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Crank-Nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative

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Journal of computational Physics 2013年 242卷 670-681页

作者： Wang, Dongling Xiao, Aiguo Yang, Wei Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China

In this paper, the Crank-Nicolson (CN) difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative is studied. The existence of this difference solution is proved by the Brouwer fixed point theorem. The stability and convergence of the CN scheme are discussed in the L2 norm. When the fractional order is two, all those results are in accord with the difference scheme developed for the classical non-fractional coupled nonlinear Schrödinger equations. Some numerical examples are also presented. © 2013 Elsevier Inc.

关键词： Nonlinear equations

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Numerical methods for fractional variational problems depending on indefinite integrals

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Journal of computational and Nonlinear Dynamics 2013年第2期8卷 021018页

作者： Wang, Dongling Xiao, Aiguo Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China

In this paper, the fractional variational integrators for fractional variational problems depending on indefinite integrals in terms of the Caputo derivative are developed. The corresponding fractional discrete Euler-Lagrange equations are derived. Some fractional variational integrators are presented based on the Grünwald-Letnikov formula. The fractional variational errors are discussed. Some numerical examples are given to illustrate these results. Copyright © 2013 by ASME.

关键词： Numerical methods

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Fractal and complex network analyses of protein molecular dynamics

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Physica A: Statistical Mechanics and its Applications 2014年 416卷 21-32页

作者： Yuan-Wu Zhou Jin-Long Liu Zu-Guo Yu Zhi-Qin Zhao Vo Anh Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan Hunan 411105 China Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China Civil Construction Engineering Department Guangxi University of Science and Technology Liuzhou 545006 China School of Mathematical Sciences Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia

Based on protein molecular dynamics, we investigate the fractal properties of energy, pressure and volume time series using the multifractal detrended fluctuation analysis (MF-DFA) and the topological and fractal properties of their converted horizontal visibility graphs (HVGs). The energy parameters of protein dynamics we considered are bonded potential, angle potential, dihedral potential, improper potential, kinetic energy, Van der Waals potential, electrostatic potential, total energy and potential energy. The shape of the h ( q ) curves from MF-DFA indicates that these time series are multifractal. The numerical values of the exponent h ( 2 ) of MF-DFA show that the series of total energy and potential energy are non-stationary and anti-persistent; the other time series are stationary and persistent apart from series of pressure (with H ≈ 0.5 indicating the absence of long-range correlation). The degree distributions of their converted HVGs show that these networks are exponential. The results of fractal analysis show that fractality exists in these converted HVGs. For each energy, pressure or volume parameter, it is found that the values of h ( 2 ) of MF-DFA on the time series, exponent λ of the exponential degree distribution and fractal dimension d B of their converted HVGs do not change much for different proteins (indicating some universality). We also found that after taking average over all proteins, there is a linear relationship between 〈 h ( 2 ) 〉 (from MF-DFA on time series) and 〈 d B 〉 of the converted HVGs for different energy, pressure and volume.

关键词： Protein molecular dynamics Multifractal detrended fluctuation analysis Horizontal visibility graph Fractal analysis Degree distribution

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Sinc-Chebyshev collocation method for a class of fractional diffusion-wave equations

The Scientific World Journal

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The Scientific World Journal 2014年第1期2014卷 143983页

作者： Mao, Zhi Xiao, Aiguo Yu, Zuguo Shi, Long Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China Mathematics and Information Engineering Department Tongren University Tongren Guizhou 554300 China

This paper is devoted to investigating the numerical solution for a class of fractional diffusion-wave equations with a variable coefficient where the fractional derivatives are described in the Caputo sense. The approach is based on the collocation technique where the shifted Chebyshev polynomials in time and the sinc functions in space are utilized, respectively. The problem is reduced to the solution of a system of linear algebraic equations. Through the numerical example, the procedure is tested and the efficiency of the proposed method is confirmed. © 2014 Zhi Mao et al.

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Fractal and multifractal properties of a family of fractal networks

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Journal of Statistical Mechanics: Theory and Experiment 2014年第2期2014卷

作者： Li, Bao-Gen Yu, Zu-Guo Zhou, Yu Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan HUNAN 411105 China School of Mathematical Sciences Queensland University of Technology Brisbane GPO Box 2434 Q4001 Australia

In this work, we study the fractal and multifractal properties of a family of fractal networks introduced by Gallos et al (2007 Proc. Nat. Acad. Sci. USA 104 7746). In this fractal network model, there is a parameter e which is between 0 and 1, and allows for tuning the level of fractality in the network. Here we examine the multifractal behavior of these networks, the dependence relationship of the fractal dimension and the multifractal parameters on parameter e. First, we find that the empirical fractal dimensions of these networks obtained by our program coincide with the theoretical formula given by Song et al (2006 Nature Phys. 2 275). Then from the shape of the τ(q) and D(q) curves, we find the existence of multifractality in these networks. Last, we find that there exists a linear relationship between the average information dimension D(1) and the parameter e. © 2014 IOP Publishing Ltd and SISSA Medialab srl.

关键词： network dynamics networks random graphs

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Lower Bounds for Eigenvalues of the Stokes Operator

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Advances in Applied Mathematics and Mechanics 2013年第1期5卷 1-18页

作者： Jun Hu Yunqing Huang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China

In this paper,we propose a condition that can guarantee the lower bound property of the discrete eigenvalue produced by the finite element method for the Stokes *** check and prove this condition for four nonconformin... 详细信息

关键词： Lower bound eigenvalue Stokes operator

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A directed continuous time random walk model with jump length depending on waiting time

The Scientific World Journal

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The Scientific World Journal 2014年第1期2014卷 182508页

作者： Shi, Long Yu, Zuguo Mao, Zhi Xiao, Aiguo Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education Xiangtan University Xiangtan Hunan 411105 China Institute of Mathematics and Physics Central South University of Forest and Technology Changsha Hunan 410004 China School of Mathematical Sciences Queensland University of Technology Brisbane QLD 4001 G.P.O. Box 2434 Australia

In continuum one-dimensional space, a coupled directed continuous time random walk model is proposed, where the random walker jumps toward one direction and the waiting time between jumps affects the subsequent jump. In the proposed model, the Laplace-Laplace transform of the probability density function P (x, t) of finding the walker at position x at time t is completely determined by the Laplace transform of the probability density function (t) of the waiting time. In terms of the probability density function of the waiting time in the Laplace domain, the limit distribution of the random process and the corresponding evolving equations are derived. © 2014 Long Shi et al.

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The subordinated processes controlled by a family of subordinators and corresponding Fokker-Planck type equations

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Journal of Statistical Mechanics: Theory and Experiment 2014年第12期2014卷

作者： Shi, Long Yu, Zu-Guo Huang, Hai-Lan Mao, Zhi Xiao, Ai-Guo Hunan Key Laboratory for Computation and Simulation in Science Ministry of Education Xiangtan University Xiangtan Hunan 411105 China School of Science Central South University of Forest and Technology Changsha Hunan 410004 China School of Mathematical Sciences Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia

In this work, we consider subordinated processes controlled by a family of subordinators which consist of a power function of a time variable and a negative power function of an α-stable random variable. The effect of parameters in the subordinators on the subordinated process is discussed. By suitable variable substitutions and the Laplace transform technique, the corresponding fractional Fokker-Planck-type equations are derived. We also compute their mean square displacements in a free force field. By choosing suitable ranges of parameters, the resulting subordinated processes may be subdiffusive, normal diffusive or superdiffusive. © 2014 IOP Publishing Ltd and SISSA Medialab srl.

关键词： diffusion

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Discovery of New Metastable Patterns in Diblock Copolymers

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Communications in computational Physics 2013年第7期14卷 443-460页

作者： Kai Jiang Chu Wang Yunqing Huang Pingwen Zhang LMAM and School of Mathematical Sciences Peking UniversityBeijing 100871P.R.China Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105P.R.China

The ordered patterns formed bymicrophase-separated block copolymer systems demonstrate periodic symmetry,and all periodic structures belong to one of 230 space *** on this fact,a strategy of estimating the initial values of self-consistent field theory to discover ordered patterns of block copolymers is *** particular,the initial period of the computational box is estimated by the Landau-Brazovskii model as *** planting the strategy into the whole-space discrete method,several new metastable patterns are discovered in diblock copolymers.

关键词： Metastable patterns diblock copolymers self-consistent field theory(SCFT) space group theory initial values

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Supercloseness of the Divergence-Free Finite Element Solutions on Rectangular Grids

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Communications in Mathematics and Statistics 2013年第2期1卷 143-162页

作者： Yunqing Huang Shangyou Zhang Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan UniversityXiangtan 411105China Department of Mathematical Sciences University of DelawareNewarkDE 19716USA

By the standard theory,the stable Qk+1,k−Qk,k+1/Qdck divergence-free element converges with the optimal order of approximation for the Stokes equations,but only order k for the velocity in H1-norm and the pressure in *** is due to one polynomial degree less in y direction for the first component of velocity,which is a Qk+1,k polynomial of x and *** this manuscript,we will show by supercloseness of the divergence free element that the order of convergence is truly k+1,for both velocity and *** special solutions(if the interpolation is also divergence-free),a two-order supercloseness is shown to *** tests are provided confirming the accuracy of the theory.

关键词： Mixed finite element Stokes equations Divergence-free element Quadrilateral element Rectangular grids Supercloseness Superconvergence

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