the control variable.Moreover,using a recovery operator,we also derive some superconvergence results for the control variable.Finally,a numerical example is given to demonstrate the theoretical results.
