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VANISHING VISCOSITY LIMIT FOR THE 3D INCOMPRESSIBLE MICROPOLAR EQUATIONS IN A BOUNDED DOMAIN

Acta Mathematica Scientia 2023年第2期43卷 959-974页

作者：储洋洋肖跃龙 Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105China

In this paper,we investigate the vanishing viscosity limit of the 3D incompressible micropolar equations in bounded domains with boundary *** is shown that there exist global weak solutions of the micropolar equations in a general bounded smooth *** particular,we establish the uniform estimate of the strong solutions for when the boundary is ***,we obtain the rate of convergence of viscosity solutions to the inviscid solutions as the viscosities tend to zero(i.e.,(ε,χ,γ,κ)→0).

关键词： incompressible micropolar equations initial-and boundary-valuc problcm van-ishingviscositylimit

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The Convergence of Euler-Maruyama Method of Nonlinear Variable-Order Fractional Stochastic Differential Equations

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Advances in Applied Mathematics and Mechanics 2023年第4期15卷 852-879页

作者： Shanshan Xu Lin Wang Wenqiang Wang School of Mathematics and Computational Science&Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtanHunan 411105China

In this paper,we first prove the existence and uniqueness theorem of the solution of nonlinear variable-order fractional stochastic differential equations(VFSDEs).We futher constructe the Euler-Maruyama method to solve the equations and prove the convergence in mean and the strong convergence of the *** particular,when the fractional order is no longer varying,the conclusions obtained are consistent with the relevant conclusions in the existing ***,the numerical experiments at the end of the article verify the correctness of the theoretical results obtained.

关键词： Variable-order Caputo fractional derivative Stochastic differential equations Euler-Maruyama method convergence multiplicative noise

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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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GENERALIZED JACOBI SPECTRAL GALERKIN METHOD FOR FRACTIONAL-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

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Journal of computational Mathematics 2024年第2期42卷 355-371页

作者： Yanping Chen Zhenrong Chen Yunqing Huang School of Mathematical Sciences South China Normal UniversityGuangzhou 510631China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China

For fractional Volterra integro-differential equations(FVIDEs)with weakly singular kernels,this paper proposes a generalized Jacobi spectral Galerkin *** basis functions for the provided method are selected generalized Jacobi functions(GJFs),which can be utilized as natural basis functions of spectral methods for weakly singular FVIDEs when appropriately *** developed method's spectral rate of convergence is determined using the L^(∞)-norm and the weighted L^(2)-*** results indicate the usefulness of the proposed method.

关键词： Generalized Jacobi spectral Galerkin method Fractional-order Volterra integ-ro-differential equations Weakly singular kernels Convergence analysis

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RECONSTRUCTION-BASED A POSTERIORI ERROR ESTIMATES FOR THE L1 METHOD FOR TIME FRACTIONAL PARABOLIC PROBLEMS

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Journal of computational Mathematics 2025年第2期43卷 345-368页

作者： Jiliang Cao Aiguo Xiao Wansheng Wang Department of Mathematics Shanghai Normal UniversityShanghai 200234China Hunan Key Laboratory for Computation and Simulation in Science and Engineering&Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan UniversityXiangtan 411105China

In this paper,we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations,whose solutions have generally the initial *** derive optimal order a posteriori error estimates,the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are *** using these continuous,piecewise time reconstructions,the upper and lower error bounds depending only on the discretization parameters and the data of the problems are *** numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results,with the convergence ofαorder for the nonsmooth case on a uniform *** recover the optimal convergence order 2-αon a nonuniform mesh,we further develop a time adaptive algorithm by means of barrier function recently *** numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.

关键词： Time fractional parabolic differential equations A posteriori error estimates Ll method Fractional integral reconstruction Quadratic reconstruction

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Optimal convergence analysis of an energy dissipation property virtual element method for the nonlinear Benjamin-Bona-Mahony-Burgers equation

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Computers and Mathematics with Applications 2025年 192卷 37-53页

作者： Chen, Yanping Liu, Wanxiang Qin, Fangfang Liang, Qin School of Science Nanjing University of Posts and Telecommunications Nanjing210023 China Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational Science Xiangtan University Xiangtan411105 China

A novel arbitrary high-order energy-stable fully discrete schemes are proposed for the nonlinear Benjamin-Bona-Mahony-Burgers equation based on linearized Crank-Nicolson scheme in time and the virtual element discretization in space. Two skew-symmetric discrete forms are introduced to preserve energy dissipation of the numerical scheme. Furthermore, by utilizing the L2 projection to approximate the nonlinear term and estimating the error of the discrete bilinear forms carefully, the optimal error estimate of the numerical scheme in the H1-norm is obtained. Finally, several numerical examples on various mesh types are provided to demonstrate the energy stability, optimal convergence and high efficiency of the method. © 2025 Elsevier Ltd

关键词： Convergence of numerical methods

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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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THE A POSTERIORI ERROR ESTIMATOR OF SDG METHOD FOR VARIABLE COEFFICIENTS TIME-HARMONIC MAXWELL'S EQUATIONS

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Journal of computational Mathematics 2023年第2期41卷 263-286页

作者： Wei Yang Xin Liu Bin He Yunqing Huang Hunan Key Laboratory for Computation and Simulation in Science and Engineering School of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105China Department of Mathematics Lanzhou Jiaotong UniversityLanzhou 730070China

In this paper,we study the a posteriori error estimator of SDG method for variable coefficients time-harmonic Maxwell's *** propose two a posteriori error estimators,one is the recovery-type estimator,and the other is the residual-type *** first propose the curl-recovery method for the staggered discontinuous Galerkin method(SDGM),and based on the super-convergence result of the postprocessed solution,an asymptotically exact error estimator is *** residual-type a posteriori error estimator is also proposed,and it's reliability and effectiveness are proved for variable coefficients time-harmonic Maxwell's *** efficiency and robustness of the proposed estimators is demonstrated by the numerical experiments.

关键词： Maxwell’s equations A posteriori error estimation Staggered discontinuous Galerkin

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Parallel finite element computation of incompressible magnetohydrodynamics based on three iterations

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Applied Mathematics and Mechanics(English Edition) 2022年第1期43卷 141-154页

作者： Qili TANG Yunqing HUANG Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtanHunan Province411105China

Based on local algorithms,some parallel finite element(FE)iterative methods for stationary incompressible magnetohydrodynamics(MHD)are *** approaches are on account of two-grid skill include two major phases:find the FE solution by solving the nonlinear system on a globally coarse mesh to seize the low frequency component of the solution,and then locally solve linearized residual subproblems by one of three iterations(Stokes-type,Newton,and Oseen-type)on subdomains with fine grid in parallel to approximate the high frequency *** error estimates with regard to two mesh sizes and iterative steps of the proposed algorithms are *** numerical examples are implemented to verify the algorithm.

关键词： local and parallel algorithm finite element(FE)method iteration stationary incompressible magnetohydrodynamics(MHD)

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