This article deals with the existence of solutions of nonlinear fractional pantograph equations. Such model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous med...This article deals with the existence of solutions of nonlinear fractional pantograph equations. Such model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous media. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the main result obtained in this article.展开更多
In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square...In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.展开更多
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
This paper is concerned with the superconvergent points of the continuous Galerkin solutions for delay differential equations of pantograph type. We prove the local nodal superconvergence of continuous Galerkin soluti...This paper is concerned with the superconvergent points of the continuous Galerkin solutions for delay differential equations of pantograph type. We prove the local nodal superconvergence of continuous Galerkin solutions under uniform meshes and locate all the superconvergent points based on the supercloseness between the continuous Galerkin solution U and the interpolation Hhu of the exact solution u. The theoretical results are illustrated by numerical examples.展开更多
This paper is concerned with numerical stability of nonlinear systems of pantograph equations. Numerical methods based on （k, l）-algebraically stable Runge-Kutta methods are suggested. Global and asymptotic stabilit...This paper is concerned with numerical stability of nonlinear systems of pantograph equations. Numerical methods based on （k, l）-algebraically stable Runge-Kutta methods are suggested. Global and asymptotic stability conditions for the presented methods are derived.展开更多
This paper is concerned with the application of generalized polynomial chaos （gPC） method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given fo...This paper is concerned with the application of generalized polynomial chaos （gPC） method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given for the error arising from finite-dimensional noise （FDN） assumption, projection error, aliasing error and discretization error. In the end, with several numerical experiments, the theoretical results are further illustrated.展开更多
This paper deals with the numerical solution of initial value problems for pantograph differential equations with variable delays. We investigate the stability of one leg θ-methods in the numerical solution of these ...This paper deals with the numerical solution of initial value problems for pantograph differential equations with variable delays. We investigate the stability of one leg θ-methods in the numerical solution of these problems. Sufficient conditions for the asymptotic stability of θ-methods are given by Fourier analysis and Ergodic theory.展开更多
In this paper, the Adomian Decomposition Method (ADM) and the Differential Transform Method (DTM) are applied to solve the multi-pantograph delay equations. The sufficient conditions are given to assure the convergenc...In this paper, the Adomian Decomposition Method (ADM) and the Differential Transform Method (DTM) are applied to solve the multi-pantograph delay equations. The sufficient conditions are given to assure the convergence of these methods. Several examples are presented to demonstrate the efficiency and reliability of the ADM and the DTM;numerical results are discussed, compared with exact solution. The results of the ADM and the DTM show its better performance than others. These methods give the desired accurate results only in a few terms and in a series form of the solution. The approach is simple and effective. These methods are used to solve many linear and nonlinear problems and reduce the size of computational work.展开更多
基金UGC New Delhi for providing BSR fellowshipproject MTM2010-16499 from the MICINN of Spain
文摘This article deals with the existence of solutions of nonlinear fractional pantograph equations. Such model can be considered suitable to be applied when the corresponding process occurs through strongly anomalous media. The results are obtained using fractional calculus and fixed point theorems. An example is provided to illustrate the main result obtained in this article.
基金Supported by the NSF of the Higher Education Institutions of Jiangsu Province(10KJD110006)Supported by the grant of Jiangsu Institute of Education(Jsjy2009zd03)Supported by the Qing Lan Project of Jiangsu Province(2010)
文摘In this paper,we present the semi-implicit Euler（SIE）numerical solution for stochastic pantograph equations with jumps and prove that the SIE approximation solution converges to the exact solution in the mean-square sense under the Local Lipschitz condition.
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Acknowledgments. The second author is supported by NSFC （Nos. 11571027, 91430215）, by Beijing Nova Program （No. 2151100003150140） and by the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions （No. CIT＆TCD201504012）. The third author is supported by the Natural Science Foundation of Fujian Province of China （No.2013J05015）, by NSFC （No.11301437）, and by the Fundamental Research ~nds for the Central Universities （No. 20720150004）.
文摘This paper is concerned with the superconvergent points of the continuous Galerkin solutions for delay differential equations of pantograph type. We prove the local nodal superconvergence of continuous Galerkin solutions under uniform meshes and locate all the superconvergent points based on the supercloseness between the continuous Galerkin solution U and the interpolation Hhu of the exact solution u. The theoretical results are illustrated by numerical examples.
基金This work is supported by the National Natural Science Foundation of China（No. 10271100）.
文摘This paper is concerned with numerical stability of nonlinear systems of pantograph equations. Numerical methods based on （k, l）-algebraically stable Runge-Kutta methods are suggested. Global and asymptotic stability conditions for the presented methods are derived.
基金Supported by the National Natural Science Foundation of China(No.11501427,11571128)
文摘This paper is concerned with the application of generalized polynomial chaos （gPC） method to nonlinear random pantograph equations. An error estimation of gPC method is derived. The global error analysis is given for the error arising from finite-dimensional noise （FDN） assumption, projection error, aliasing error and discretization error. In the end, with several numerical experiments, the theoretical results are further illustrated.
文摘This paper deals with the numerical solution of initial value problems for pantograph differential equations with variable delays. We investigate the stability of one leg θ-methods in the numerical solution of these problems. Sufficient conditions for the asymptotic stability of θ-methods are given by Fourier analysis and Ergodic theory.
文摘In this paper, the Adomian Decomposition Method (ADM) and the Differential Transform Method (DTM) are applied to solve the multi-pantograph delay equations. The sufficient conditions are given to assure the convergence of these methods. Several examples are presented to demonstrate the efficiency and reliability of the ADM and the DTM;numerical results are discussed, compared with exact solution. The results of the ADM and the DTM show its better performance than others. These methods give the desired accurate results only in a few terms and in a series form of the solution. The approach is simple and effective. These methods are used to solve many linear and nonlinear problems and reduce the size of computational work.
