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SHARP POINTWISE-IN-TIME ERROR ESTIMATE OF L1 SCHEME FOR NONLINEAR SUBDIFFUSION EQUATIONS

Journal of Computational Mathematics 2024年第3期42卷 662-678页

作者： Dongfang Li Hongyu Qin Jiwei Zhang 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 School of Mathematics and Statistics Wuhan UniversityWuhan 430072China

An essential feature of the subdiffusion equations with theα-order time fractional derivative is the weak singularity at the initial *** weak regularity of the solution is usually characterized by a regularity parameterσ∈(0,1)∪(1,2).Under this general regularity assumption,we present a rigorous analysis for the truncation errors and develop a new tool to obtain the stability results,i.e.,a refined discrete fractional-type Grönwall inequality(DFGI).After that,we obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion *** present results fill the gap on some interesting convergence results of L1 scheme onσ∈(0,α)∪(α,1)∪(1,2].Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.

关键词： Sharp pointwise-in-time error estimate Ll scheme Nonlinear subdiffusion equations Non-smooth solutions

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Boundary Control of Coupled Wave Systems with Spatially-Varying Coefficients

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Journal of Systems Science & Complexity 2022年第4期35卷 1310-1329页

作者： FENG Xiaodan ZHANG Zhifei School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan430074China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan430074China

This paper considers the stabilization of the coupled wave systems with spatially-varying *** authors design a state feedback controller by backstepping *** contrast to the previous work in the literature,the kernel equations become more complicated and the main difficulty lies in proving the existence and uniqueness of the solution to the kernel ***,using the backstepping approach,the authors verify the kernel equations,which is a system of coupled hyperbolic equations with spatially-varying ***,the existence and uniqueness of the kernel matrices is ***,the authors use a Lyapunov function to get the exponential stabilization of the closed-loop system.A numerical example is presented to illustrate the effectiveness of the proposed controller.

关键词： Backstepping boundary control coupled wave systems exponential stabilization spatially-varying coefficients

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Two-step Runge–Kutta methods for Volterra integro-differential equations

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International Journal of Computer Mathematics 2024年第1期101卷 37-55页

作者： Wen, Jiao Huang, Chengming Guan, Hongbo College of Mathematics and Information Science Zhengzhou University of Light Industry Zhengzhou China 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

In this paper, we investigate two-step Runge–Kutta methods to solve Volterra integro-differential equations. Two-step Runge–Kutta methods increase the order of convergence in comparing the classical Runge–Kutta method without extra computational cost. First, the local order conditions and convergence theorem are derived. Then, stability properties of two-step Runge–Kutta methods corresponding to the basic and convolution test equations are analysed. Furthermore, one-stage method with order four and two-stage method with order six are constructed and we plot the stability regions. Numerical examples are presented to confirm the theoretical analyses. © 2024 Informa UK Limited, trading as Taylor & Francis Group.

关键词： Stability

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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

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Phase-field-based lattice Boltzmann method for containerless freezing

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Physical Review E 2024年第3期110卷 035301页

作者： Jiangxu Huang Lei Wang Zhenhua Chai Baochang Shi School of Mathematics and Statistics School of Mathematics and Physics Institute of Interdisciplinary Research for Mathematics and Applied Science Hubei Key Laboratory of Engineering Modeling and Scientific Computing

In this paper we first propose a phase-field model for the containerless freezing problems, in which the volume expansion or shrinkage of the liquid caused by the density change during the phase change process is considered by adding a mass source term to the continuum equation. Then a phase-field-based lattice Boltzmann (LB) method is further developed to simulate solid-liquid phase change phenomena in multiphase systems. We test the developed LB method by the problem of conduction-induced freezing in a semi-infinite space, the three-phase Stefan problem, and the droplet solidification on a cold surface, and the numerical results are in agreement with the analytical and experimental solutions. In addition, the LB method is also used to study the rising bubbles with solidification. The results of the present method not only accurately capture the effect of bubbles on the solidification process, but also are in agreement with the previous work. Finally, a parametric study is carried out to examine the influences of some physical parameters on the sessile droplet solidification, and it is found that the time of droplet solidification increases with the increase of droplet volume and contact angle.

关键词： Drops & bubbles Liquid-solid phase transition

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DISCRETE ENERGY ANALYSIS OF THE THIRD-ORDER VARIABLE-STEP BDF TIME-STEPPING FOR DIFFUSION EQUATIONS

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Journal of Computational Mathematics 2023年第2期41卷 325-344页

作者： Hong-lin Liao Tao Tang Tao Zhou Department of Mathematics Nanjing University of Aeronautics and AstronauticsNanjing 211106China Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles(NUAA) MIITNanjing 211106China Division of Science and Technology BNU-HKBU United International CollegeZhuhai 519087China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China

This is one of our series works on discrete energy analysis of the variable-step BDF *** this part,we present stability and convergence analysis of the third-order BDF(BDF3)schemes with variable steps for linear diffusion equations,see,e.g.,[SIAM ***.,58:2294-2314]and[***.,90:1207-1226]for our previous works on the BDF2 *** this aim,we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877,by which we can establish a discrete energy dissipation ***-robust stability and convergence analysis in the L^(2) norm are then *** the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step *** also present numerical tests to support our theoretical results.

关键词： Diffusion equations Variable-step third-order BDF scheme Discrete gradient structure Discrete orthogonal convolution kernels Stability and convergence

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REQUIRED NUMBER OF ITERATIONS FOR SPARSE SIGNAL RECOVERY VIA ORTHOGONAL LEAST SQUARES

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Journal of Computational Mathematics 2023年第1期41卷 1-17页

作者： Haifeng Li Jing Zhang Jinming Wen Dongfang Li Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control College of Mathematics and Information ScienceHenan Normal UniversityXinxiang 453007China College of Information Science and Technology Jinan UniversityGuangzhou 510632China 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

In countless applications,we need to reconstruct a K-sparse signal x∈R n from noisy measurements y=Φx+v,whereΦ∈R^(m×n)is a sensing matrix and v∈R m is a noise *** least squares(OLS),which selects at each step the column that results in the most significant decrease in the residual power,is one of the most popular sparse recovery *** this paper,we investigate the number of iterations required for recovering x with the OLS *** show that OLS provides a stable reconstruction of all K-sparse signals x in[2.8K]iterations provided thatΦsatisfies the restricted isometry property(RIP).Our result provides a better recovery bound and fewer number of required iterations than those proposed by Foucart in 2013.

关键词： Sparse signal recovery Orthogonal least squares(OLS) Restricted isometry property(RIP)

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GLOBAL STRONG SOLUTION FOR THE STOCHASTIC TAMED CHEMOTAXIS-NAVIER-STOKES SYSTEM IN R3

arXiv

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arXiv 2024年

作者： Xu, Fan Zhang, Lei Liu, Bin School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

In this work, we consider the 3D Cauchy problem for a coupled system arising from the biomathematics, which consists of a chemotaxis model with cubic logistic source and the stochastic tamed Navier-Stokes equations (STCNS, for short). Our main goal is to establish the existence and uniqueness of global strong solution (strong in both the probabilistic and PDE senses) for the 3D STCNS system with large initial data. To achieve this, we first introduce a triple approximation scheme by virtue of the Friedrichs mollifier, frequency truncation operators and cut-off functions, which makes it possible for constructing sufficiently smooth approximation solutions, and facilitating the effective use of stochastic entropy-energy estimates. Then, building on the entropy-energy estimates, we further derive some higher-order energy estimates, which together with the stochastic compactness method permits us to construct the strong solution for the STCNS system. Copyright © 2024, The Authors. All rights reserved.

关键词： Navier Stokes equations

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NEW GLOBAL CARLEMAN ESTIMATES AND NULL CONTROLLABILITY FOR FORWARD/BACKWARD SEMI-LINEAR PARABOLIC SPDES

arXiv

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arXiv 2024年

作者： Zhang, Lei Xu, Fan Liu, Bin School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, we study the null controllability for some linear and semi-linear parabolic SPDEs involving both the state and the gradient of the state. To start with, an improved global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with general random coefficients and L2-valued source terms is derived. Based on this, we further develop a new global Carleman estimate for linear forward (resp. backward) parabolic SPDEs with H-1-valued source terms, which enables us to deal with the global null controllability for linear backward (resp. forward) parabolic SPDEs with gradient terms. As byproduct, a special energy-type estimate for the controlled system that explicitly depends on the parameters λ, μ and the weighted function θ is obtained. Furthermore, by employing a fixed-point argument, we extend the previous linear controllability results to some semi-linear backward (resp. forward) parabolic SPDEs. Copyright © 2024, The Authors. All rights reserved.

关键词： Controllability

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GLOBAL WELL-POSEDNESS FOR THE LANDAU-LIFSHITZ-BARYAKHTAR EQUATION IN 3

arXiv

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arXiv 2024年

作者： Xu, Fan Liu, Bin School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China

This paper establishes the global well-posedness of the Landau-Lifshitz-Baryakhtar (LLBar) equation in the whole space 3. The study first demonstrates the existence and uniqueness of global strong solutions using the weak compactness approach. Furthermore, the existence and uniqueness of classical solutions, as well as arbitrary smooth solutions, are derived through a bootstrap argument. The proofs for the existence of these three types of global solutions are based on Friedrichs mollifier approximation and energy estimates, with the structure of the LLBar equation playing a crucial role in the derivation of the results. Copyright © 2024, The Authors. All rights reserved.

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