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A new and sharper bound for Legendre expansion of differentiable functions

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arXiv 2018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

In this paper, we provide a new and sharper bound for the Legendre coefficients of differentiable functions and then derive a new error bound of the truncated Legendre series in the uniform norm. The key idea of proof relies on integration by parts and a sharp Bernstein-type inequality for the Legendre polynomial. An illustrative example is provided to demonstrate the sharpness of our new results.41A25, 41A10 Copyright © 2018, The Authors. All rights reserved.

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Positive ground state solutions for fractional Laplacian system with one critical exponent and one subcritical exponent

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arXiv 2018年

作者： Zhen, Maoding He, Jinchun Xu, Haoyuan Yang, Meihua 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

In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent (formula presented), where (−∆)s is the fractional Laplacian, 0 2s, λ ∗−1 and 2∗ = N2−N2s is the Sobolev critical exponent. By using the Nehari manifold, we show that there exists a µ0 ∈ (0, 1), such that when 0 µ0, there exists a λµ,ν (formula presented) such that if λ > λµ,ν, the system has a positive ground state solution, if λ µ,ν, the system has no ground state solution. Copyright © 2018, The Authors. All rights reserved.

关键词： Laplace transforms

Compensated projected Euler method for stochastic differential equations with jumps under global monotonicity condition

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arXiv 2018年

This paper presents and analyzes the compensated projected Euler-Maruyama method for stochastic differential equations with jumps under a global monotonicity condition. Compared with existing conditions, this condition allows the jump-diffusion coefficient to be growth superlinearly. Moreover, the method is proved to be convergent with strongly order 1/2 on the discrete time level. Finally, some numerical experiments are carried out to confirm the theoretical results. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

Fast IMEX time integration of nonlinear stiff fractional differential equations

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arXiv 2019年

作者： Zhou, Yongtao Suzuki, Jorge L. Zhang, Chengjian Zayernouri, Mohsen School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Department of Mechanical Engineering Michigan State University East LansingMI48824 United States Department of Mechanical Engineering Michigan State University East LansingMI48824 United States 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 Department of Mechanical Engineering Department of Statistics and Probability Michigan State University East LansingMI48824 United States

Efficient long-time integration of nonlinear fractional differential equations is significantly challenging due to the integro-differential nature of the fractional operators. In addition, the inherent non-smoothness introduced by the inverse power-law kernels deteriorates the accuracy and efficiency of many existing numerical methods. We develop two efficient first- and second-order implicit-explicit (IMEX) methods for accurate time-integration of stiff/nonlinear fractional differential equations with fractional order α ∈ (0, 1] and prove their convergence and linear stability properties. The developed methods are based on a linear multi-step fractional Adams-Moulton method (FAMM), followed by the extrapolation of the nonlinear force terms. In order to handle the singularities nearby the initial time, we employ Lubich-like corrections to the resulting fractional operators. The obtained linear stability regions of the developed IMEX methods are larger than existing IMEX methods in the literature. Furthermore, the size of the stability regions increase with the decrease of fractional order values, which is suitable for stiff problems. We also rewrite the resulting IMEX methods in the language of nonlinear Toeplitz systems, where we employ a fast inversion scheme to achieve a computational complexity of O(N log N), where N denotes the number of time-steps. Our computational results demonstrate that the developed schemes can achieve global first- and second-order accuracy for highly-oscillatory stiff/nonlinear problems with singularities. Copyright © 2019, The Authors. All rights reserved.

关键词： Matrix algebra

Third-order discrete unified gas kinetic scheme for continuum and rarefied flows: Low-speed isothermal case

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Physical Review E 2018年第2期97卷 023306-023306页

作者： Chen Wu Baochang Shi Chang Shu Zhen Chen State Key Laboratory of Coal Combustion Huazhong University of Science and Technology Wuhan 430074 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China Department of Mechanical Engineering National University of Singapore 10 Kent Ridge Crescent Singapore 119260 Singapore

An efficient third-order discrete unified gas kinetic scheme (DUGKS) is presented in this paper for simulating continuum and rarefied flows. By employing a two-stage time-stepping scheme and the high-order DUGKS flux reconstruction strategy, third order of accuracy in both time and space can be achieved in the present method. It is also analytically proven that the second-order DUGKS is a special case of the present method. Compared with the high-order lattice Boltzmann equation-based methods, the present method is capable to deal with the rarefied flows by adopting the Newton-Cotes quadrature to approximate the integrals of moments. Instead of being constrained by the second order (or lower order) of accuracy in the time-splitting scheme as in the conventional high-order Runge-Kutta-based kinetic methods, the present method solves the original Boltzmann equation, which overcomes the limitation in time accuracy. Typical benchmark tests are carried out for comprehensive evaluation of the present method. It is observed in the tests that the present method is advantageous over the original DUGKS in accuracy and capturing delicate flow structures. Moreover, the efficiency of the present third-order method is also shown in simulating rarefied flows.

关键词： Boltzmann theory

Lattice Boltzmann model for high-order nonlinear partial differential equations

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Physical Review E 2018年第1期97卷 013304-013304页

作者： Zhenhua Chai Nanzhong He Zhaoli Guo Baochang Shi School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China State Key Laboratory of Coal Combustion Huazhong University of Science and Technology Wuhan 430074 China School of Mathematics and Computer Wuhan Textile University Wuhan 430200 China

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ+∑k=1mαk∂xkΠk(ϕ)=0 (1≤k≤m≤6), αk are constant coefficients, Πk(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009); H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)] for high-order nonlinear partial differential equations.

关键词： Lattice-Boltzmann methods

The Onsager-Machlup Function as Lagrangian for the Most Probable Path of a Jump-Diffusion Process

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arXiv 2018年

作者： Chao, Ying Duan, Jinqiao School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

This work is devoted to deriving the Onsager-Machlup function for a class of stochastic dynamical systems under (non-Gaussian) Lévy noise as well as (Gaussian) Brownian noise, and examining the corresponding most probable paths. This Onsager-Machlup function is the Lagrangian giving the most probable path connecting metastable states for jump-diffusion processes. This is done by applying the Girsanov transformation for measures induced by jump-diffusion processes. Moreover, we have found this Lagrangian function is consistent with the result in the special case of diffusion processes. Finally, we apply this new Onsager-Machlup function to investigate dynamical behaviors analytically and numerically in several examples. These include the transitions from one metastable state to another metastable state in a double-well system, with numerical experiments illustrating most probable transition paths for various noise parameters. Copyright © 2018, The Authors. All rights reserved.

关键词： Lagrange multipliers

A parameter estimator based on Smoluchowski-Kramers approximation 1

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arXiv 2018年

作者： He, Ziying Duan, Jinqiao Cheng, Xiujun Center for Mathematical Sciences School of Mathematics and Statistics Hubei Key Laboratory for Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China Department of Applied Mathematics Illinois Institute of Technology ChicagoIL60616 United States

We devise a simplified parameter estimator for a second order stochastic differential equation by a first order system based on the Smoluchowski-Kramers approximation. We establish the consistency of the estimator by using Γ-convergence theory. We further illustrate our estimation method by an experimentally studied movement model of a colloidal particle immersed in water under conservative force and constant diffusion. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

A remark on the central model method for the weak Palis conjecture of higher dimensional singular flows

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arXiv 2018年

作者： Xiao, Qianying Zhang, Yiwei School of Mathematical Sciences Peking University Beijing100871 China School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing HuaZhong University of Science and Technology 1037 Luoyu Road Wuhan430074 China

For a generic vector field robustly without horseshoes, and an aperiodic chain recurrent class with singularities whose saddle values have different signs, the extended rescaled Poincaré map is associated with a central model. We estimate such central model and show it must have chain recurrent cen-tral segments over the singularities. This obstructs the application of central model to create horseshoes, and indicates that, differing from C1 diffeomor-phisms, solo using central model method is insufficient as a strategy to prove weak Palis conjecture for higher dimensional ( 4) singular flows. Our compu-tation is actually based on simplified way of addressing blowup construction. As a byproduct, we are applicable to directly compute the extended rescaled Poincaré map upto second order derivatives, which we believe has its inde-pendent interests. Copyright © 2018, The Authors. All rights reserved.

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