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Adapted Runge-Kutta Methods for Nonlinear First-Order Delay BVPs with Time-variable Delay

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Acta Mathematicae Applicatae Sinica 2025年第2期41卷 400-413页

作者： Cheng-jian ZHANG Yang WANG Hao HAN School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China

This paper deals with numerical solutions for nonlinear first-order boundary value problems(BVPs) with time-variable delay. For solving this kind of delay BVPs, by combining Runge-Kutta methods with Lagrange interpolation, a class of adapted Runge-Kutta(ARK) methods are developed. Under the suitable conditions, it is proved that ARK methods are convergent of order min{p, μ+ν +1}, where p is the consistency order of ARK methods and μ, ν are two given parameters in Lagrange interpolation. Moreover, a global stability criterion is derived for ARK methods. With some numerical experiments, the computational accuracy and global stability of ARK methods are further testified.

关键词： delay boundary value problems adapted Runge-Kutta method Lagrange interpolation error analysis global stability

A DECOUPLED,LINEARLY IMPLICIT AND UNCONDITIONALLY ENERGY STABLE SCHEME FOR THE COUPLED CAHN-HILLIARD SYSTEMS

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Journal of Computational Mathematics 2025年第3期43卷 708-730页

作者： Dan Zhao Dongfang Li Yanbin Tang Jinming Wen School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China College of Information Science and Technology Jinan UniversityGuangzhou 510632China

We present a decoupled,linearly implicit numerical scheme with energy stability and mass conservation for solving the coupled Cahn-Hilliard *** time-discretization is done by leap-frog method with the scalar auxiliary variable(SAV)*** only needs to solve three linear equations at each time step,where each unknown variable can be solved *** is shown that the semi-discrete scheme has second-order accuracy in the temporal *** convergence results are proved by a rigorous analysis of the boundedness of the numerical solution and the error estimates at different *** examples are presented to further confirm the validity of the methods.

关键词： Coupled Cahn-Hilliard system Leap-frog method Scalar auxiliary variable Error estimate

ERROR ANALYSIS OF FRACTIONAL COLLOCATION METHODS FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH NONCOMPACT OPERATORS

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Journal of Computational Mathematics 2025年第3期43卷 690-707页

作者： Zheng Ma Chengming Huang Anatoly A.Alikhanov School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China North-Caucasus Center for Mathematical Research North-Caucasus Federal UniversityStavropol 355017Russia North-Eastern Federal University Yakutsk 677000Russia

This paper is concerned with the numerical solution of Volterra integro-differential equations with noncompact *** focus is on the problems with weakly singular *** handle the initial weak singularity of the solution,a fractional collocation method is applied.A rigorous hp-version error analysis of the numerical method under a weighted H1-norm is carried *** result shows that the method can achieve high order convergence for such *** experiments are also presented to confirm the effectiveness of the proposed method.

关键词： Volterra integro-differential equation Noncompact operator Nonsmooth solution Collocation method Fractional polynomial hp-version error analysis

A HIGH ORDER SCHEME FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH THE CAPUTO-HADAMARD DERIVATIVE

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Journal of Computational Mathematics 2025年第3期43卷 615-640页

作者： Xingyang Ye Junying Cao Chuanju Xu School of Science Jimei UniversityXiamen 361021China School of Data Science and Information Engineering Guizhou Minzu UniversityGuiyang 550025China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005China

In this paper,we consider numerical solutions of the fractional diffusion equation with theαorder time fractional derivative defined in the Caputo-Hadamard sense.A high order time-stepping scheme is constructed,analyzed,and numerically *** contribution of the paper is twofold:1)regularity of the solution to the underlying equation is investigated,2)a rigorous stability and convergence analysis for the proposed scheme is performed,which shows that the proposed scheme is 3+αorder *** numerical examples are provided to verify the theoretical statement.

关键词： Caputo-Hadamard derivative Fractional differential equations High order scheme Stability and convergence analysis

On the deterioration of convergence rate of spectral differentiations for functions with singularities

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arXiv 2025年

作者： Wang, Haiyong School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

Spectral differentiations are basic ingredients of spectral methods. In this work, we analyze the pointwise rate of convergence of spectral differentiations for functions containing singularities and show that the deteriorations of the convergence rate at the endpoints, singularities and other points in the smooth region exhibit different patterns. As the order of differentiation increases by one, we show for functions with an algebraic singularity that the convergence rate of spectral differentiation by Jacobi projection deteriorates two orders at both endpoints and only one order at each point in the smooth region. The situation at the singularity is more complicated and the convergence rate either deteriorates two orders or does not deteriorate, depending on the parity of the order of differentiation, when the singularity locates in the interior of the interval and deteriorates two orders when the singularity locates at the endpoint. Extensions to some related problems, such as the spectral differentiation using Chebyshev interpolation, are also discussed. Our findings justify the error localization property of Jacobi approximation and differentiation and provide some new insight into the convergence behavior of Jacobi spectral *** Codes 65D25, 41A10, 41A25 Copyright © 2025, The Authors. All rights reserved.

关键词： Chebyshev approximation

Incomplete crossing and semi-topological horseshoes

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arXiv 2025年

作者： Cheng, Junfeng Yang, Xiao-Song School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

This paper enriches the topological horseshoe theory using finite subshift theory in symbolic dynamical systems, and develops an elementary framework addressing incomplete crossing and semi-horseshoes. Two illustrative examples are provided: one from the perturbed Duffing system and another from a polynomial system proposed by Chen, demonstrating the prevalence of semi-horseshoes in chaotic systems. Moreover, the semi-topological horseshoe theory enhances the detection of chaos and improves the accuracy of topological entropy estimation. Copyright © 2025, The Authors. All rights reserved.

关键词： Topology

ENERGY DECAY OF NONLOCAL VISCOELASTIC EQUATIONS WITH NONLINEAR DAMPING AND POLYNOMIAL NONLINEARITY

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arXiv 2025年

作者： Peng, Qingqing Liu, Yikan School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Mathematics Kyoto University Kitashirakawa-Oiwakecho Sakyo-ku Kyoto606-8502 Japan

This paper is concerned with the energy decay of a viscoelastic variable coefficient wave equation with nonlocality in time as well as nonlinear damping and polynomial nonlinear terms. Using the Lyapunov method, we establish a polynomial energy decay for the solution under relatively weak assumptions regarding the kernel of the nonlocal term. More specifically, we improve the decay rate of the energy by additionally imposing a certain convexity assumption on the kernel. Several examples are provided to confirm such improvements to faster polynomial or even exponential *** Codes 35L20, 35L70, 35B40 Copyright © 2025, The Authors. All rights reserved.

关键词： Damping

EXPONENTIAL STABILITY FOR AN INFINITE MEMORY WAVE EQUATION WITH FRICTIONAL DAMPING AND LOGARITHMIC NONLINEAR TERMS

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arXiv 2025年

This article is concerned with the energy decay of an infinite memory wave equation with a logarithmic nonlinear term and a frictional damping term. The problem is formulated in a bounded domain in d (d ≥ 3) with a smooth boundary, on which we prescribe a mixed boundary condition of the Dirichlet and the acoustic types. We establish an exponential decay result for the energy with a general material density ρ(x) under certain assumptions on the involved coefficients. The proof is based on a contradiction argument, the multiplier method and some microlocal analysis techniques. In addition, if ρ(x) takes a special form, our result even holds without the damping effect, that is, the infinite memory effect alone is strong enough to guarantee the exponential stability of the *** Codes 35L20, 35L70, 35B40 Copyright © 2025, The Authors. All rights reserved.

关键词： Damping

Clustering-based Low Rank Approximation Method

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arXiv 2025年

作者： Zhu, Yujun Zhu, Jie Arshad, Hizba Wang, Zhongming Ming, Ju School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan China Department of Mathematics and Statistics Florida International University MiamiFL United States

We propose a clustering-based generalized low rank approximation method, which takes advantage of appealing features from both the generalized low rank approximation of matrices (GLRAM) and cluster analysis. It exploits a more general form of clustering generators and similarity metrics so that it is more suitable for matrix-structured data relative to conventional partitioning methods. In our approach, we first pre-classify the initial matrix collection into several small subset clusters and then sequentially compress the matrices within the clusters. This strategy enhances the numerical precision of the low-rank approximation. In essence, we combine the ideas of GLRAM and clustering into a hybrid algorithm for dimensionality reduction. The proposed algorithm can be viewed as the generalization of both techniques. Theoretical analysis and numerical experiments are established to validate the feasibility and effectiveness of the proposed algorithm. © 2025, CC BY-NC-ND.

关键词： Matrix algebra