The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth conditio...The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.展开更多
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
The analytic and discretized dissipativity of nonlinear infinite-delay systems of the form x＇（t） = g（x（t）,x（qt））（q∈ （0, 1）, t 〉 0） is investigated. A sufficient condition is presented to ensure that the...The analytic and discretized dissipativity of nonlinear infinite-delay systems of the form x＇（t） = g（x（t）,x（qt））（q∈ （0, 1）, t 〉 0） is investigated. A sufficient condition is presented to ensure that the above nonlinear system is dissipative. It is proved the backward Euler method inherits the dissipativity of the underlying system. Numerical examples are given to confirm the theoretical results.展开更多
The paper introduces a new class of numerical schemes for the approximate solutions of stochastic pantograph equations. As an effective technique to implement implicit stochastic methods, strong predictor-corrector me...The paper introduces a new class of numerical schemes for the approximate solutions of stochastic pantograph equations. As an effective technique to implement implicit stochastic methods, strong predictor-corrector methods （PCMs） are designed to handle scenario simulation of solutions of stochastic pantograph equations. It is proved that the PCMs are strong convergent with order 1/2.Linear M^-stabiiity of stochastic pantograph equationsand the PCMs are researched in the paper. Sufficient conditions of MS-unstability of stochastic pantograph equations and MS-stability of the PCMs are obtained, respectively. Numerical experiments demonstrate these theoretical results.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
In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stoc...In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.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
基金support from the National Natural Science Foundation of China(70871046,71171091,71191091)Fundamental Research Funds for the Central Universities(2011QN167)
文摘The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.
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This work is supported by the National Natural Science Foundation of China(Grant No.10571147).
文摘The analytic and discretized dissipativity of nonlinear infinite-delay systems of the form x＇（t） = g（x（t）,x（qt））（q∈ （0, 1）, t 〉 0） is investigated. A sufficient condition is presented to ensure that the above nonlinear system is dissipative. It is proved the backward Euler method inherits the dissipativity of the underlying system. Numerical examples are given to confirm the theoretical results.
文摘The paper introduces a new class of numerical schemes for the approximate solutions of stochastic pantograph equations. As an effective technique to implement implicit stochastic methods, strong predictor-corrector methods （PCMs） are designed to handle scenario simulation of solutions of stochastic pantograph equations. It is proved that the PCMs are strong convergent with order 1/2.Linear M^-stabiiity of stochastic pantograph equationsand the PCMs are researched in the paper. Sufficient conditions of MS-unstability of stochastic pantograph equations and MS-stability of the PCMs are obtained, respectively. Numerical experiments demonstrate these theoretical results.
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In this paper, we derive the stochastic maximum principle for optimal control problems of the forward-backward Markovian regime-switching system. The control system is described by an anticipated forward-backward stochastic pantograph equation and modulated by a continuous-time finite-state Markov chain. By virtue of classical variational approach, duality method, and convex analysis, we obtain a stochastic maximum principle for the optimal control.
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.
