一类半线性抛物型方程全离散Chebyshev拟谱逼近的大时间性态

In the paper, the nonperidic initial value problem for a class of semilinear parabolic equations is considered. We construct the full discrete Chebyshev pseudospectral scheme and analyze the error of approximate solution for it. We obtain the error estimation on large time using the local continuation method and the existence of approximate global attractor.     

高维广义BBM方程的Chebyshev拟谱方法

在非线性长波的研究中[1],提出并研究了BBM方程.由于这类方程在很多数学物理问题中出现,如双温热传导的冷却过程,液体在碎石中的渗流问题等,因而引起了人们的关注.这类方程的数值方法,已有许多工作,但主要是采用差分法和有限元法.[2]使用.Fourier谱方法讨论了一维广义BBM方程,我们在[3]中也用Fourier谱和拟谱方法讨论了高维广义BBM方程.然而对于非周期情况,Fourier谱方法无法使用.在     

带弱阻尼的非线性Schrodinger方程谱逼近的大时间性态

表文讨论带弱阻尼的非线性Schrōdinger方程周期初值问题当采用谱方法求解时近似解的大时间误差估计、近似吸引子fn的存在和它们的弱上半连续性(fn，f)→0．     

非线性KdV-Schr(o)dinger方程Fourier谱逼近的大时间性态

近几年来,对具弱阻尼的非线性发展方程的研究越来越受到人们的关注.大部分情况下,由于精确解无法得到,我们只有通过求数值解来研究方程解的性质.本文讨论具弱阻尼的非线性KdV-Schroedinger方程Fourier谱逼近的大时间性态问题.我们构造了方程的Fourier近似谱格式,并对方程的近似解作了相应的先验估计及方程近似解与精确解之间的误差估计.最后,证明了近似吸引子AN的存在性及其弱上半连续性dω(AN,A)→0.     

广义KdV—Burgers方程吸引子的谱逼近

１引言近年来．随着对无限维动力系统研究的深入,人们对非线性发展方程解的渐近性态了解得越来越多．例如对某些耗散的非线性发展方程,象Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程、Ｋｕｒａｍｏｔｏ－Ｓｉｖａｓｈｉｎ－ｓｋｙ方程等都存在整体的吸引子．系统的渐近性质和系统的复杂性完全由整体吸引子所确定（详细请参见［３］）．与此同时,这类系统的有限维逼近也是人们非常关心的问题,在这方面已有许多工作,如Ｊ．Ｋ．Ｈａｌｅ等人在［５］中基于有限元方法研究了某些非线性发展方程．得到了近似吸引子是上半连续的;Ｃ．Ｍ．Ｅｌｌｏｔｔａ…     

带弱阻尼的非线性Schrdinger方程谱逼近的大时间性态

本文讨论带弱阻尼的非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程周期初值问题当采用谱方法求解时近似解的大时间误差估计、近似吸引子ＡＮ的存在和它们的弱上半连续性ｄｗ（ＡＮ，Ａ）→０．     

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     

无界域上半线性强阻尼波动方程的全离散有理谱逼近

本文运用Chebyshev有理谱方法来讨论半线性强阻尼波动方程.通过建立时间、空间方向全离散的Chebyshev有理谱格式,证明了由此格式所确定的离散算子半群存在整体吸引子,并从理论上建立了在有限时间上近似解的误差估计.     

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。     

半直线上BBM方程初边值问题的修正有理谱耦合方法

本文考虑使用修正的有理谱方法处理半直线上的BBM方程初边值问题.对非线性项使用Chebyshev有理插值显式处理,而线性项使用修正Legendre有理谱方法隐式处理.这种处理既可以节约运算又可以保持良好的稳定性.数值例子表明了算法的有效性     

Existence and Orbital Stability of Solitary-Wave Solutions for Higher-Order BBM Equations

This paper discusses the existence and stability of solitary-wave solutions of a general higher-order Benjamin-Bona-Mahony (BBM) equation, which involves pseudo-differential operators for the linear part. One of such equations can be derived from water-wave problems as second-order approximate equations from fully nonlinear governing equations. Under some conditions on the symbols of pseudo-differential operators and the nonlinear terms, it is shown that the general higher-order BBM equation has solitary-wave solutions. Moreover, under slightly more restrictive conditions, the set of solitary-wave solutions is orbitally stable. Here, the equation has a nonlinear part involving the polynomials of solution and its derivatives with different degrees (not homogeneous), which has not been studied before. Numerical stability and instability of solitary-wave solutions for some special fifth-order BBM equations are also given.     

一维非齐次BBM方程周期边界问题的整体吸引子

研究了一维非齐次方程BBM方程ut-uxxt-αφ(u)x=g(x)+βf(u)+γuxx(α>0,β>0,γ>0),u(x+2π,t)=u(x,t),u(x,0)=u0(x)的周期边界问题.利用Sobolev插值不等式,对解做关于时间t的一致性先验估计,证明了该问题的整体吸引子的存在性.     

THE LONG-TIME BEHAVIOR OF SPECTRAL APPROXIMATE FOR KLEIN-GORDON-SCHROEDINGER EQUATIONS

Klein-Gordon-Schroedinger (KGS) equations are very important in physics. Some papers studied their well-posedness and numerical solution [1-4], and another works investigated the existence of global attractor in R^n and Ω包含于R^n (n≤3) [5-6,11-12]. In this paper, we discuss the dynamical behavior when we apply spectral method to find numerical approximation for periodic initial value problem of KGS equations. It includes the existence of approximate attractor AN, the upper semi-continuity on A which is a global attractor of initial problem and the upper bounds of Hausdorff and fractal dimensions for A and AN,etc.     

Least squares and Chebyshev fitting for parameter estimation in ODEs

We discuss the problem of determining parameters in mathematical models described by ordinary differential equations. This problem is normally treated by least squares fitting. Here some results from nonlinear mean square approximation theory are outlined which highlight the problems associated with nonuniqueness of global and local minima in this fitting procedure. Alternatively, for Chebyshev fitting and for the case of a single differential equation, we extend and apply the theory of [17, 18] which ensures a unique global best approximation. The theory is applied to two numerical examples which show how typical difficulties associated with mean square fitting can be avoided in Chebyshev fitting.This paper is presented as an outcome of the LMS Durham Symposium convened by Professor C.T.H. Baker on 4th-14th July 1992 with support from the SERC under Grant reference number GR/H03964.     

Rational Chebyshev collocation method for solving higher‐order linear ordinary differential equations

A collocation method to find an approximate solution of higher‐order linear ordinary differential equation with variable coefficients under the mixed conditions is proposed. This method is based on the rational Chebyshev (RC) Tau method and Taylor‐Chebyshev collocation methods. The solution is obtained in terms of RC functions. Also, illustrative examples are included to demonstrate the validity and applicability of the technique, and performed on the computer using a program written in maple9. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1130–1142, 2011     

The Use of Chebyshev Polynomials in the Space–Time Least-Squares Spectral Element Method

Chebyshev polynomials of the first kind are employed in a space–time least-squares spectral element formulation applied to linear and nonlinear hyperbolic scalar equations. No stabilization techniques are required to render a stable, high order accurate scheme. In parts of the domain where the underlying exact solution is smooth, the scheme exhibits exponential convergence with polynomial enrichment, whereas in parts of the domain where the underlying exact solution contains discontinuities the solution displays a Gibbs-like behavior. An edge detection method is employed to determine the position of the discontinuity. Piecewise reconstruction of the numerical solution retrieves a monotone solution. Numerical results will be given in which the capabilities of the space–time formulation to capture discontinuities will be demonstrated.