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Fully Discrete Galerkin Finite Element Method for the Cubic Nonlinear Schr(o)dinger Equation

高等学校计算数学学报（英文版） 2017年第3期10卷 671-688页

作者： Jianyun Wang Yunqing Huang School of Science Hunan University of Technology Zhuzhou 412007 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan 411105 China

This paper is concemed with numerical method for a two-dimensional timedependent cubic nonlinear Schr(o)dinger *** approximations are obtained by the Galerkin finite element method in space in conjunction with the backward Euler method and the Crank-Nicolson method in time,*** prove optimal L2 error estimates for two fully discrete schemes by using elliptic projection ***,a numerical example is provided to verify our theoretical results.

关键词： Finite element method nonlinear Schr(o)dinger equations backward Euler scheme Crank-Nicolson scheme

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The Schur complements for SDD1 matrices and their application to linear complementarity problems

arXiv

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arXiv 2025年

作者： Hu, Yang Liu, Jianzhou Zeng, Wenlong School of Mathematics and Computational Science Xiangtan University Hunan Xiangtan411105 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan China Department of Mathematics Shanghai University Shanghai200444 China

In this paper we propose a new scaling method to study the Schur complements of SDD1 matrices. Its core is related to the non-negative property of the inverse M-matrix, while numerically improving the Quotient formula. Based on the Schur complement and a novel norm splitting manner, we establish an upper bound for the infinity norm of the inverse of SDD1 matrices, which depends solely on the original matrix entries. We apply the new bound to derive an error bound for linear complementarity problems of B1-matrices. Additionally, new lower and upper bounds for the determinant of SDD1 matrices are presented. Numerical experiments validate the effectiveness and superiority of our results. © 2025, CC BY.

关键词： Matrix algebra

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AN EFFICIENT NUMERICAL METHOD FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO DERIVATIVES

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Journal of computational Mathematics 2016年第2期34卷 113-134页

作者： Shuiping Yang Aiguo Xiao Department of Mathematics Huizhou University Guangdong 516007 China School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China

In this paper, we study the Hermite cubic spline collocation method with two parame- ters for solving a initial value problem （IVP） of nonlinear fractional differential equations with two Caputo derivatives. The convergence and nonlinear stability of the method are established. Some illustrative examples are provided to verify our theoretical results. The numerical results also indicate that the convergence order is min{4 - α, 4 - β}, where 0 〈β〈 αa 〈 1 are two parameters associated with the fractional differential equations.

关键词： Fractional differential equations Caputo derivatives Spline collocation method Convergence Stability.

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Numerical methods for fractional variational problems depending on indefinite integrals

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Journal of computational and Nonlinear Dynamics 2013年第2期8卷 021018页

作者： Wang, Dongling Xiao, Aiguo Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China

In this paper, the fractional variational integrators for fractional variational problems depending on indefinite integrals in terms of the Caputo derivative are developed. The corresponding fractional discrete Euler-Lagrange equations are derived. Some fractional variational integrators are presented based on the Grünwald-Letnikov formula. The fractional variational errors are discussed. Some numerical examples are given to illustrate these results. Copyright © 2013 by ASME.

关键词： Numerical methods

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Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative

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Journal of computational Physics 2013年 242卷 670-681页

作者： Wang, Dongling Xiao, Aiguo Yang, Wei Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China

In this paper, the Crank-Nicolson (CN) difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative is studied. The existence of this difference solution is proved by the Brouwer fixed point theorem. The stability and convergence of the CN scheme are discussed in the L2 norm. When the fractional order is two, all those results are in accord with the difference scheme developed for the classical non-fractional coupled nonlinear Schrödinger equations. Some numerical examples are also presented. © 2013 Elsevier Inc.

关键词： Nonlinear equations

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An accelerated scheme with high quality mesh based on Lloyd iteration

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Journal of Central South University 2012年第10期19卷 2797-2802页

作者：秦衡峰王艺李明富周后明 School of Mechanical Engineering Xiangtan University Key Laboratory of Computation and Simulation in Science and Engineering of Hunan Province(Xiangtan University)

High quality mesh plays an important role for finite element methods in science computation and numerical *** the mesh quality is good or not,to some extent,it determines the calculation results of the accuracy and *** from classic Lloyd iteration algorithm which is convergent slowly,a novel accelerated scheme was presented,which consists of two core parts:mesh points replacement and local edges Delaunay *** using it,almost all the equilateral triangular meshes can be generated based on centroidal Voronoi tessellation(CVT).Numerical tests show that it is significantly effective with time consuming decreasing by 40%.Compared with other two types of regular mesh generation methods,CVT mesh demonstrates that higher geometric average quality increases over 0.99.

关键词： Lloyd iteration mesh generation Delaunay triangulation high quality mesh centroidal Voronoi tessellation

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A Conservative SAV-RRK Finite Element Method for the Nonlinear Schrodinger Equation

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Advances in Applied Mathematics and Mechanics 2023年第3期15卷 583-601页

作者： Jun Yang Nianyu Yi School of Mathematics and Computational Science Xiangtan UniversityXiangtanHunan 411105China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China

*** this paper,we propose,analyze and numerically validate a conservative finite element method for the nonlinear Schrodinger equation.A scalar auxiliary variable(SAV)is introduced to reformulate the nonlinear Schrodinger equation into an equivalent system and to transform the energy into a quadratic *** use the standard continuous finite element method for the spatial discretization,and the relaxation Runge-Kutta method for the time *** mass and energy conservation laws are shown for the semi-discrete finite element scheme,and also preserved for the full-discrete scheme with suitable relaxation coefficient in the relaxation Runge-Kutta *** examples are presented to demonstrate the accuracy of the proposed method,and the conservation of mass and energy in long time simulations.

关键词： Schrodinger equation mass conservation energy conservation finite element method relaxation Runge-Kutta scalar auxiliary variable

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Dynamic coupled thermo-hydro-mechanical problem for heterogeneous deep-sea sediments under vibration of mining vehicle

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Applied Mathematics and Mechanics(English Edition) 2023年第4期44卷 603-622页

作者： Wei ZHU Xingkai MA Xinyu SHI Wenbo MA Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105Hunan ProvinceChina School of Mathematics and Computational Science Xiangtan UniversityXiangtan 411105Hunan ProvinceChina School of Mechanical Engineering and Mechanics Xiangtan UniversityXiangtan 411105Hunan ProvinceChina

Due to the influence of deep-sea environment,deep-sea sediments are usually heterogeneous,and their moduli of elasticity and density change as depth *** with the characteristics of deep-sea sediments,the thermo-hydro-mechanical coupling dynamic response model of heterogeneous saturated porous sediments can be established to study the influence of elastic modulus,density,frequency,and load amplitude changes on the *** on the Green-Lindsay generalized thermoelasticity theory and Darcy’s law,the thermo-hydro-mechanical coupled dynamic response model and governing equations of heterogeneous deep-sea sediments with nonlinear elastic modulus and density are *** analytical solutions of dimensionless vertical displacement,vertical stress,excess pore water pressure,and temperature are obtained by means of normal modal analysis,which are depicted *** results show that the changes of elastic modulus and density have few effects on vertical displacement,vertical stress,and temperature,but have great effects on excess pore water *** the mining machine vibrates,the heterogeneity of deep-sea sediments has great influence on vertical displacement,vertical stress,and excess pore water pressure,but has few effects on *** addition,the vertical displacement,vertical stress,and excess pore water pressure of heterogeneous deep-sea sediments change more *** variation trends of physical quantities for heterogeneous and homogeneous deep-sea sediments with frequency and load amplitude are basically the *** results can provide theoretical guidance for deep-sea mining engineering construction.

关键词： heterogeneous deep-sea sediment coupled thermo-hydro-mechanical Green-Lindsay generalized thermoelastic theory normal modal anlalysis dynamic re-sponse

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Neural network methods based on efficient optimization algorithms for solving impulsive differential equations

IEEE Transactions on Artificial Intelligence

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IEEE Transactions on Artificial Intelligence 2024年第3期5卷 1067-1076页

作者： Xing, Baixue Liu, Hongliang Tang, Xiao Shi, Long School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China School of Computational Science and Electronics Hunan Institute of Engineering Xiangtan411105 China

In view of the fact that impulsive differential equations have the discreteness due to the impulse phenomenon, this article proposes a single hidden layer neural networkmethod-based extreme learning machine and a physics-informed neural network method combined with learning rate attenuation strategy to solve linear impulsive differential equations and nonlinear impulsive differential equations, respectively. For the linear impulsive differential equations, first, the interval is segmented according to the impulse points, and a single hidden layer neural network model is constructed, the weight parameters of the hidden layer are randomly set, the optimal output parameters, and solution of the first segment are obtained by the extreme learning machine algorithm, then we calculate the initial value of the second segment according to the jumping equation and the remaining segments are solved in turn in the same way. Although the single hidden layer neural network method proposed can solve linear equations with high accuracy, it is not suitable for solving nonlinear equations. Therefore, we propose the physics-informed neural network combined with a learning rate attenuation strategy to solve the nonlinear impulsive differential equations, then the Adam algorithm and L-BFGS algorithm are combined to find the optimal approximate solution of each segment. Numerical examples show that the single hidden layer neural network method with Legendre polynomials as the activation function and the physics-informed neural network method combined with learning rate attenuation strategy can solve linear and nonlinear impulsive differential equations with higher accuracy. Impact Statement-It is difficult to obtain the analytical solutions of impulsive differential equations because of the existence of impulse points, and the current numerical methods are complicated and demanding. In recent years, artificial neural network methods have been widely used due to its simplicity and efficie

关键词： Approximation algorithms

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Dependence Analysis of the Solutions on the Parameters of Fractional Delay Differential Equations

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Advances in Applied Mathematics and Mechanics 2011年第5期3卷 586-597页

作者： Shuiping Yang Aiguo Xiao Xinyuan Pan School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and EngineeringXiangtan UniversityHunan 411105China Department of Mathematics Huizhou UniversityGuangdong 516007China

In this paper,we investigate the dependence of the solutions on the parameters(order,initial function,right-hand function)of fractional delay differential equations(FDDEs)with the Caputo fractional *** results includi... 详细信息

关键词： Fractional delay differential equation Caputo fractional derivative dependence

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