Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are establi...Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.展开更多
This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. We investigate the dissipativity properties of （k, l）- algebraic...This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. We investigate the dissipativity properties of （k, l）- algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid. The finite- dimensional and infinite-dimensional dissipativity results of （k, /）-algebraically stable Runge-Kutta methods are obtained.展开更多
The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method （...The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method （LWPM）. Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge-Kutta method of order four （RK-4）. Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.展开更多
基金Project supported by the National Natural Science Foundation of China(No.11432010)the Doctoral Program Foundation of Education Ministry of China(No.20126102110023)+2 种基金the 111Project of China(No.B07050)the Fundamental Research Funds for the Central Universities(No.310201401JCQ01001)the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University(No.CX201517)
文摘Projected Runge-Kutta （R-K） methods for constrained Hamiltonian systems are proposed. Dynamic equations of the systems, which are index-3 differential-algebraic equations （DAEs） in the Heisenberg form, are established under the framework of Lagrangian multipliers. R-K methods combined with the technique of projections are then used to solve the DAEs. The basic idea of projections is to eliminate the constraint violations at the position, velocity, and acceleration levels, and to preserve the total energy of constrained Hamiltonian systems by correcting variables of the position, velocity, acceleration, and energy. Numerical results confirm the validity and show the high precision of the proposed method in preserving three levels of constraints and total energy compared with results reported in the literature.
基金supported by National Natural Science Foundation of China (No. 11171125,91130003)Natural Science Foundation of Hubei (No. 2011CDB289)Youth Foundation of Naval University of Engineering (No.HGDQNJJ10003)
文摘This paper is concerned with the numerical dissipativity of multistep Runge-Kutta methods for nonlinear Volterra delay-integro-differential equations. We investigate the dissipativity properties of （k, l）- algebraically stable multistep Runge-Kutta methods with constrained grid and an uniform grid. The finite- dimensional and infinite-dimensional dissipativity results of （k, /）-algebraically stable Runge-Kutta methods are obtained.
文摘The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method （LWPM）. Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge-Kutta method of order four （RK-4）. Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.
