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Convergence Analysis of the Numerical Solution for Cathode Design of Aero-engine Blades in Electrochemical Machining 被引量:17
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作者 Li Zhiyong Niu Zongwei 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2007年第6期570-576,共7页
As a main difficult problem encountered in electrochemical machining (ECM), the cathode design is tackled, at present, with various numerical analysis methods such as finite difference, finite element and boundary e... As a main difficult problem encountered in electrochemical machining (ECM), the cathode design is tackled, at present, with various numerical analysis methods such as finite difference, finite element and boundary element methods. Among them, the finite element method presents more flexibility to deal with the irregularly shaped workpieces. However, it is very difficult to ensure the convergence of finite element numerical approach. This paper proposes an accurate model and a finite element numerical approach of cathode design based on the potential distribution in inter-electrode gap. In order to ensure the convergence of finite element numerical approach and increase the accuracy in cathode design, the cathode shape should be iterated to eliminate the design errors in computational process. Several experiments are conducted to verify the machining accuracy of the designed cathode. The experimental results have proven perfect convergence and good computing accuracy of the proposed finite element numerical approach by the high surface quality and dimensional accuracy of the machined blades. 展开更多
关键词 electrochemical machining aero-engine blade cathode design convergence analysis
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Convergence analysis of the corrected Uzawa algorithm for symmetric saddle point problems 被引量:2
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作者 LU Jun-feng 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2014年第1期29-35,共7页
For the large sparse saddle point problems, Pan and Li recently proposed in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31(3): 231-242] a corrected Uzawa algorithm based on a nonlinear Uzawa algorithm with two no... For the large sparse saddle point problems, Pan and Li recently proposed in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31(3): 231-242] a corrected Uzawa algorithm based on a nonlinear Uzawa algorithm with two nonlinear approximate inverses, and gave the detailed convergence analysis. In this paper, we focus on the convergence analysis of this corrected Uzawa algorithm, some inaccuracies in [H. K. Pan, W. Li, Math. Numer. Sinica, 2009, 31(3): 231-242] are pointed out, and a corrected convergence theorem is presented. A special case of this modified Uzawa algorithm is also discussed. 展开更多
关键词 Saddle point problem Uzawa algorithm convergence analysis
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A Mathematical Model of Real-Time Simulation and the Convergence Analysis on Real-Time Runge-Kutta Algorithms 被引量:1
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作者 Song Xiaoqiu, Li Bohu, Liu Degui, Yuan ZhaodingBeijing Institute of Computer Application and Simulation Technology, P. O. Box 142-213, Beijing 100854, China 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 1991年第1期129-139,共11页
In this paper, a mathematical model of real-time simulation is given, and the problem of convergence on real-time Runge-Kutta algorithms is analysed. At last a theorem on the relation between the order of compensation... In this paper, a mathematical model of real-time simulation is given, and the problem of convergence on real-time Runge-Kutta algorithms is analysed. At last a theorem on the relation between the order of compensation and the convergent order of real-time algorithm is proved. 展开更多
关键词 Real-time simulation Runge-Kutta algorithm convergence analysis.
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Land degradation sensitivity assessment and convergence analysis in Korla of Xinjiang, China
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作者 DING Jinchen CHEN Yunzhi +1 位作者 WANG Xiaoqin CAO Meiqin 《Journal of Arid Land》 SCIE CSCD 2020年第4期594-608,共15页
Land degradation has a major impact on environmental and socio-economic sustainability. Scientific methods are necessary to monitor the risk of land degradation. In this study, the environmental sensitive area index(E... Land degradation has a major impact on environmental and socio-economic sustainability. Scientific methods are necessary to monitor the risk of land degradation. In this study, the environmental sensitive area index(ESAI) was utilized to assess land degradation sensitivity and convergence analysis in Korla, a typical oasis city in Xinjiang of China, which is located on the northeast border of the Tarim Basin. A total of 18 indicators depicting soil, climate, vegetation, and management qualities were used to illustrate spatial-temporal patterns of land degradation sensitivity from 1994 to 2018. We investigated the causes of spatial convergence and divergence based on the Ordinary Least Squares(OLS) and Geographically Weighted Regression(GWR) models. The results show that the branch of the Tianshan Mountains and oasis plain had a low sensitivity to land degradation, while the Tarim Basin had a high risk of land degradation. More than two-thirds of the study area can be categorized as "critical" sensitivity classes. The largest percentage(32.6%) of fragile classes was observed for 2006. There was no significant change in insensitive or low-sensitivity areas, which accounted for less than 0.4% of the entire observation period. The ESAI of the four time periods(1994–1998, 1998–2006, 2006–2010, and 2010–2018) formed a series of convergence patterns. The convergence patterns of 1994–1998 and 1998–2006 can be explained by the government's efforts to "Returning Farmland to Forests" and other governance projects. In 2006–2010, the construction of afforested work intensified, but industrial development and human activities affected the convergence pattern. The pattern of convergence in most regions between 2010 and 2018 can be attributed to the government's implementation of a series of key ecological protection projects, which led to a decrease in sensitivity to land degradation. The results of this study altogether suggest that the ESAI convergence analysis is an effective early warning method for land degradation sensitivity. 展开更多
关键词 land degradation quality index convergence analysis remote sensing environmental sensitive area index Korla
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Numerical solutions of two-dimensional nonlinear integral equations via Laguerre Wavelet method with convergence analysis
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作者 K.Maleknejad M.Soleiman Dehkordi 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2021年第1期83-98,共16页
In this paper,the approximate solutions for two different type of two-dimensional nonlinear integral equations:two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm i... In this paper,the approximate solutions for two different type of two-dimensional nonlinear integral equations:two-dimensional nonlinear Volterra-Fredholm integral equations and the nonlinear mixed Volterra-Fredholm integral equations are obtained using the Laguerre wavelet method.To do this,these two-dimensional nonlinear integral equations are transformed into a system of nonlinear algebraic equations in matrix form.By solving these systems,unknown coefficients are obtained.Also,some theorems are proved for convergence analysis.Some numerical examples are presented and results are compared with the analytical solution to demonstrate the validity and applicability of the proposed method. 展开更多
关键词 he two-dimensional nonlinear integral equations the nonlinear mixed Volterra-Fredholm inte-gral equations two-dimensional Laguerre wavelet Orthogonal polynomial convergence analysis the Darboux problem.
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Convergence analysis for delay Volterra integral equation
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作者 ZHENG Wei-shan 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2022年第2期306-316,共11页
In this article we use Chebyshev spectral collocation method to deal with the Volterra integral equation which has two kinds of delay items. We use linear transformation to make the interval into a fixed interval [-1,... In this article we use Chebyshev spectral collocation method to deal with the Volterra integral equation which has two kinds of delay items. We use linear transformation to make the interval into a fixed interval [-1, 1]. Then we use the Gauss quadrature formula to approximate the solution. With the help of lemmas, we get the result that the numerical error decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norm. Some numerical experiments are given to confirm our theoretical prediction. 展开更多
关键词 Chebyshev spectral collocation method DELAY Gauss quadrature formula convergence analysis
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Convergence Analysis of a Self-Stabilizing Algorithm for Minor Component Analysis
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作者 Haidi Dong Yingbin Gao Gang Liu 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2020年第6期1585-1592,共8页
The M?ller algorithm is a self-stabilizing minor component analysis algorithm.This research document involves the study of the convergence and dynamic characteristics of the M?ller algorithm using the deterministic di... The M?ller algorithm is a self-stabilizing minor component analysis algorithm.This research document involves the study of the convergence and dynamic characteristics of the M?ller algorithm using the deterministic discrete time(DDT)methodology.Unlike other analysis methodologies,the DDT methodology is capable of serving the distinct time characteristic and having no constraint conditions.Through analyzing the dynamic characteristics of the weight vector,several convergence conditions are drawn,which are beneficial for its application.The performing computer simulations and real applications demonstrate the correctness of the analysis’s conclusions. 展开更多
关键词 convergence analysis deterministic discrete time(DDT) dynamic characteristic Moller algorithm
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Theoretical convergence analysis of complex Gaussian kernel LMS algorithm
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作者 Wei Gao Jianguo Huang +1 位作者 Jing Han Qunfei Zhang 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2016年第1期39-50,共12页
With the vigorous expansion of nonlinear adaptive filtering with real-valued kernel functions,its counterpart complex kernel adaptive filtering algorithms were also sequentially proposed to solve the complex-valued no... With the vigorous expansion of nonlinear adaptive filtering with real-valued kernel functions,its counterpart complex kernel adaptive filtering algorithms were also sequentially proposed to solve the complex-valued nonlinear problems arising in almost all real-world applications.This paper firstly presents two schemes of the complex Gaussian kernel-based adaptive filtering algorithms to illustrate their respective characteristics.Then the theoretical convergence behavior of the complex Gaussian kernel least mean square(LMS) algorithm is studied by using the fixed dictionary strategy.The simulation results demonstrate that the theoretical curves predicted by the derived analytical models consistently coincide with the Monte Carlo simulation results in both transient and steady-state stages for two introduced complex Gaussian kernel LMS algonthms using non-circular complex data.The analytical models are able to be regard as a theoretical tool evaluating ability and allow to compare with mean square error(MSE) performance among of complex kernel LMS(KLMS) methods according to the specified kernel bandwidth and the length of dictionary. 展开更多
关键词 nonlinear adaptive filtering complex Gaussian kernel convergence analysis non-circular data kernel least mean square(KLMS).
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The fast method and convergence analysis of the fractional magnetohydrodynamic coupled flow and heat transfer model for the generalized second-grade fluid 被引量:1
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作者 Xiaoqing Chi Hui Zhang Xiaoyun Jiang 《Science China Mathematics》 SCIE CSCD 2024年第4期919-950,共32页
In this paper,we first establish a new fractional magnetohydrodynamic(MHD)coupled flow and heat transfer model for a generalized second-grade fluid.This coupled model consists of a fractional momentum equation and a h... In this paper,we first establish a new fractional magnetohydrodynamic(MHD)coupled flow and heat transfer model for a generalized second-grade fluid.This coupled model consists of a fractional momentum equation and a heat conduction equation with a generalized form of Fourier law.The second-order fractional backward difference formula is applied to the temporal discretization and the Legendre spectral method is used for the spatial discretization.The fully discrete scheme is proved to be stable and convergent with an accuracy of O(τ^(2)+N-r),whereτis the time step-size and N is the polynomial degree.To reduce the memory requirements and computational cost,a fast method is developed,which is based on a globally uniform approximation of the trapezoidal rule for integrals on the real line.The strict convergence of the numerical scheme with this fast method is proved.We present the results of several numerical experiments to verify the effectiveness of the proposed method.Finally,we simulate the unsteady fractional MHD flow and heat transfer of the generalized second-grade fluid through a porous medium.The effects of the relevant parameters on the velocity and temperature are presented and analyzed in detail. 展开更多
关键词 fractional MHD coupled flow and heat transfer model generalized second-grade fuid fast method convergence analysis numerical simulation
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Convergence Analysis of Kernel Learning FBSDE Filter
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作者 Yunzheng Lyu Feng Bao 《Communications in Mathematical Research》 CSCD 2024年第3期313-342,共30页
Kernel learning forward backward stochastic differential equations(FBSDE)filter is an iterative and adaptive meshfree approach to solve the non-linear filtering problem.It builds from forward backward SDE for Fokker-P... Kernel learning forward backward stochastic differential equations(FBSDE)filter is an iterative and adaptive meshfree approach to solve the non-linear filtering problem.It builds from forward backward SDE for Fokker-Planker equation,which defines evolving density for the state variable,and employs kernel density estimation(KDE)to approximate density.This algo-rithm has shown more superior performance than mainstream particle filter method,in both convergence speed and efficiency of solving high dimension problems.However,this method has only been shown to converge empirically.In this paper,we present a rigorous analysis to demonstrate its local and global convergence,and provide theoretical support for its empirical results. 展开更多
关键词 Forward backward stochastic differential equations kernel density estima-tion nonlinear filtering problems convergence analysis.
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A Framework of Convergence Analysis of Mini-batch Stochastic Projected Gradient Methods 被引量:1
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作者 Jian Gu Xian-Tao Xiao 《Journal of the Operations Research Society of China》 EI CSCD 2023年第2期347-369,共23页
In this paper,we establish a unified framework to study the almost sure global convergence and the expected convergencerates of a class ofmini-batch stochastic(projected)gradient(SG)methods,including two popular types... In this paper,we establish a unified framework to study the almost sure global convergence and the expected convergencerates of a class ofmini-batch stochastic(projected)gradient(SG)methods,including two popular types of SG:stepsize diminished SG and batch size increased SG.We also show that the standard variance uniformly bounded assumption,which is frequently used in the literature to investigate the convergence of SG,is actually not required when the gradient of the objective function is Lipschitz continuous.Finally,we show that our framework can also be used for analyzing the convergence of a mini-batch stochastic extragradient method for stochastic variational inequality. 展开更多
关键词 Stochastic projected gradient method Variance uniformly bounded convergence analysis
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Convergence Analysis of Splitting-Up Algorithm of the Zakai’s Equation with Correlated Noises
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作者 LUO Xue PAN Ting DONG Wenhui 《Journal of Systems Science & Complexity》 SCIE EI CSCD 2023年第3期922-946,共25页
In the field of nonlinear filtering(NLF),it is well-known that the unnormalized conditional density of the states satisfies the Zakai’s equation.The splitting-up algorithm has been first studied in the independent no... In the field of nonlinear filtering(NLF),it is well-known that the unnormalized conditional density of the states satisfies the Zakai’s equation.The splitting-up algorithm has been first studied in the independent noises case by Bensoussan,et al.(1990).In this paper,the authors extend this convergence analysis of the splitting-up algorithm to the correlated noises’case.Given a time discretization,one splits the solution of the Zakai’s equation into two interlacing processes(with possibly computational advantage).These two processes correspond respectively to the prediction and updating.Under certain conditions,the authors show that both processes tend to the solution of the Zakai’s equation,as the time step goes to zero.The authors specify the conditions imposed on the way of splitting-up to guarantee the convergence.The major technical difficulty in the correlated noises’case,compared with the independent case,is to control the gradient of the second process in some sense.To illustrate the potentially computational advantage of the schemes based on the splitting-up ways,the authors experiment on a toy NLF model using the feedback particle filter(FPF)developed based on the splitting-up method and the sampling importance and resampling(SIR)as comparison.The FPF outperforms in both accuracy and efficiency. 展开更多
关键词 convergence analysis correlated noises nonlinear filtering splitting-up algorithm
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A Convergence Analysis of a Structure-Preserving Gradient Flow Method for the All-Electron Kohn-Sham Model
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作者 Yedan Shen Ting Wang +1 位作者 Jie Zhou Guanghui Hu 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE CSCD 2023年第3期597-621,共25页
In[Dai et al.,Multi.Model.Simul.18(4)(2020)],a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory,based on which a linearized method was deve... In[Dai et al.,Multi.Model.Simul.18(4)(2020)],a structure-preserving gradient flow method was proposed for the ground state calculation in Kohn-Sham density functional theory,based on which a linearized method was developed in[Hu et al.,EAJAM.13(2)(2023)]for further improving the numerical efficiency.In this paper,a complete convergence analysis is delivered for such a linearized method for the all-electron Kohn-Sham model.Temporally,the convergence,the asymptotic stability,as well as the structure-preserving property of the linearized numerical scheme in the method is discussed following previous works,while spatially,the convergence of the h-adaptive mesh method is demonstrated following[Chen et al.,Multi.Model.Simul.12(2014)],with a key study on the boundedness of the Kohn-Sham potential for the all-electron Kohn-Sham model.Numerical examples confirm the theoretical results very well. 展开更多
关键词 Kohn-Sham density functional theory gradient flow model structure-preserving linear scheme convergence analysis
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A New Framework of Convergence Analysis for Solving the General Nonlinear Schrodinger Equation using the Fourier Pseudo-Spectral Method in Two Dimensions
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作者 Jialing Wang Tingchun Wang Yushun Wang 《Advances in Applied Mathematics and Mechanics》 SCIE 2023年第3期786-813,共28页
This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the n... This paper aims to build a new framework of convergence analysis of conservative Fourier pseudo-spectral method for the general nonlinear Schr¨odinger equation in two dimensions,which is not restricted that the nonlinear term is mere cubic.The new framework of convergence analysis consists of two steps.In the first step,by truncating the nonlinear term into a global Lipschitz function,an alternative numerical method is proposed and proved in a rigorous way to be convergent in the discrete L2 norm;followed in the second step,the maximum bound of the numerical solution of the alternative numerical method is obtained by using a lifting technique,as implies that the two numerical methods are the same one.Under our framework of convergence analysis,with neither any restriction on the grid ratio nor any requirement of the small initial value,we establish the error estimate of the proposed conservative Fourier pseudo-spectral method,while previous work requires the certain restriction for the focusing case.The error bound is proved to be of O(h^(r)+t^(2))with grid size h and time step t.In fact,the framework can be used to prove the unconditional convergence of many other Fourier pseudo-spectral methods for solving the nonlinear Schr¨odinger-type equations.Numerical results are conducted to indicate the accuracy and efficiency of the proposed method,and investigate the effect of the nonlinear term and initial data on the blow-up solution. 展开更多
关键词 Framework of convergence analysis general nonlinear Schr¨odinger equation Fourier pseudo-spectral method conservation laws unconditional convergence blow-up solution
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An Upwind Mixed Finite Volume Element-fractional Step Method and Convergence Analysis for Three-dimensional Compressible Contamination Treatment from Nuclear Waste
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作者 Chang-feng LI Yi-rang YUAN Huai-ling SONG 《Acta Mathematicae Applicatae Sinica》 SCIE CSCD 2023年第4期808-829,共22页
In this paper the authors discuss a numerical simulation problem of three-dimensional compressible contamination treatment from nuclear waste. The mathematical model, a nonlinear convection-diffusion system of four PD... In this paper the authors discuss a numerical simulation problem of three-dimensional compressible contamination treatment from nuclear waste. The mathematical model, a nonlinear convection-diffusion system of four PDEs, determines four major physical unknowns: the pressure, the concentrations of brine and radionuclide, and the temperature. The pressure is solved by a conservative mixed finite volume element method, and the computational accuracy is improved for Darcy velocity. Other unknowns are computed by a composite scheme of upwind approximation and mixed finite volume element. Numerical dispersion and nonphysical oscillation are eliminated, and the convection-dominated diffusion problems are solved well with high order computational accuracy. The mixed finite volume element is conservative locally, and get the objective functions and their adjoint vector functions simultaneously. The conservation nature is an important character in numerical simulation of underground fluid. Fractional step difference is introduced to solve the concentrations of radionuclide factors, and the computational work is shortened significantly by decomposing a three-dimensional problem into three successive one-dimensional problems. By the theory and technique of a priori estimates of differential equations, we derive an optimal order estimates in L2norm. Finally, numerical examples show the effectiveness and practicability for some actual problems. 展开更多
关键词 compressible nuclear waste contamination in porous media upwind mixed finite volume elementfractional step conservation of mass and energy convergence analysis numerical example
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ON SPECTRAL METHODS FOR VOLTERRA INTEGRAL EQUATIONS AND THE CONVERGENCE ANALYSIS 被引量:34
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作者 Tao Tang Xiang XU Jin Cheng 《Journal of Computational Mathematics》 SCIE CSCD 2008年第6期825-837,共13页
The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equatio... The main purpose of this work is to provide a novel numerical approach for the Volterra integral equations based on a spectral approach. A Legendre-collocation method is proposed to solve the Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical results confirm the theoretical prediction of the exponential rate of convergence. The result in this work seems to be the first successful spectral approach (with theoretical justification) for the Volterra type equations. 展开更多
关键词 Legendre-spectral method Second kind Volterra integral equations convergence analysis.
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Convergence Analysis of a Block-by-Block Method for Fractional Differential Equations 被引量:11
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作者 Jianfei Huang Yifa Tang Luis Vázquez 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE 2012年第2期229-241,共13页
The block-by-block method,proposed by Linz for a kind of Volterra integral equations with nonsingular kernels,and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations... The block-by-block method,proposed by Linz for a kind of Volterra integral equations with nonsingular kernels,and extended by Kumar and Agrawal to a class of initial value problems of fractional differential equations(FDEs)with Caputo derivatives,is an efficient and stable scheme.We analytically prove and numerically verify that this method is convergent with order at least 3 for any fractional order indexα>0. 展开更多
关键词 Fractional differential equation Caputo derivative block-by-block method convergence analysis
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Convergence analysis of Jacobi spectral collocation methods for Abel-Volterra integral equations of second kind 被引量:9
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作者 Xianjuan LI Tao TANG 《Frontiers of Mathematics in China》 SCIE CSCD 2012年第1期69-84,共16页
This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel p(t, s) = (t - s)^-μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 20... This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel p(t, s) = (t - s)^-μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233:938 950], the error analysis for this approach is carried out for 0 〈 μ 〈 1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-ype but also establish the error estimates under a more general regularity assumption on the exact solution. 展开更多
关键词 Jacobi spectral collocation method Abel-Volterra integralequation convergence analysis
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ON THE CONVERGENCE ANALYSIS OF ALGORITHMS FOR GENERALIZED SET-VALUED VARIATIONAL INCLUSIONS IN BANACH SPACES 被引量:4
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作者 ZENGLuchuan 《Journal of Systems Science & Complexity》 SCIE EI CSCD 2004年第1期64-74,共11页
In this paper, we are devoted to the convergence analysis of algorithms forgeneralized set-valued variational inclusions in Banach spaces. Our results improve, extend,and develop the earlier and recent corresponding r... In this paper, we are devoted to the convergence analysis of algorithms forgeneralized set-valued variational inclusions in Banach spaces. Our results improve, extend,and develop the earlier and recent corresponding results. 展开更多
关键词 generalized set-valued variational inclusion ALGORITHM convergence analysis
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ADAPTIVE QUADRILATERAL AND HEXAHEDRAL FINITE ELEMENT METHODS WITH HANGING NODES AND CONVERGENCE ANALYSIS 被引量:3
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作者 Xuying Zhao Shipeng Mao Zhong-Ci Shi 《Journal of Computational Mathematics》 SCIE CSCD 2010年第5期621-644,共24页
In this paper we study the convergence of adaptive finite element methods for the gen- eral non-attine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes. Based on several basic ... In this paper we study the convergence of adaptive finite element methods for the gen- eral non-attine equivalent quadrilateral and hexahedral elements on 1-irregular meshes with hanging nodes. Based on several basic ingredients, such as quasi-orthogonality, estimator reduction and D6fler marking strategy, convergence of the adaptive finite element methods for the general second-order elliptic partial equations is proved. Our analysis is effective for all conforming Qm elements which covers both the two- and three-dimensional cases in a unified fashion. 展开更多
关键词 Finite element method Adaptive algorithm Hanging node 1-irregular mesh convergence analysis.
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