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Finite difference and sinc-collocation approximations to a class of fractional diffusion-wave equations

Journal of Applied Mathematics 2014年第1期2014卷

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

We propose an efficient numerical method for a class of fractional diffusion-wave equations with the Caputo fractional derivative of order α. This approach is based on the finite difference in time and the global sinc collocation in space. By utilizing the collocation technique and some properties of the sinc functions, the problem is reduced to the solution of a system of linear algebraic equations at each time step. Stability and convergence of the proposed method are rigorously analyzed. The numerical solution is of 3 - α order accuracy in time and exponential rate of convergence in space. Numerical experiments demonstrate the validity of the obtained method and support the obtained theoretical results. © 2014 Zhi Mao et al.

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Topological properties and fractal analysis of a recurrence network constructed from fractional Brownian motions

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Physical Review E 2014年第3期89卷 032814-032814页

作者： Jin-Long Liu Zu-Guo Yu Vo Anh Hunan Key Laboratory for Computation and Simulation in Science and Engineering and 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 GPO Box 2434 Brisbane Q4001 Australia

Many studies have shown that we can gain additional information on time series by investigating their accompanying complex networks. In this work, we investigate the fundamental topological and fractal properties of recurrence networks constructed from fractional Brownian motions (FBMs). First, our results indicate that the constructed recurrence networks have exponential degree distributions; the average degree exponent 〈λ〉 increases first and then decreases with the increase of Hurst index H of the associated FBMs; the relationship between H and 〈λ〉 can be represented by a cubic polynomial function. We next focus on the motif rank distribution of recurrence networks, so that we can better understand networks at the local structure level. We find the interesting superfamily phenomenon, i.e., the recurrence networks with the same motif rank pattern being grouped into two superfamilies. Last, we numerically analyze the fractal and multifractal properties of recurrence networks. We find that the average fractal dimension 〈dB〉 of recurrence networks decreases with the Hurst index H of the associated FBMs, and their dependence approximately satisfies the linear formula 〈dB〉≈2−H, which means that the fractal dimension of the associated recurrence network is close to that of the graph of the FBM. Moreover, our numerical results of multifractal analysis show that the multifractality exists in these recurrence networks, and the multifractality of these networks becomes stronger at first and then weaker when the Hurst index of the associated time series becomes larger from 0.4 to 0.95. In particular, the recurrence network with the Hurst index H=0.5 possesses the strongest multifractality. In addition, the dependence relationships of the average information dimension 〈D(1)〉 and the average correlation dimension 〈D(2)〉 on the Hurst index H can also be fitted well with linear functions. Our results strongly suggest that the recurrence network inherits the basic characteristic

关键词： TOPOLOGICAL property FRACTAL analysis BROWNIAN motion CORRELATION (Statistics) RECURSIVE sequences (Mathematics) COMPUTER networks

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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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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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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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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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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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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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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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Adaptive mesh refinement and superconvergence for two-dimensional interface problems

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SIAM Journal on Scientific Computing 2014年第4期36卷 A1478-A1499页

作者： Wei, Huayi Chen, Long Huang, Yunqing Zheng, Bin Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Lab. of Intelligent Comp. and Info. Proc. Comp. Min. of Educ. School of Math. and Computational Sci. Xiangtan University Xiangtan Hunan 411105 China Department of Mathematics University of California at Irvine Irvine 92697 CA United States Northwest National Lab Richland 99352 WA United States

Adaptive mesh refinement and the Börgers algorithm are combined to generate a body-fitted mesh which can resolve the interface with fine geometric details. Standard linear finite element method based on such body-fitted meshes is applied to the elliptic interface problem and proven to be superclose to the linear interpolation of the exact solution. Based on this superconvergence result, a maximal norm error estimate of order one and half is obtained without using the discrete maximum principle. The data structure and meshing algorithms, including local refinement and coarsening, are very simple. In particular, no tree structure is needed. An efficient solver for solving the resulting linear algebraic systems is also developed and shown be robust with respect to both the problem size and the jump of the diffusion coefficients. © 2014 Society for Industrial and Applied Mathematics.

关键词： Adaptive mesh refinement Body-fitted mesh Superconvergence

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