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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

Generalized lattice boltzmann method: modeling, analysis, and elements

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arXiv 2019年

作者： Shi, Baochang Chai, Zhenhua 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 first present a unified framework for the modelling of generalized lattice Boltzmann method (GLBM). We then conduct a comparison of the four popular analysis methods (Chapman-Enskog analysis, Maxwell iteration, direct Taylor expansion and recurrence equations approaches) that have been used to obtain the macroscopic Navier-Stokes equations and nonlinear convection-diffusion equations from the GLBM, and show that from mathematical point of view, these four analysis methods are equivalent to each other. Finally, we give some elements that are needed in the implementation of the GLBM, and also find that some available LB models can be obtained from this GLBM. Copyright © 2019, The Authors. All rights reserved.

关键词： Nonlinear equations

CENTRAL LIMIT THEOREM FOR TEMPORAL AVERAGE OF BACKWARD EULER–MARUYAMA METHOD

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arXiv 2023年

作者： Diancong, J.I.N. 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 work focuses on the temporal average of the backward Euler–Maruyama (BEM) method, which is used to approximate the ergodic limit of stochastic ordinary differential equations with super-linearly growing drift coefficients. We give the central limit theorem (CLT) of the temporal average, which characterizes the asymptotics in distribution of the temporal average. When the deviation order is smaller than the optimal strong order, we directly derive the CLT of the temporal average through that of original equations and the uniform strong order of the BEM method. For the case that the deviation order equals to the optimal strong order, the CLT is established via the Poisson equation associated with the generator of original equations. Numerical experiments are performed to illustrate the theoretical results. Copyright © 2023, The Authors. All rights reserved.

关键词： Poisson equation

Complex generalized Gauss-Radau quadrature rules for Hankel transforms of integer order

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arXiv 2024年

作者： Wang, Haiyong Wu, Menghan 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

Complex Gaussian quadrature rules for oscillatory integral transforms have the advantage that they can achieve optimal asymptotic order. However, their existence for Hankel transform can only be guaranteed when the order of the transform belongs to [0, 1/2]. In this paper we consider the construction of generalized Gauss-Radau quadrature rules for Hankel transform. We show that, if adding certain value and derivative information at the left endpoint, then complex generalized Gauss-Radau quadrature rules for Hankel transform of integer order can be constructed with theoretical guarantees. Orthogonal polynomials that are closely related to such quadrature rules are investigated and their existence for even degrees is proved. Numerical experiments are presented to confirm our *** Codes 65R10, 65D32 © 2024, CC BY-NC-ND.

关键词： Integral equations

Multiple-relaxation-time lattice Boltzmann model-based four-level finite-difference scheme for one-dimensional diffusion equations

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Physical Review E 2021年第1期104卷 015312-015312页

作者： Yuxin Lin Ning Hong Baochang Shi Zhenhua Chai School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan 430074 China School of General Education Wuchang University of Technology Wuhan 430223 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074 China

In this paper, we first present a multiple-relaxation-time lattice Boltzmann (MRT-LB) model for one-dimensional diffusion equation where the D1Q3 (three discrete velocities in one-dimensional space) lattice structure is considered. Then through the theoretical analysis, we derive an explicit four-level finite-difference scheme from this MRT-LB model. The results show that the four-level finite-difference scheme is unconditionally stable, and through adjusting the weight coefficient ω0 and the relaxation parameters s1 and s2 corresponding to the first and second moments, it can also have a sixth-order accuracy in space. Finally, we also test the four-level finite-difference scheme through some numerical simulations and find that the numerical results are consistent with our theoretical analysis.

关键词： Lattice-Boltzmann methods

Analysis of error localization of Chebyshev spectral approximations

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arXiv 2021年

Chebyshev spectral methods are widely used in numerical computations. When the underlying function has a singularity, it has been observed by L. N. Trefethen in 2011 that its Chebyshev interpolants exhibit an error localization property, that is, their errors in a neighborhood of the singularity are obviously larger than elsewhere. In this paper, we first present a pointwise error analysis for Chebyshev projections of functions with a singularity and prove that the rate of convergence of Chebyshev projections of degree n at each point away from the singularity is one power of n faster than that of at the singularity. This gives a rigorous justification for the error localization of Chebyshev projections. We then extend the framework of our analysis to Chebyshev interpolants, Chebyshev spectral differentiations and Legendre projections and justify their error localization using similar arguments. As a result, we find that Chebyshev spectral differentiations converge faster than their best counterparts except in a neighborhood of the singularity and, in the particular case where the singularity is located in the interior of interval, they converge even faster than their best counterparts in the maximum *** Codes 41A10, 41A25, 41A50 © 2021, CC BY-NC-SA.

关键词： Numerical methods

Convergence analysis of Laguerre approximations for analytic functions

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arXiv 2023年

Laguerre spectral approximations play an important role in the development of efficient algorithms for problems in unbounded domains. In this paper, we present a comprehensive convergence rate analysis of Laguerre spectral approximations for analytic functions. By exploiting contour integral techniques from complex analysis, we prove that Laguerre projection and interpolation methods of degree n converge at the root-exponential rate O(exp(−2ρ√n)) with ρ > 0 when the underlying function is analytic inside and on a parabola with focus at the origin and vertex at z = −ρ2. As far as we know, this is the first rigorous proof of root-exponential convergence of Laguerre approximations for analytic functions. Several important applications of our analysis are also discussed, including Laguerre spectral differentiations, Gauss-Laguerre quadrature rules, the scaling factor and the Weeks method for the inversion of Laplace transform, and some sharp convergence rate estimates are derived. Numerical experiments are presented to verify the theoretical *** Codes 41A25, 41A10 © 2023, CC BY-NC-SA.

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

作者： Li, Min Huang, Chengming 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 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

Carleman and observability estimates for stochastic beam equation

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arXiv 2017年

In this paper, we establish a weight identity for stochastic beam equation by means of the multiplier method. Based on this identity, we first establish the global Carleman estimate for the special system with zero initial value and end value, then a revised Carleman estimate for stochastic beam equation is established through a cutoff technique. Finally, we use the revised Carleman estimate to get the required boundary observability estimate. Copyright © 2017, The Authors. All rights reserved.

关键词： Stochastic systems