解高维广义BBM方程的谱方法和拟谱方法

在非线性色散介质的长波研究中,Benjanin,Bona和Mahony等人提出并讨论了BBM方程。这类方程在许多数学物理问题中出现,如热力学中的双温热传导问题、在岩石裂缝中的渗流问题等,因而引起了人们的重视。之后,Goldstein,Avrin,郭柏灵等进一步研究了高维广义BBM方程。这类方程的数值分析很多,但主要是差分法和有限元法,如[9-10],[11]在一维情形下用谱方法和拟谱方法作了研究。本文讨论高维     

广义BBM方程有理Chebyshev谱逼近的大时间性态

本文考虑广义BBM方程的初值问题,建立了方程的有理Chebyshev谱格式,给出了谱格式的误差估计,并证明了原问题和近似问题所生成的算子半群分别具有整体吸引子A和AN,且AN关于A 是上半连续的．     

高维广义Kuramoto-Sivashinsky型方程Fourier拟谱方法的大时间误差估计

在很多物理问题中出现如下方程:Kuramoto在研究反应扩散系统耗散结构时导出了上述方程,Sivashinsky在模拟火焰传播时也得到了它.此外,它还出现在粘性层流和Navier-Stokes方程的分枝解中.在[5-8]中,作者研究了一维情形下周期初值问题的整体吸引子和分枝解;[9]提出了广义KS型方程;[10-14]中研究了它的光滑解的存在性和t→+∞时的渐近性     

多维广义SRLW方程的Chebyshev拟谱方法分析

考虑了一类多维的广义对称正则长波(SRLW)方程的齐次初边值问题Chebyshev拟谱逼近,构造了全离散的Chebyshev拟谱格式,给出了这种格式近似解的收敛性和最优误差估计。     

广义SRLW方程的拟谱配点方法

1.引言 我们考虑求解下列非线性波动方程周期初值问题：     

高维广义BBM方程组的初边值问题

本文应用先验估计和Galerkin方法证明了高维广义BBM方程组的初边值问题在L~∞(0,T;H~3(Ω)∩H_0~1(Ω)),(s≥2)中整体解的存在性和正则性,并得到了整体解在||·||_(H~3×L~∞)范数下的稳定性和光滑解的唯一性。     

SRLW方程的Fourier拟谱方法

本文研究用带抑制算子的Foutier拟谱方法求解SRLW方程,我们证明格式的稳定性,并给出最佳误差估计.     

混合广义Jacobi和Chebyshev谱配置法求解时间分数阶对流扩散方程

研究时间Caputo分数阶对流扩散方程的高效高阶数值方法.对于给定的时间分数阶偏微分方程,在时间和空间方向分别采用基于移位广义Jacobi函数为基底和移位Chebyshev多项式运算矩阵的谱配置法进行数值求解.这样得到的数值解可以很好地逼近一类在时间方向非光滑的方程解.最后利用一些数值例子来说明该数值方法的有效性和准确性.     

A Generalized Chebyshev Method for Non-linear Parabolic Equations in One Space Variable

The derivation and implementation of a generalized Chebyshevmethod is described for the numerical solution of non-linearparabolic equations in one space dimension. The solution isobtained by using the method of lines and is approximated inthe space variable by piecewise Chebyshev polynomial expansions.These expansions are normally few in number and of high order.It is shown that the method can be derived from a perturbedform of the original equation. A numerical example is givento illustrate its performance compared with the finite elementand finite difference method. A comparison of various Chebyshev methods is made by applyingthem to two-point eigenproblems. It is shown by analysis andnumerical examples that the approach used to derive the generalizedChebyshev method is comparable, in terms of the accuracy obtained,with existing Chebyshev methods.     

A Pseudospectral Method for Solving Navier-Stokes Equations

In this paper, we propose a new kind of pseudospectral schemes with a restraint operator to solve the periodic problem of Navier-Stokes equations. The generalized stability of the schemes is analysed and convergence is proved. Numerical results are presented also.     

The Fourier Pseudospectral Method for Two-Dimensional Vorticity Equations

In this paper we develop a Fourier pseudospectral method forsolving two-dimensional vorticity equations. We prove the generalizedstability of the schemes and give convergence estimations dependingon the smoothness of the solution of the vorticity equations. Spectral methods have been applied widely to the partial differentialequations of fluid dynamics [4–11]. Guo Ben-yu proposeda technique to estimate strictly the error of the spectral schemesfor the K.D.V.-Burgers equation, the two-dimensional vorticityequations, and the Navier-Stokes equations [5,6,8]. On the otherhand, the authors [7,10] developed a pseudospectral method byusing Riesz spherical means to get better results. In this paper,we generalize this method to two-dimensional vorticity equations.The generalized stability and the convergence are proved. Thenumerical results show the advantage of such a method.     

The Chebyshev Spectral Method for Burgers-Like Equations

The Chebyshev polynomials have good approximation properties which are not affected by boundary values. They have higher resolution near the boundary than in the interior and are suitable for problems in which the solution changes rapidly near the boundary. Also, they can be calculated by FFT. Thus they are used mostly for initial-boundary value problems for P.D.E.'s (see [1, 3-4, 6, 8-11]). Maday and Quarterom discussed the convergence of Legendre and Chebyshev spectral approximations to the steady Burgers equation. In this paper we consider Burgers-like equations.$$\begin{cases}∂_iu+F(u)_x-vu_{zx}=0, & -1≤x≤1, 0＜t≤T \\ u (-1,t) =u (1,t) =0, & 0≤t≤T & (0.1)\\ u (x,0) =u_0(x), & -1≤x≤1\end{cases}$$ where $F\in C(R)$ and there exists a positive function $A\in C(R)$ and a constant $p＞1$ such that $$|F(z+y)-F(z)|\leq A(z)(|y|+|y|^p).$$ We develop a Chebyshev spectral scheme and a pseudospectral scheme for solving (0.1) and establish their generalized stability and convergence.     

Finite Dimensional Behavior for Weakly Damped Generalized KdV-Burgers Equations

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一类推广的BBM方程的强吸引子

主要研究了推广的Benjamin-Bona-Mahony（BBM）方程的解的渐近行为,通过证明半群的渐近紧性证明了方程在H)_per²（Ω）中全局吸引子的存在性.