A Fourier-Chebyshev pseudospectral scheme is proposed for three-dimensionalvorticily equation with unilaterally periodic boundary condition. The generalized stability and convergence are analysed. The numerical result...A Fourier-Chebyshev pseudospectral scheme is proposed for three-dimensionalvorticily equation with unilaterally periodic boundary condition. The generalized stability and convergence are analysed. The numerical results are presented.展开更多
The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are est...The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are established, which play important roles in the related numerical methods for unbounded domains. As an example, the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation. The stability and convergence of the suggested schemes are proved. Numerical results demonstrate the high accuracy of these approaches.展开更多
Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application,a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck...Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application,a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efficiency of the scheme and conform well to theoretical analysis.展开更多
In this paper, spectral and pseudospectral methods are applied to both time and space variables for parabolic equations. Spectral and pseudospectral schemes are given, and error estimates are obtained for approximate ...In this paper, spectral and pseudospectral methods are applied to both time and space variables for parabolic equations. Spectral and pseudospectral schemes are given, and error estimates are obtained for approximate solutions.展开更多
Several mixed Legendre spectral-pseudospectral approximations and Chebyshev-Legendre ap- proximations are proposed for estimating parameters in differential equations. They are easy to be performed, and have the spect...Several mixed Legendre spectral-pseudospectral approximations and Chebyshev-Legendre ap- proximations are proposed for estimating parameters in differential equations. They are easy to be performed, and have the spectral accuracy. The numerical results coincide with those of the theoretical analysis. It is easy to generalize the proposed methods to multiple-dimensional prob- lems.展开更多
In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interv...In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method＇s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.展开更多
This paper presents a fully spectral discretization method for solving KdV equations with periodic boundary conditions.Chebyshev pseudospectral approximation in the time direction and Fourier Galerkin approximation in...This paper presents a fully spectral discretization method for solving KdV equations with periodic boundary conditions.Chebyshev pseudospectral approximation in the time direction and Fourier Galerkin approximation in the spatial direction.The expansion coefficients are determined by minimizing an object funictional.Rapid convergence of the method is proved.展开更多
The nuclear norm convex relaxation method is proposed to force the rank constraint in the identification of the continuous-time( CT) Hammerstein system. The CT Hammerstein system is composed of a linear time invariant...The nuclear norm convex relaxation method is proposed to force the rank constraint in the identification of the continuous-time( CT) Hammerstein system. The CT Hammerstein system is composed of a linear time invariant( LTI) system and a static nonlinear function( the linear part is followed by the nonlinear part). The nonlinear function is approximated by the pseudospectral basis functions, which have a better performance than Hinge functions and Radial Basis functions. After the approximation on the nonlinear function, the CT Hammerstein system has been transformed into a multiple-input single-output( MISO) linear model system with the differential pre-filters. However, the coefficients of static nonlinearity and the numerators of the linear transfer function are coupled together to challenge the parameters identification of the Hammerstein system. This problem is solved by replacing the one-rank constraint of the regularization optimization with the nuclear norm convex relaxation. Finally, a numerical example is given to verify the accuracy and the efficiency of the method.展开更多
文摘A Fourier-Chebyshev pseudospectral scheme is proposed for three-dimensionalvorticily equation with unilaterally periodic boundary condition. The generalized stability and convergence are analysed. The numerical results are presented.
基金the Science Foundation of the Science and Technology Commission of Shanghai Municipality(No.075105118)the Shanghai Leading Academic Discipline Project(No.T0401)the Fund for E-institute of Shanghai Universities(No.E03004)
文摘The Laguerre spectral and pseudospectral methods are investigated for multidimensional nonlinear partial differential equations. Some results on the modified Laguerre orthogonal approximation and interpolation are established, which play important roles in the related numerical methods for unbounded domains. As an example, the modified Laguerre spectral and pseudospectral methods are proposed for two-dimensional Logistic equation. The stability and convergence of the suggested schemes are proved. Numerical results demonstrate the high accuracy of these approaches.
文摘Results on the composite generalized Laguerre-Legendre interpolation in unbounded domains are established. As an application,a composite Laguerre-Legendre pseudospectral scheme is presented for nonlinear Fokker-Planck equations on the whole line. The convergence and the stability of the proposed scheme are proved. Numerical results show the efficiency of the scheme and conform well to theoretical analysis.
文摘In this paper, spectral and pseudospectral methods are applied to both time and space variables for parabolic equations. Spectral and pseudospectral schemes are given, and error estimates are obtained for approximate solutions.
基金the Chinese Key project on Basic Research (No.1999032804), and Shanghai Natural Science Foundation (No.00JC14D57).
文摘Several mixed Legendre spectral-pseudospectral approximations and Chebyshev-Legendre ap- proximations are proposed for estimating parameters in differential equations. They are easy to be performed, and have the spectral accuracy. The numerical results coincide with those of the theoretical analysis. It is easy to generalize the proposed methods to multiple-dimensional prob- lems.
基金supported by the National Natural Science Foundation of China (No.61203022)the Aeronautical Science Foundation of China (2012CZ51029)
文摘In view of generating optimal trajectories of Bolza problems, standard Chebyshev pseudospectral （PS） method makes the points＇ accumulation near the extremities and rarefaction of nodes close to the center of interval, which causes an ill-condition of differentiation matrix and an oscillation of the optimal solution. For improvement upon the difficulties, a mapped Chebyshev pseudospectral method is proposed. A conformal map is applied to Chebyshev points to move the points closer to equidistant nodes. Condition number and spectral radius of differentiation matrices from both methods are presented to show the improvement. Furthermore, the modification keeps the Chebyshev pseudospectral method＇s advantage, the spectral convergence rate. Based on three numerical examples, a comparison of the execution time, convergence and accuracy is presented among the standard Chebyshev pseudospectral method, other collocation methods and the proposed one. In one example, the error of results from mapped Chebyshev pseudospectral method is reduced to 5% of that from standard Chebyshev pseudospectral method.
文摘This paper presents a fully spectral discretization method for solving KdV equations with periodic boundary conditions.Chebyshev pseudospectral approximation in the time direction and Fourier Galerkin approximation in the spatial direction.The expansion coefficients are determined by minimizing an object funictional.Rapid convergence of the method is proved.
文摘The nuclear norm convex relaxation method is proposed to force the rank constraint in the identification of the continuous-time( CT) Hammerstein system. The CT Hammerstein system is composed of a linear time invariant( LTI) system and a static nonlinear function( the linear part is followed by the nonlinear part). The nonlinear function is approximated by the pseudospectral basis functions, which have a better performance than Hinge functions and Radial Basis functions. After the approximation on the nonlinear function, the CT Hammerstein system has been transformed into a multiple-input single-output( MISO) linear model system with the differential pre-filters. However, the coefficients of static nonlinearity and the numerators of the linear transfer function are coupled together to challenge the parameters identification of the Hammerstein system. This problem is solved by replacing the one-rank constraint of the regularization optimization with the nuclear norm convex relaxation. Finally, a numerical example is given to verify the accuracy and the efficiency of the method.
