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

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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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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AutoFE-Pointer: Auto-weighted feature extractor based on pointer network for DNA methylation prediction

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International Journal of Biological Macromolecules 2025年 311卷 143668页

作者： Feng, Xin Xin, Ruihao Wu, Jiezhang Zheng, Jiaxin Wang, Chenyan Yu, Cuinan School of Science Jilin Institute of Chemical Technology Jilin130000 China State Key Laboratory of Inorganic Synthesis and Preparative Chemistry College of Chemistry Jilin University Changchun130012 China College of Information and Control Engineering Jilin Institute of Chemical Technology Jilin130000 China College of Computer Science and Technology Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education Jilin University Changchun130012 China Civil Aviation General Hospital Beijing100123 China School of Engineering Westlake University Zhejiang Hangzhou310030 China Hunan Bojii Life Technology Co. Ltd. Hunan Province Changsha410000 China

DNA methylation is a critical epigenetic modification that plays a central role in gene regulation, cellular differentiation, and the development of various diseases, including cancers. Aberrant methylation patterns have emerged as both biomarkers and mechanistic drivers in pathogenesis, underscoring the urgent need for precise and efficient predictive tools. Although some deep learning techniques have advanced methylation prediction, most existing models are trained independently on single-species datasets. This species-specific approach limits efficiency because each model can only handle one dataset at a time, often at the expense of predictive performance. Additionally, the state-of-the-art deep learning models tend to have enormous parameter counts and computational overhead, making them impractical for integration into local offline software applications. To overcome these challenges, we propose AutoFE-Pointer, a lightweight and novel framework that harnesses an improved softened pointer network to dynamically extract and weight features from diverse DNA sequences. AutoFE-Pointer is designed to simultaneously process 17 different benchmark datasets spanning multiple species, achieving superior performance compared to models that are trained individually on single-species data. In doing so, it not only offers state-of-the-art predictive accuracy and robust cross-species generalization but also significantly reduces computational demands, facilitating its deployment in local offline environments. This breakthrough represents a significant advancement in the field of epigenetic modeling and computational biology. © 2025

关键词： Enterprise software

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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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Particle Scale Numerical simulation on Momentum and Heat Transfer of Two Tandem Spheroids:an IB-LBM Study

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Advances in Applied Mathematics and Mechanics 2018年第6期10卷 1497-1526页

作者： Chunhai Ke Shi Shu Hao Zhang Haizhuan Yuan Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105HunanChina School of Metallurgy Northeastern UniversityShenyang 110819LiaoningChina

The cold fluid flowing over two hot spheroids placed in a tandem arrangement was numerically studied via a GPU-based immersed boundary-lattice Boltzmann method(IB-LBM)*** drag coefficient and average Nusselt number of both the two spheroids were obtained with the main influencing factors *** validate the IB-LBM model,several numerical case studies containing one and two spheres were firstly conducted to reach the good agreement with the previously reported ***,a number of simulations were further carried out which were designed by changing the particle aspect ratio(1.0≤Ar≤4.0)and inter particle distance(1.5≤ι≤7.0,whereι=L/D,D stands for the volume-equivalent sphere diameter)as well as the Reynolds number(10≤Re≤200).Their influence on the momentum and heat transfer characteristics between the solid and fluid phases was fully *** results show that,for all the considered Reynolds numbers and aspect ratios,the individual and total drag coefficients and average Nusselt number increase with the inter particle *** inter particle distance has greater influence on the drag coefficient and average Nusselt number of the trailing particle than the leading *** drag coefficient and average Nusselt number of the trailing particle are far less than the leading one under the same working *** prediction correlations for the drag coefficient and average Nusselt number of both the two spheroids were established with low *** last,the influence of the relative incidence angles between the two tandem spheroids on the momentum and heat transfer was *** is shown that the relative incidence angles play significant roles due to the change of the frontal area of the leading spheroid with these angles.

关键词： Drag coefficient average Nusselt number IB-LBM spheroids inter particle distance relative incidence angle

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

IEEE Transactions on Artificial Intelligence

引用

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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A lowest order divergence-free finite element on rectangular grids

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中国数学前沿 2011年第2期6卷 253-270页

作者： Yunqing HUANG Shangyou ZHANG Department of Mathematical Sciences University of Delaware Newark DE 19716 USA Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan 411105 China

It is shown that the conforming Q2,1;1,2-Q'1 mixed element is stable, and provides optimal order of approximation for the Stokes equations on rectangular grids. Here, Q2,1;1,2 = Q2,1 × Q1,2, and Q2,1 denotes the space of continuous piecewise-polynomials of degree 2 or less in the x direction but of degree 1 in the y direction. Q'1 is the space of discontinuous bilinear polynomials, with spurious modes filtered. To be precise, Q'1 is the divergence of the discrete velocity space Q2,1;1,2. Therefore, the resulting finite element solution for the velocity is divergence-free pointwise, when solving the Stokes equations. This element is the lowest order one in a family of divergence-free element, similar to the families of the Bernardi-Raugel element and the RaviartThomas element.

关键词： Mixed finite element Stokes divergence-free element quadrilateral element rectangular grid

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Invariant-preserving schemes with arbitrarily high-order accuracy for the two-component Camassa–Holm system via Hamiltonian boundary value methods

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Communications in Nonlinear science and Numerical simulation 2025年 150卷

作者： Qian Luo Aiguo Xiao Min Li Xiaoqiang Yan School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Hunan 411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Hunan 411105 China School of Mathematics and Physics China University of Geosciences Wuhan 430074 China National Center for Applied Mathematics in Hunan Xiangtan University Hunan 411105 China

In this paper, we present an arbitrarily high-order numerical scheme for the two-component Camassa–Holm system, ensuring the preservation of three invariants: energy and two Casimir functions. The spatial discretization is achieved using Fourier–Galerkin methods, resulting in a semi-discrete system which retains a Hamiltonian structure and approximates the invariants of the original continuous system. Subsequently, an energy-preserving Runge–Kutta method, such as Hamiltonian boundary value methods, is applied to the semi-discrete system, yielding a fully discrete scheme of arbitrarily high order. The proposed scheme is validated through numerical simulations, demonstrating its accuracy and capability in capturing different types of solutions of the two-component Camassa–Holm equation.

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OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES

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Journal of computational Mathematics 2006年第3期24卷 365-372页

作者： Yun-qing Huang Wei Li Fang Su Hunan Key Laboratory for Computation and Simulation in Science and Engineering Institute for Computational and Applied Mathematics Xiangtan University Xiangtan 4 11105 China Department of Computational Science Xiangtan University Xiangtan 4 11105 China

In this paper, we provide a theoretical method（PUFEM）, which belongs to the analysis of the partition of unity finite element family of meshfree methods. The usual error analysis only shows the order of error estimate to the same as the local approximations[12]. Using standard linear finite element base functions as partition of unity and polynomials as local approximation space, in l-d case, we derive optimal order error estimates for PUFEM interpolants. Our analysis show that the error estimate is of one order higher than the local approximations. The interpolation error estimates yield optimal error estimates for PUFEM solutions of elliptic boundary value problems.

关键词： Meshless methods Partition of unity finite element method（PUFEM） Error estimate.

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