In this paper we propose some waveform relaxation (WR) methods for solving large systems of initial value problems. Nonlinear ODEs, linear ODEs, semi-explicit DAEs and linear DAEs are discussed. The accuracy increase ...In this paper we propose some waveform relaxation (WR) methods for solving large systems of initial value problems. Nonlinear ODEs, linear ODEs, semi-explicit DAEs and linear DAEs are discussed. The accuracy increase for WR methods is investigated.展开更多
This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems.In this solution procedure,no smal...This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems.In this solution procedure,no small parameter is assumed.The harmonic residue of balance equation is separated in two parts at each step.The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement.The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again.Three kinds of different differential equations involving general,fractional and delay ordinary differential systems are given as numerical examples respectively.Highly accurate limited cycle frequency and amplitude are captured.The results match well with the exact solutions or numerical solutions for a wide range of control parameters.Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems.Moreover,the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.展开更多
This work seeks to describe intra-solution particle movement system. It makes use of data obtained from simulations of patients on efavirenz. A system of ordinary differential equations is used to model movement state...This work seeks to describe intra-solution particle movement system. It makes use of data obtained from simulations of patients on efavirenz. A system of ordinary differential equations is used to model movement state at some particular concentration. The movement states’ description is found for the primary and secondary level. The primary system is found to be predominantly an unstable system while the secondary system is stable. This is derived from the state of dynamic eigenvalues associated with the system. The saturated solution-particle is projected to be stable both for the primary potential and secondary state. A volume conserving linear system has been suggested to describe the dynamical state of movement of a solution particle.展开更多
A model for the amount of pollution in lakes connected with some rivers is introduced.In this model,it is supposed the density of pollution in a lake has memory.The model leads to a system of fractional differential e...A model for the amount of pollution in lakes connected with some rivers is introduced.In this model,it is supposed the density of pollution in a lake has memory.The model leads to a system of fractional differential equations.This system is transformed into a system of Volterra integral equations with memory kernels.The existence and regularity of the solutions are investigated.A high-order numerical method is introduced and analyzed and compared with an explicit method based on the regularity of the solution.Validation examples are supported,and some models are simulated and discussed.展开更多
量子化状态系统(Quantized State System,QSS)在求解一般常微分方程(Ordinary Differential Equation,ODE)系统时,比传统基于时间离散的积分方法更具优势,但QSS方法不适合求解刚性ODE系统,为此提出一种基于量子化状态系统的步进校正优...量子化状态系统(Quantized State System,QSS)在求解一般常微分方程(Ordinary Differential Equation,ODE)系统时,比传统基于时间离散的积分方法更具优势,但QSS方法不适合求解刚性ODE系统,为此提出一种基于量子化状态系统的步进校正优化算法(Step-correction Optimization Algorithm Based on QSS,SCOA based-on QSS),它结合QSS方法及隐式算法中梯形积分法的思想,以有效提高刚性ODE系统的求解精度和效率。通过对3个典型刚性ODE算例的仿真求解,结果表明,SCOA based-on QSS算法总体上比其他算法更具优势,同时在适当减小量子大小时能显著提高仿真精度。展开更多
In this paper, we establish a result of Leray-Schauder degree on the order interval which is induced by a pair of strict lower and upper solutions for a system of second-order ordinary differential equations. As appli...In this paper, we establish a result of Leray-Schauder degree on the order interval which is induced by a pair of strict lower and upper solutions for a system of second-order ordinary differential equations. As applications, we prove the global existence of positive solutions for a multi-parameter system of second-order ordinary differential equations with respect to parameters. The discussion is based on the result of Leray- Schauder degree on the order interval and the fixed point index theory in cones.展开更多
A class of parallel implicit Runge-Kutta formulas is constructed for multiprocessor system. A family of parallel implicit two-stage fourth order Runge-Kutta formulas is given. For these formulas, the convergence is pr...A class of parallel implicit Runge-Kutta formulas is constructed for multiprocessor system. A family of parallel implicit two-stage fourth order Runge-Kutta formulas is given. For these formulas, the convergence is proved and the stability analysis is given. The numerical examples demonstrate that these formulas can solve an extensive class of initial value problems for the ordinary differential equations.展开更多
The local analytical methods for digital simulation of the dynamical control system arepresented to overcome the difficulties due to the stiffness, the oscillation with high fre-quency and the discontinuity. The idea ...The local analytical methods for digital simulation of the dynamical control system arepresented to overcome the difficulties due to the stiffness, the oscillation with high fre-quency and the discontinuity. The idea of computation of the analogue computer is intro-duced into the computation of digital simulation, the system is decomposed into somesubsystems, each of which can run independently, and, with the input informations ap-proached to, the local analytical expressions for the solutions of these subsystems are cons-tructed. Finally, a simple analysis about the accuracy estimate and the stability of the algo-rithms are given. The computational results show that the CPU time expended on thedigital simulation of a flying control system is only 1/20 of that needed by the traditionalnumerical methods.展开更多
This paper studies exact synchronization and asymptotic synchronization problems for a controlled linear system of ordinary differential equations. In this paper, we build up necessary and sufficient conditions for ex...This paper studies exact synchronization and asymptotic synchronization problems for a controlled linear system of ordinary differential equations. In this paper, we build up necessary and sufficient conditions for exact synchronization and asymptotic synchronization problems. When a system is not controllable but exactly synchronizable, it can be asymptotically synchronized in any given rate and the state of exact synchronization is given. However, when a system is not controllable and can be asymptotically synchronized in any given rate,it may not be exactly synchronizable.展开更多
The data-driven methods extract the feature information from data to build system models, which enable estimation and identification of the systems and can be utilized for prognosis and health management(PHM). However...The data-driven methods extract the feature information from data to build system models, which enable estimation and identification of the systems and can be utilized for prognosis and health management(PHM). However, most data-driven models are still black-box models that cannot be interpreted. In this study, we use the neural ordinary differential equations(ODEs), especially the inherent computational relationships of a system added to the loss function calculation, to approximate the governing equations. In addition, a new strategy for identifying the local parameters of the system is investigated, which can be utilized for system parameter identification and damage detection. The numerical and experimental examples presented in the paper demonstrate that the strategy has high accuracy and good local parameter identification. Moreover, the proposed method has the advantage of being interpretable. It can directly approximate the underlying governing dynamics and be a worthwhile strategy for system identification and PHM.展开更多
文摘In this paper we propose some waveform relaxation (WR) methods for solving large systems of initial value problems. Nonlinear ODEs, linear ODEs, semi-explicit DAEs and linear DAEs are discussed. The accuracy increase for WR methods is investigated.
基金supported by the Natural Science Foundation of Shandong Province,China(Grant Nos.ZR2011AQ022 and ZR2012AL03)
文摘This paper presents a simple and rigorous solution procedure of residue harmonic balance for predicting the accurate approximation of certain autonomous ordinary differential systems.In this solution procedure,no small parameter is assumed.The harmonic residue of balance equation is separated in two parts at each step.The first part has the same number of Fourier terms as the present order of approximation and the remaining part is used in the subsequent improvement.The corrections are governed by linear ordinary differential equation so that they can be solved easily by means of harmonic balance method again.Three kinds of different differential equations involving general,fractional and delay ordinary differential systems are given as numerical examples respectively.Highly accurate limited cycle frequency and amplitude are captured.The results match well with the exact solutions or numerical solutions for a wide range of control parameters.Comparison with those available shows that the residue harmonic balance solution procedure is very effective for these autonomous differential systems.Moreover,the present method works not only in predicting the amplitude but also the frequency of bifurcated period solution for delay ordinary differential equation.
文摘This work seeks to describe intra-solution particle movement system. It makes use of data obtained from simulations of patients on efavirenz. A system of ordinary differential equations is used to model movement state at some particular concentration. The movement states’ description is found for the primary and secondary level. The primary system is found to be predominantly an unstable system while the secondary system is stable. This is derived from the state of dynamic eigenvalues associated with the system. The saturated solution-particle is projected to be stable both for the primary potential and secondary state. A volume conserving linear system has been suggested to describe the dynamical state of movement of a solution particle.
文摘A model for the amount of pollution in lakes connected with some rivers is introduced.In this model,it is supposed the density of pollution in a lake has memory.The model leads to a system of fractional differential equations.This system is transformed into a system of Volterra integral equations with memory kernels.The existence and regularity of the solutions are investigated.A high-order numerical method is introduced and analyzed and compared with an explicit method based on the regularity of the solution.Validation examples are supported,and some models are simulated and discussed.
文摘量子化状态系统(Quantized State System,QSS)在求解一般常微分方程(Ordinary Differential Equation,ODE)系统时,比传统基于时间离散的积分方法更具优势,但QSS方法不适合求解刚性ODE系统,为此提出一种基于量子化状态系统的步进校正优化算法(Step-correction Optimization Algorithm Based on QSS,SCOA based-on QSS),它结合QSS方法及隐式算法中梯形积分法的思想,以有效提高刚性ODE系统的求解精度和效率。通过对3个典型刚性ODE算例的仿真求解,结果表明,SCOA based-on QSS算法总体上比其他算法更具优势,同时在适当减小量子大小时能显著提高仿真精度。
基金Supported in part by the National Natural Science Foundation (No.11101404) of Chinathe Fundamental Research Funds for the Central Universities (No.lzujbky-2012-11)
文摘In this paper, we establish a result of Leray-Schauder degree on the order interval which is induced by a pair of strict lower and upper solutions for a system of second-order ordinary differential equations. As applications, we prove the global existence of positive solutions for a multi-parameter system of second-order ordinary differential equations with respect to parameters. The discussion is based on the result of Leray- Schauder degree on the order interval and the fixed point index theory in cones.
基金Project supported by the National Natural Science Foundation of China
文摘A class of parallel implicit Runge-Kutta formulas is constructed for multiprocessor system. A family of parallel implicit two-stage fourth order Runge-Kutta formulas is given. For these formulas, the convergence is proved and the stability analysis is given. The numerical examples demonstrate that these formulas can solve an extensive class of initial value problems for the ordinary differential equations.
文摘The local analytical methods for digital simulation of the dynamical control system arepresented to overcome the difficulties due to the stiffness, the oscillation with high fre-quency and the discontinuity. The idea of computation of the analogue computer is intro-duced into the computation of digital simulation, the system is decomposed into somesubsystems, each of which can run independently, and, with the input informations ap-proached to, the local analytical expressions for the solutions of these subsystems are cons-tructed. Finally, a simple analysis about the accuracy estimate and the stability of the algo-rithms are given. The computational results show that the CPU time expended on thedigital simulation of a flying control system is only 1/20 of that needed by the traditionalnumerical methods.
基金supported by National Natural Science Foundation of China (Grant Nos.11771344 and 11701138)Natural Science Foundation of Hebei Province of China (Grant No. A2020202033)。
文摘This paper studies exact synchronization and asymptotic synchronization problems for a controlled linear system of ordinary differential equations. In this paper, we build up necessary and sufficient conditions for exact synchronization and asymptotic synchronization problems. When a system is not controllable but exactly synchronizable, it can be asymptotically synchronized in any given rate and the state of exact synchronization is given. However, when a system is not controllable and can be asymptotically synchronized in any given rate,it may not be exactly synchronizable.
基金Project supported by the National Natural Science Foundation of China (Nos. 12132010 and12021002)the Natural Science Foundation of Tianjin of China (No. 19JCZDJC38800)。
文摘The data-driven methods extract the feature information from data to build system models, which enable estimation and identification of the systems and can be utilized for prognosis and health management(PHM). However, most data-driven models are still black-box models that cannot be interpreted. In this study, we use the neural ordinary differential equations(ODEs), especially the inherent computational relationships of a system added to the loss function calculation, to approximate the governing equations. In addition, a new strategy for identifying the local parameters of the system is investigated, which can be utilized for system parameter identification and damage detection. The numerical and experimental examples presented in the paper demonstrate that the strategy has high accuracy and good local parameter identification. Moreover, the proposed method has the advantage of being interpretable. It can directly approximate the underlying governing dynamics and be a worthwhile strategy for system identification and PHM.
