Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation

Fractional differential equations have recently been applied in various areas of engineering, science, finance, applied mathematics, bio-engineering and others. However, many researchers remain unaware of this field. In this paper, an efficient numerical method for solving the fractional Advection-dispersion equation (ADE) is considered. The fractional derivative is described in the Caputo sense. The method is based on Chebyshev approximations. The properties of Chebyshev polynomials are used to reduce ADE to a system of ordinary differential equations, which are solved using the finite difference method (FDM). Moreover, the convergence analysis and an upper bound of the error for the derived formula are given. Numerical solutions of ADE are presented and the results are compared with the exact solution.

Comparison of finite difference and pseudo-spectral methods in forward modelling based on metal ore model of random media

With more applications of seismic exploration in metal ore exploration,forward modelling of seismic wave has become more important in metal ore. Finite difference method and pseudo-spectral method are two important methods of wave-field simulation. Results of previous studies show that both methods have distinct advantages and disadvantages: Finite difference method has high precision but its dispersion is serious; pseudospectral method considers both computational efficiency and precision but has less precision than finite-difference. The authors consider the complex structural characteristics of the metal ore,furthermore add random media in order to simulate the complex effects produced by metal ore for wave field. First,the study introduced the theories of random media and two forward modelling methods. Second,it compared the simulation results of two methods on fault model. Then the authors established a complex metal ore model,added random media and compared computational efficiency and precision. As a result,it is found that finite difference method is better than pseudo-spectral method in precision and boundary treatment,but the computational efficiency of pseudospectral method is slightly higher than the finite difference method.

An Accurate Numerical Solution for the Modified Equal Width Wave Equation Using the Fourier Pseudo-Spectral Method

In this study, the numerical solution for the Modified Equal Width Wave (MEW) equation is presented using Fourier spectral method that use to discretize the space variable and Leap-frog method scheme for time dependence. Test problems including the single soliton wave motion, interaction of two solitary waves and interaction of three solitary waves will use to validate the proposed method. The three invariants of the motion are evaluated to determine the conservation properties of the generated scheme. Finally, a Maxwellian initial condition pulse is then studied. The L2 and L∞ error norms are computed to study the accuracy and the simplicity of the presented method.

Stability of weighted spectral distribution in a pseudo tree-like network model

The comparison of networks with different orders strongly depends on the stability analysis of graph features in evolving systems. In this paper, we rigorously investigate the stability of the weighted spectral distribution（i.e., a spectral graph feature） as the network order increases. First, we use deterministic scale-free networks generated by a pseudo treelike model to derive the precise formula of the spectral feature, and then analyze the stability of the spectral feature based on the precise formula. Except for the scale-free feature, the pseudo tree-like model exhibits the hierarchical and small-world structures of complex networks. The stability analysis is useful for the classification of networks with different orders and the similarity analysis of networks that may belong to the same evolving system.

Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.

Pseudo-Spectral Method for Space Fractional Diffusion Equation

This paper presents a numerical scheme for space fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this approach, using the Guass-Lobatto nodes, the unknown function is approximated by orthogonal polynomials or interpolation polynomials. Then, by using pseudo-spectral method, the SFDE is reduced to a system of ordinary differential equations for time variable t. The high order Runge-Kutta scheme can be used to solve the system. So, a high order numerical scheme is derived. Numerical examples illustrate that the results obtained by this method agree well with the analytical solutions.

Two-Level Block Decompositions for Solving Helmholtz Equation via Chebyshev Pseudo Spectral Method

In this paper, we consider solving the Helmholtz equation in the Cartesian domain , subject to homogeneous Dirichlet boundary condition, discretized with the Chebyshev pseudo-spectral method. The main purpose of this paper is to present the formulation of a two-level decomposition scheme for decoupling the linear system obtained from the discretization into independent subsystems. This scheme takes advantage of the homogeneity property of the physical problem along one direction to reduce a 2D problem to several 1D problems via a block diagonalization approach and the reflexivity property along the second direction to decompose each of the 1D problems to two independent subproblems using a reflexive decomposition, effectively doubling the number of subproblems. Based on the special structure of the coefficient matrix of the linear system derived from the discretization and a reflexivity property of the second-order Chebyshev differentiation matrix, we show that the decomposed submatrices exhibits a similar property, enabling the system to be decomposed using reflexive decompositions. Explicit forms of the decomposed submatrices are derived. The decomposition not only yields more efficient algorithm but introduces coarse-grain parallelism. Furthermore, it preserves all eigenvalues of the original matrix.

Jacobi-Sobolev Orthogonal Polynomialsand Spectral Methods for Elliptic Boundary Value Problems

Generalized Jacobi polynomials with indexes α,β∈ R are introduced and some basic properties are established. As examples of applications,the second- and fourth-order elliptic boundary value problems with Dirichlet or Robin boundary conditions are considered,and the generalized Jacobi spectral schemes are proposed. For the diagonalization of discrete systems,the Jacobi-Sobolev orthogonal basis functions are constructed,which allow the exact solutions and the approximate solutions to be represented in the forms of infinite and truncated Jacobi series. Error estimates are obtained and numerical results are provided to illustrate the effectiveness and the spectral accuracy.

Synthesis and Crystal Structure of a Ni(Ⅱ) Complex of 2'-[4-N,N'-(Dimethylaminobenzyl- idene)]-3,5-dihydroxybenzoylhydrazide and Its Spectral Properties

A new nickel（II） complex [NiL2]（DMF）4 （HL = 2＇-[4-N,N＇-（dimethylamino- benzylidene）]-3,5-dihydroxybenzoylhydrazide） has been synthesized and its structure was characterized by IR,UV-Vis,1H NMR and elemental analysis. The fluorescence emission mechanism of the complex was discussed by fluorimetric spectra. The single-crystal structure was determined by X-ray diffraction analysis. The crystal belongs to the triclinic system,space group P1 with a = 0.9918（4）,b = 1.1297（4）,c = 1.1331（4） nm,α = 76.860（7）,β = 75.105（6）,γ = 89.022（7）°,V = 1.1936（8） nm^3,Z = 1,μ = 0.472 mm^-1,Dc = 1.319 g/cm^3,F（000） = 502 and Rint = 0.0913. In the title compound,the nickel（II） atom is four-coordinated with two nitrogen atoms from amide and two oxygen atoms from keto group. The complex is centrosymmetric with nickel located at the center.

Pseudo-Glaucoma after Hypertensive Retinopathy

A 35-year-old Japanese woman showed the typical fundus appearance of acute hypertensive retinopathy, including multiple cotton-wool spots, retinal hemorrhages, star exudates, and disc edema bilaterally. Her systemic blood pressure was high (246/158 mmHg). Nephrologists diagnosed her with secondary hypertension due to immunoglobulin A nephropathy. After 2 years, all signs of acute retinopathy resolved and only multiple striated or sectorial darkening of the fundus (retinal nerve fiber layer defects [NFLD]) remained. Computerized analysis of the macular thickness measurement by optical coherence tomography alerted the risk of glaucoma. NFLD is a hallmark of early glaucoma that is more reliable than IOP elevations since normal tension glaucoma is popular in Japan. To differentiate this case from true glaucoma, information regarding her past history is critical for her future ophthalmologists.

CFS-PML边界条件在PSTD算法中的实现与性能分析

文中将具有复频率参数的PML(CFS PML)边界条件引入到时域伪谱 (PSTD)算法中 ,并利用卷积PML(CPML)方法予以实现 ,CPML的吸收性能作为时间的函数被给出。与传统PML相比 ,该边界条件不仅可以吸收传播模 ,而且还可以吸收低频凋落模。仿真结果也表明 ,CPML中参数α ,Kmax在一定范围内取值可使该边界的反射误差得到最小。

速比离散型车辆加速-滑行式节油巡航策略

巡航过程的节油驾驶是汽车经济性驾驶技术的重要组成,其核心问题是节油巡航策略的辨识。为此,定量化地求解了速比离散型变速器车辆的加速-滑行式(Pulse-and-glide,PnG)节油巡航策略。将节油策略的辨识构建为一最优控制问题;由于发动机的油耗呈强非线性,且PnG策略的加速阶段与滑行阶段动力学特性不一致,导致该问题属于非线性切换型最优控制。采用拼接策略将该非连续问题转化为加速和滑行两段光滑子问题的组合,通过伪谱法实现对积分型性能函数和微分型状态空间方程的代数转化,从而将原最优控制问题转化为非线性规划问题,并进而实现发动机输出转矩、变速器档位、模式切换时刻等参数的数值优化。结果表明:对所研究的车辆,平均巡航速度在中速范围内,PnG策略相对匀速巡航具有显著的节油效果,最高节油率接近25%。最后分析PnG策略实现节油的物理机理,并对节油效果进行了实车试验验证。

基于Benjamin-Feir不稳定性的畸形波模拟

通过引入四阶非线性薛定谔方程,基于随机波列演化的Benjamin-Feir不稳定性,采用伪谱方法建立了二维深水波浪数值水槽来模拟海洋中的畸形波现象.为了验证该数值模型的有效性,计算了二维水槽中边带扰动随机波列的传播变形,通过比较数值和试验结果发现该模型可以很好地再现畸形波现象.

拟谱-微分求积混合方法求解一类双曲电报方程

拟谱方法和微分求积法是两类重要的无网格法,二者都已在科学和工程计算中获得了广泛应用。采用拉格朗日插值多项式作为二者的试函数,且采用同一种网格点分布,指出了在空间域上,微分求积法是拟谱方法的一种特殊形式。在此基础上,结合二者各自的特点,提出了拟谱-微分求积混合方法用于求解一类双曲电报方程。理论分析和数值测试表明,新方法在空间域上具有谱精度收敛性,在时间域上是A-稳定的,比较适合于求解多维电报方程。

FENE-P流体混合层中拟序结构的研究

采用FENE_P模型对高雷诺数下二维粘弹性混合过程中拟序结构的演变进行了数值研究· 由于对聚合物应力场采用了适当的滤波措施 ,得到了较AH更宽参数范围内FENE_P流体混合层流场中拟序结构的运动特性 ,计算结果表明 :在流场中加入聚合物 ,基波和次谐波的发展受到抑制 ,涡量扩散加强 ,减慢了配对时两涡核的旋转运动 ,这种影响随Weissenberg数的增大而减小 ,但却随参数b的增大而加强· 另外 ,与相同溶液粘度下的牛顿流体相比 ,配对过程中涡量完全合并的时刻延迟了·

助推-滑翔飞行器弹道最优控制研究

从多阶段多约束最优控制的角度对助推-滑翔飞行器弹道最优控制问题进行研究。首先建立了无量纲化的弹道动力学方程组并提炼出总体设计参数;其次使用Radau伪谱方法将多约束的弹道最优控制问题转化为非线性规划问题,通过引入连接点概念处理多阶段不连续问题,并采用序列二次规划方法进行求解;最后通过数值算例验证了方法的实用性和有效性。数值优化算例表明,通过多阶段Radau伪谱方法可较好处理助推-滑翔飞行器全射程最优问题,所得计算结果与理论最优解一致。

Klein-Gordon-Schrdinger方程的辛Fourier拟谱格式(英文)

主要讨论Klein-Gordon-Schrdinger方程的Fourier拟谱辛格式,包括中点公式和Strmer/Verlet格式.首先构造一个哈密尔顿方程,针对此哈密尔顿方程,在空间方向用Fourier拟谱离散得到一个有限维的哈密尔顿系统,对此有限维系统在时间方向用Strmer/Verlet方法离散得到KGS方程的完全显式的辛格式.中点格式虽然是隐式的但效率也很高,且具有质量守恒律.数值实验表明,辛格式能够在长时间内很好地模拟各类孤立波.

基于分数阶Wigner-Ville分布的地震信号谱分解

本文将经典Wigner-Ville分布(WVD)拓展到分数阶Wigner-Ville分布(FrWVD),并将其引入到复杂地震信号的时频分析和谱分解中。利用分数阶傅里叶变换的旋转特性和广义时频带宽积理论可以获得比经典WVD更高的时频分辨率,提出的最优分数阶伪WVD能更好地解决交叉项抑制问题。理论信号仿真及实际地震资料的处理结果验证了该方法的有效性。同时,分数域频谱成像的引入为油气储层流体识别提供了一条新的途径。

组合KdV-mKdV方程的多辛Fourier拟谱格式

基于Hamilton空间体系下的多辛理论,提出组合KdV-mKdV方程的一个多辛方程组.通过离散此方程组,得到原方程的一个多辛Fourier拟谱格式,以及格式的全离散多辛守恒律.由数值结果可知,多辛Fourier拟谱格式能很好地模拟孤立波运动的波形,不出现振荡现象,且在空间方向具有较高的精度和收敛阶.