半直线上三阶微分方程的Laguerre-Petrov-Galerkin谱方法

对半无界区域上的三阶方程提出了Laguerre-Petrov-Galerkin谱逼近方法,选取了相同的试探空间和检验空间.通过构造该空间上的基函数,离散问题所对应的线性系统的系数矩阵是半稀疏的.数值算例验证了该方法的有效性和高精度.     

非线性对流-扩散方程的多区域拟谱方法

提出了非线性对流-扩散方程的多区域拟谱方法．在每个子区间上,该格式整体上按Legendre－Galerkin方法形成,但对于非线性项采用在 Legendre/Chebyshev－Gauss-Lobatto点上的配置法处理．通过选取适当的基函数,使得系数矩阵稀疏,并且可以并行计算,提高运算效率．给出了该方法的稳定性和收敛性分析,获得了按L2-模的最佳误差估计．最后给出单区域和多区域方法的数值算例,并加以比较．     

三阶微分方程解的增长性

In this paper,the precise estimation of the order and hyper-order of solutions of a class of three order homogeneous and non-homogeneous linear differential equations are obtained. The results of M. Ozawa (1980), G. Gundersen (1988) and J. K. Langley ( 1986 ) are improved.     

一类分数阶多项延迟微分方程的Jacobi谱配置方法

本文利用Jacobi谱配置方法数值求解了一类分数阶多项延迟微分方程,并证明了该方法是收敛的,通过若干数值算例验证了相应的理论结果,结果表明Jacobi谱配置方法求解这类方程是非常高效的,同时也为这类分数阶延迟微分方程的数值求解提供了新的选择,对分数阶泛函方程的数值方法的研究有一定的指导意义.     

三阶非线性泛函微分方程振动的比较准则

本文利用比较法,与一阶时滞微分方程的振动性相比较,研究一类三阶非线性泛函微分方程的振动性,建立该类方程振动的新比较振动准则.本文结果推广和改进了最近文献中的一些新结果.     

半直线上三阶方程的Legendre-Laguerre耦合谱元法

针对建立在半直线上的三阶微分方程,提出Legendre-Laguerre耦合谱元法.通过构造满足试探函数空间和检验函数空间的基函数,分解得到的线性系统的系数矩阵是稀疏的,可以有效地进行求解.数值例子验证了方法的有效性和高精度.     

一类变延迟微分方程谱方法的收敛性

本文主要应用谱方法求解一类线性变系数变延迟微分方程,构造相应的数值方法,证明其收敛性,并给出两个具有代表性的数值算例.这些结果表明应用谱方法求解延迟微分方程可以获得谱收敛与谱精度的计算效果.     

基于常微分方程边值问题的Chebyshev谱方法

常微分方程边值问题的数值解法有多种,其中较常用的是化边值问题为初值问题解法以及边值问题差分解法.常微分方程边值问题数值解的Chebyshev谱方法是近年来出现的一种新解法.作为应用例子,分别采用Chebyshev谱方法、化边值问题为初值问题解法、以及边值问题差分解法对一类二阶常微分方程边值问题进行数值求解,并对数值解的精确性及计算时间定量地比较,从而说明Chebyshev解法是精度很高的一种快捷解法.     

三阶非线性中立型微分方程的振动性

研究三阶中立型时滞微分方程(r(t)[x(t)+p(t)x(σ(t))]″)′+q(t)f(x(t),x[q(t)])h(x′(t))=0的振动性和渐进性.给出了方程一切解振动或者渐近趋向于零的若干充分条件.     

三阶线性常微分方程Sinc方程组的结构预处理方法

三阶线性常微分方程在天文学和流体力学等学科的研究中有着广泛的应用.本文介绍求解三阶线性常微分方程由Sinc方法离散所得到的线性方程组的结构预处理方法.首先, 我们利用Sinc方法对三阶线性常微分方程进行离散,证明了离散解以指数阶收敛到原问题的精确解.针对离散后线性方程组的系数矩阵的特殊结构, 提出了结构化的带状预处理子,并证明了预处理矩阵的特征值位于复平面上的一个矩形区域之内.然后, 我们引入新的变量将三阶线性常微分方程等价地转化为由两个二阶线性常微分方程构成的常微分方程组, 并利用Sinc方法对降阶后的常微分方程组进行离散.离散后线性方程组的系数矩阵是分块2×2的, 且每一块都是Toeplitz矩阵与对角矩阵的组合.为了利用Krylov子空间方法有效地求解离散后的线性方程组,我们给出了块对角预处理子, 并分析了预处理矩阵的性质.最后, 我们对降阶后二阶线性常微分方程组进行了一些比较研究.数值结果证实了Sinc方法能够有效地求解三阶线性常微分方程.     

A first-second order splitting method for a third-order partial differential equation

A splitting of a third order partial differential equation into a first-order and a second-order one is proposed as the basis for a mixed finite element method to approximate its solution. A time-continuous numerical method is described and error estimates for its solution are demonstrated. Finally, a full discretization is described based on backward Euler finite differences in time, and error estimates for the resulting approximation are established. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 89–96, 1998     

Legendre spectral Galerkin method for second-kind Volterra integral equations

The Legendre spectral Galerkin method for the Volterra integral equations of the second kind is proposed in this paper. We provide a rigorous error analysis for the proposed method, which indicates that the numerical errors (in the L ₂ norm) will decay exponentially provided that the kernel function and the source function are sufficiently smooth. Numerical examples are given to illustrate the theoretical results.      

Uniform convergence of the legendre spectral method for the Zakharov equations

The Legendre spectral and pseudospectral approximations are proposed for the standard Zakharov equations with initial boundary conditions. Optimal H¹ error estimate of the method is given for both semidiscrete and fully discrete schemes. The uniform convergence for the parameter ε relative to the acoustic speed is proved. Moreover, the multidomain Legendre spectral scheme is also constructed, which can be implemented in parallel. Finally, numerical results in single domain and multidomain verify the high accuracy of the Legendre spectral method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Comparative study on order-reduced methods for linear third-order ordinary differential equations

The linear third-order ordinary differential equation (ODE) can be transformed into a system of two second-order ODEs by introducing a variable replacement, which is different from the common order-reduced approach. We choose the functions p(x) and q(x) in the variable replacement to get different cases of the special order-reduced system for the linear third-order ODE. We analyze the numerical behavior and algebraic properties of the systems of linear equations resulting from the sinc discretizations of these special second-order ODE systems. Then the block-diagonal preconditioner is used to accelerate the convergence of the Krylov subspace iteration methods for solving the discretized system of linear equation. Numerical results show that these order-reduced methods are effective for solving the linear third-order ODEs.     

Space‐time spectral method for two‐dimensional semilinear parabolic equations

In this paper, a high‐order accurate numerical method for two‐dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high‐order accurate in both space and time. Optimal a priori error bound is derived in the L²‐norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd.     

Conditional linearizability criteria for a system of third-order ordinary differential equations

We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

第二类Volterra型积分方程的Chebyshev-Legendre谱配置方法

提出了一种新的求解第二类线性Volterra型积分方程的Chebyshev谱配置方法.该方法分别对方程中积分部分的核函数和未知函数在Chebyshev-Gauss-Lobatto点上进行插值,通过Chebyshev-Legendre变换,把插值多项式表示成Legendre级数形式,从而将积分转换为内积的形式,再利用Legendre多项式的正交性进行计算.利用Chebyshev插值算子在不带权范数意义下的逼近结果,对该方法在理论上给出了L∞范数意义下的误差估计,并通过数值算例验证了算法的有效性和理论分析的正确性.     

某些非线性三阶常微分方程的几个存在性结论

考察了某些非线性三阶常微分方程的存在性．主要结论的条件涉及非线性项在无穷远处的增长速度．改进和推广了某些现有的结论．