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Alternative Error Bounds for Some Numerical Quadrature Rules
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作者 LU Jian-fang FENG Ping-hua 《Chinese Quarterly Journal of Mathematics》 CSCD 2011年第1期108-113,共6页
Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson's quadrature rule involving n-time(1 ≤ n ≤ 4) differentiable mappings and ... Similar to having done for the mid-point and trapezoid quadrature rules,we obtain alternative estimations of error bounds for the Simpson's quadrature rule involving n-time(1 ≤ n ≤ 4) differentiable mappings and then to the estimations of error bounds for the adaptive Simpson's quadrature rule. 展开更多
关键词 n-numerical quadrature rule Simpson's quadrature rule adaptive quadrature rule differentiable mapping error bounds
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Orthogonal Polynomials with Respect to Modified JacobiWeight and Corresponding Quadrature Rules of Gaussian Type
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作者 Marija P.Stanic Aleksandar S.Cvetkovic 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE 2011年第4期478-488,共11页
In this paper we consider polynomials orthogonal with respect to the linear functional L:P→C,defined on the space of all algebraic polynomials P by L[p]=∫_(-1)^(1)p(x)(1−x)^(α−1/2)(1+x)^(β−1/2)exp(iζx)dx,whereα,... In this paper we consider polynomials orthogonal with respect to the linear functional L:P→C,defined on the space of all algebraic polynomials P by L[p]=∫_(-1)^(1)p(x)(1−x)^(α−1/2)(1+x)^(β−1/2)exp(iζx)dx,whereα,β>−1/2 are real numbers such thatℓ=|β−α|is a positive integer,andζ∈R\{0}.We prove the existence of such orthogonal polynomials for some pairs ofαandζand for all nonnegative integersℓ.For such orthogonal polynomials we derive three-term recurrence relations and also some differential-difference relations.For such orthogonal polynomials the corresponding quadrature rules of Gaussian type are considered.Also,some numerical examples are included. 展开更多
关键词 Orthogonal polynomials modified Jacobi weight function recurrence relation Gaussian quadrature rule
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Generalized cubature quadrature Kalman filters:derivations and extensions 被引量:2
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作者 Hongwei Wang Wei Zhang +1 位作者 Junyi Zuo Heping Wang 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2017年第3期556-562,共7页
A new Gaussian approximation nonlinear filter called generalized cubature quadrature Kalman filter (GCQKF) is introduced for nonlinear dynamic systems. Based on standard GCQKF, two extensions are developed, namely squ... A new Gaussian approximation nonlinear filter called generalized cubature quadrature Kalman filter (GCQKF) is introduced for nonlinear dynamic systems. Based on standard GCQKF, two extensions are developed, namely square root generalized cubature quadrature Kalman filter (SR-GCQKF) and iterated generalized cubature quadrature Kalman filter (I-GCQKF). In SR-GCQKF, the QR decomposition is exploited to alter the Cholesky decomposition and both predicted and filtered error covariances have been propagated in square root format to make sure the numerical stability. In I-GCQKF, the measurement update step is executed iteratively to make full use of the latest measurement and a new terminal criterion is adopted to guarantee the increase of likelihood. Detailed numerical experiments demonstrate the superior performance on both tracking stability and estimation accuracy of I-GCQKF and SR-GCQKF compared with GCQKF. 展开更多
关键词 cubature rule quadrature rule Kalman filter iterated method QR decomposition nonlinear estimation target tracking
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Differential quadrature time element method for structural dynamics 被引量:3
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作者 Yu-Feng Xing Jing Guo 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2012年第3期782-792,共11页
An accurate and efficient differential quadrature time element method (DQTEM) is proposed for solving ordi- nary differential equations (ODEs), the numerical dissipation and dispersion of DQTEM is much smaller tha... An accurate and efficient differential quadrature time element method (DQTEM) is proposed for solving ordi- nary differential equations (ODEs), the numerical dissipation and dispersion of DQTEM is much smaller than that of the direct integration method of single/multi steps. Two methods of imposing initial conditions are given, which avoids the tediousness when derivative initial conditions are imposed, and the numerical comparisons indicate that the first method, in which the analog equations of initial displacements and velocities are used to directly replace the differential quadra- ture (DQ) analog equations of ODEs at the first and the last sampling points, respectively, is much more accurate than the second method, in which the DQ analog equations of initial conditions are used to directly replace the DQ analog equations of ODEs at the first two sampling points. On the contrary to the conventional step-by-step direct integration schemes, the solutions at all sampling points can be obtained simultaneously by DQTEM, and generally, one differential quadrature time element may be enough for the whole time domain. Extensive numerical comparisons validate the effi- ciency and accuracy of the proposed method. 展开更多
关键词 Differential quadrature rule Direct integrationmethod Time element Phase error. Artificial damping
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A QUADRATURE RULE FOR HADAMARD FINITE PART INTEGRALS
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作者 Samir A.Ashour 《Analysis in Theory and Applications》 1996年第4期105-110,共6页
Finite part integrals introduced by Hadamard in connection with hyperbolic partial differential equations,have been useful in a number of engineering applications.In this paper we investigate some numerical methods fo... Finite part integrals introduced by Hadamard in connection with hyperbolic partial differential equations,have been useful in a number of engineering applications.In this paper we investigate some numerical methods for computing finite-part integrals. 展开更多
关键词 A quadrature RULE FOR HADAMARD FINITE PART INTEGRALS
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A stable one-point quadrature rule for three-dimensional numerical manifold method
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作者 ZHANG Ning ZHENG Hong +2 位作者 YANG Liang WU WenAn YUAN Chi 《Science China(Technological Sciences)》 SCIE EI CAS CSCD 2024年第5期1401-1416,共16页
We present a numerically stable one-point quadrature rule for the stiffness matrix and mass matrix of the three-dimensional numerical manifold method(3D NMM).The rule simplifies the integration over irregularly shaped... We present a numerically stable one-point quadrature rule for the stiffness matrix and mass matrix of the three-dimensional numerical manifold method(3D NMM).The rule simplifies the integration over irregularly shaped manifold elements and overcomes locking issues,and it does not cause spurious modes in modal analysis.The essential idea is to transfer the integral over a manifold element to a few moments to the element center,thereby deriving a one-point integration rule by the moments and making modifications to avoid locking issues.For the stiffness matrix,after the virtual work is decomposed into moments,higher-order moments are modified to overcome locking issues in nearly incompressible and bending-dominated conditions.For the mass matrix,the consistent and lumped types are derived by moments.In particular,the lumped type has the clear advantage of simplicity.The proposed method is naturally suitable for 3D NMM meshes automatically generated from a regular grid.Numerical tests justify the accuracy improvements and the stability of the proposed procedure. 展开更多
关键词 three-dimensional numerical manifold method quadrature rule LOCKING mass lumping
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Modeling and analysis of piezoelectric beam with periodically variable cross-sections for vibration energy harvesting 被引量:7
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作者 M.HAJHOSSEINI M.RAFEEYAN 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2016年第8期1053-1066,共14页
A bimorph piezoelectric beam with periodically variable cross-sections is used for the vibration energy harvesting. The effects of two geometrical parameters on the first band gap of this periodic beam are investigate... A bimorph piezoelectric beam with periodically variable cross-sections is used for the vibration energy harvesting. The effects of two geometrical parameters on the first band gap of this periodic beam are investigated by the generalized differential quadrature rule (GDQR) method. The GDQR method is also used to calculate the forced vibration response of the beam and voltage of each piezoelectric layer when the beam is subject to a sinusoidal base excitation. Results obtained from the analytical method are compared with those obtained from the finite element simulation with ANSYS, and good agreement is found. The voltage output of this periodic beam over its first band gap is calculated and compared with the voltage output of the uniform piezoelectric beam. It is concluded that this periodic beam has three advantages over the uniform piezoelectric beam, i.e., generating more voltage outputs over a wide frequency range, absorbing vibration, and being less weight. 展开更多
关键词 vibration energy harvesting piezoelectric cantilever beam periodically variable cross-section vibration band gap forced vibration analysis generalized differential quadrature rule (GDQR)
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Chebyshev spectral variational integrator and applications 被引量:2
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作者 Zhonggui YI Baozeng YUE Mingle DENG 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2020年第5期753-768,共16页
The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rat... The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system. 展开更多
关键词 geometric numerical method spectral method variational integrator Clenshaw-Curtis quadrature rule barycentric Lagrange interpolation orbital propagation
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Error Control Strategies for Numerical Integrations in Fast Collocation Methods 被引量:2
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作者 陈仲英 巫斌 许跃生 《Northeastern Mathematical Journal》 CSCD 2005年第2期233-252,共20页
We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utiliz... We propose two error control techniques for numerical integrations in fast multiscale collocation methods for solving Fredholm integral equations of the second kind with weakly singular kernels. Both techniques utilize quadratures for singular integrals using graded points. One has a polynomial order of accuracy if the integrand has a polynomial order of smoothness except at the singular point and the other has exponential order of accuracy if the integrand has an infinite order of smoothness except at the singular point. We estimate the order of convergence and computational complexity of the corresponding approximate solutions of the equation. We prove that the second technique preserves the order of convergence and computational complexity of the original collocation method. Numerical experiments are presented to illustrate the theoretical estimates. 展开更多
关键词 Fredholm integral equation of the second kind fast collocation method quadrature rule error control
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THEORETICAL ANALYSIS OF THE REPRODUCING KERNEL GRADIENT SMOOTHING INTEGRATION TECHNIQUE IN GALERKIN MESHLESS METHODS 被引量:1
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作者 Xiaolin Li 《Journal of Computational Mathematics》 SCIE CSCD 2023年第3期501-524,共24页
Numerical integration poses greater challenges in Galerkin meshless methods than finite element methods owing to the non-polynomial feature of meshless shape functions.The reproducing kernel gradient smoothing integra... Numerical integration poses greater challenges in Galerkin meshless methods than finite element methods owing to the non-polynomial feature of meshless shape functions.The reproducing kernel gradient smoothing integration(RKGSI)is one of the optimal numerical integration techniques in Galerkin meshless methods with minimum integration points.In this paper,properties,quadrature rules and the effect of the RKGSI on meshless methods are analyzed.The existence,uniqueness and error estimates of the solution of Galerkin meshless methods under numerical integration with the RKGSI are established.A procedure on how to choose quadrature rules to recover the optimal convergence rate is presented. 展开更多
关键词 Galerkin meshless method Numerical integration quadrature rule Error estimates Element-free Galerkin method Degree of precision
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Sinc-Multistep Schemes for Forward Backward Stochastic Differential Equations
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作者 Xu Wang Weidong Zhao 《Advances in Applied Mathematics and Mechanics》 SCIE 2023年第3期737-768,共32页
In this work,by combining the multistep discretization in time and the Sinc quadrature rule for approximating the conditional mathematical expectations,we will propose new fully discrete multistep schemes called“Sinc... In this work,by combining the multistep discretization in time and the Sinc quadrature rule for approximating the conditional mathematical expectations,we will propose new fully discrete multistep schemes called“Sinc-multistep schemes”for forward backward stochastic differential equations(FBSDEs).The schemes avoid spatial interpolations and admit high order of convergence.The stability and the K-th order error estimates in time for the K-step Sinc multistep schemes are theoretically proved(1≤K≤6).This seems to be the first time for analyzing fully time-space discrete multistep schemes for FBSDEs.Numerical examples are also presented to demonstrate the effectiveness,stability,and high order of convergence of the proposed schemes. 展开更多
关键词 Forward backward stochastic differential equations multistep schemes Sinc quadrature rule error estimates
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GENUINE-OPTIMAL CIRCULANT PRECONDITIONERS FOR WIENER-HOPF EQUATIONS
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作者 Fu-rong Lin (Department of Mathematics, Shantou University, Shantou 515063,China) 《Journal of Computational Mathematics》 SCIE EI CSCD 2001年第6期629-638,共10页
Discusses the genuine-optimal circulant preconditioner for finite-section Wiener-Hopf equations. Definition of the genuine-optimal circulant preconditioner; Use of the preconditioned conjugate gradient method; Numeric... Discusses the genuine-optimal circulant preconditioner for finite-section Wiener-Hopf equations. Definition of the genuine-optimal circulant preconditioner; Use of the preconditioned conjugate gradient method; Numerical treatments for high order quadrature rules. 展开更多
关键词 Wiener-Hopf equations circulant preconditioner preconditioned conjugate gradient method quadrature rules Hilbert-Schmidt norm
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Orthogonal Laurent Polynomials and Their Zeros
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作者 周恒 王仁宏 《Journal of Mathematical Research and Exposition》 CSCD 北大核心 2006年第1期19-22,共4页
In this note, we investigate the necessity for the measure dψ being a strong distribution, which is associated with the coefficients of the recurrence relation satisfied by the orthogonal Laurent polynomials. We also... In this note, we investigate the necessity for the measure dψ being a strong distribution, which is associated with the coefficients of the recurrence relation satisfied by the orthogonal Laurent polynomials. We also give out a representation of the greatest zeros of orthogonal Laurent polynomials in the case of dψ being a strong distribution. 展开更多
关键词 orthogonal Laurent polynomials quadrature rule zeros of orthogonal Laurent polynomials.
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A SPARSE-GRID METHOD FOR MULTI-DIMENSIONAL BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS 被引量:2
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作者 Guannan Zhang Max Gunzburger Weidong Zhao 《Journal of Computational Mathematics》 SCIE CSCD 2013年第3期221-248,共28页
A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e.... A sparse-grid method for solving multi-dimensional backward stochastic differential equations (BSDEs) based on a multi-step time discretization scheme [31] is presented. In the multi-dimensional spatial domain, i.e. the Brownian space, the conditional mathe- matical expectations derived from the original equation are approximated using sparse-grid Gauss-Hermite quadrature rule and (adaptive) hierarchical sparse-grid interpolation. Error estimates are proved for the proposed fully-discrete scheme for multi-dimensional BSDEs with certain types of simplified generator functions. Finally, several numerical examples are provided to illustrate the accuracy and efficiency of our scheme. 展开更多
关键词 Backward stochastic differential equations Multi-step scheme Gauss-Hermite quadrature rule Adaptive hierarchical basis Sparse grids.
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