浅水长波近似方程组的拟小波解

以浅水长波近似方程组为例,提出了拟小波方法求解(1 1)维非线性偏微分方程组数值解,该方程用拟小波离散格式离散空间导数,得到关于时间的常微分方程组,用四阶Runge-K utta方法离散时间导数,并将其拟小波解与解析解进行比较和验证.     

解Burgers方程的一种显式稳定性方法

拟线性Burgers方程在空间离散后转化成常微分方程,再用指数积分方法求解．数值结果表明指数积分法有显式稳定性,有相应Runge-Kutta方法相同的精度．     

分数阶Burgers方程的有限元计算(英文)

建立了一维和二维分数阶Burgers方程的有限元格式.时间分数阶导数使用L1方法离散,空间方向使用有限元方法离散.通过选择合适的基函数,将离散后的方程转化成一个非线性代数方程组,并应用牛顿迭代方法求解.数值实验显示出了方法的有效性.     

带非线性源项的双侧空间分数阶扩散方程的隐式中点方法

本文用隐式中点方法离散一阶时间偏导数,并用拟紧差分算子逼近Riemann-Liouville空间分数阶偏导数,构造了求解带非线性源项的空间分数阶扩散方程的数值格式.给出了数值方法的稳定性和收敛性分析.数值试验表明数值方法是有效的.     

基于拟Shannon小波浅水长波近似方程组的数值解

本文研究了浅水长波近似方程组初边值问题的数值解.利用小波多尺度分析和区间拟Shannon小波,对浅水长波近似方程组空间导数实施空间离散,用时间步长自适应精细积分法对其变换所的非线性常微分方程组进行求解,得到了浅水长波近似方程组的数值解,并将此方法计算的结果与其解析解进行比较和验证.     

A Space-Time Polynomial Collocation Method for Solving 3D Burgers Equations北大核心CSCD

Burgers方程是一类应用广泛的非线性偏微分方程,方程中的非线性项难以处理。该文提出一种新的时空多项式配点法——多项式特解法求解三维Burgers方程。求解过程分为两步:第一步,对三维Burgers方程中的线性导数项(包括时间导数项),求出相应的多项式特解。第二步,将求出的多项式特解作为基函数,对三维Burgers方程中剩余的非线性项进行迭代求解。与时空多项式函数作为基函数对三维Burgers方程进行直接求解相比,该算法简单易行,得到的近似解精度非常高,算法极其稳定,对于教学过程中提高学生的编程能力,加深对高维Burgers方程的理解能力以及Burgers方程的实际应用具有重要意义。     

用时间方向二阶精度的混合Jacobi-球面调和拟谱方法求解Fisher型方程

提出了用时间方向二阶精度的混合Jacobi-球面调和拟谱方法求解单位球内的Fisher型方程,即在半径方向选择Gauss-Radau型插值逼近,球面方向选择球面调和插值逼近,而时间方向的导数采用中心差商离散.给出了误差估计的相关结论,并以数值结果显示所论述方法的高精度.     

偏微分方程的区间小波自适应精细积分法

利用插值小波理论构造了拟Shannon区间小波,并结合外推法给出了一种求解非线性常微分方程组的时间步长自适应精细积分法,在此基础上构造了求解非线性偏微分方程的区间小波自适应精细积分法（AIWPIM）．数值结果表明,该方法在计算精度上优于将小波和四阶Runge-Kutta法组合得到的偏微分方程的数值求解方法,而计算量则相差不大．该文方法通过Burgers方程给出,但适用于一般情形．     

有限差分-配点法求解二维Burgers方程

将重心插值配点法结合Crank-Nicolson差分格式来求解Burgers方程.首先,利用Hopf-Cole变换将Burgers方程转化为线性热传导方程;空间方向采用重心插值配点法进行离散,时间方向采用Crank-Nicolson格式离散,导出对应的线性代数方程组,并对此计算格式进行相容性分析;最后,通过数值算例验证此计算格式具有高精度和有效性.     

多维区域上椭圆型方程Dirichlet问题的勒让德多小波多层扩充算法

用小波伽辽金方法求解多维区域上椭圆型方程齐次Dirichlet问题,构造了近似解空间的两个等价的勒让德多小波基,使得快速求解离散后的线性方程组的多层扩充算法得以实现.数值算例表明该算法是有效的.     

Burgers方程的一类交替分组方法

对于Burgers方程给出了一组新的Saul'yev型非对称差分格式，并用这些差分格式构造了求解非线性Burgers方程的交替分组四点方法．该算法把剖分节点分成若干组，在每组上构造能够独立求解的差分方程．因此算法具有并行本性，能直接在并行计算机上使用．章还证明了所给算法线性绝对稳定．数值试验表明，该方法使用简便，稳定性好，有很好的精度。     

Generalization of a Relaxation Scheme for Systems of Forced Nonlinear Hyperbolic Conservation Laws with Spatially Dependent Flux Functions

A generalization of a finite difference method for calculating numerical solutions to systems of nonlinear hyperbolic conservation laws in one spatial variable is investigated. A previously developed numerical technique called the relaxation method is modified from its initial application to solve initial value problems for systems of nonlinear hyperbolic conservation laws. The relaxation method is generalized in three ways herein to include problems involving any combination of the following factors: systems of nonlinear hyperbolic conservation laws with spatially dependent flux functions, nonzero forcing terms, and correctly posed boundary values. An initial value problem for the forced inviscid Burgers' equation is used as an example to show excellent agreement between theoretical solutions and numerical calculations. An initial boundary value problem consisting of a system of four partial differential equations based on the two-layer shallow-water equations is solved numerically to display a more general applicability of the method than was previously known.     

Burgers方程的小波近似惯性流形及数值分析

研究Burgers方程小波基下小波近似惯性流形的存在性,并作低阶多分辨分析下的数值分析,数值分析表明小波近似惯性流形更能反映方程的局部行为.     

小波方法及其力学应用研究进展

小波理论在进行信号处理与函数逼近时体现出非常独特的时频局部性与多分辨分析能力,小波基函数则可兼具正交性、紧支性、低通滤波与插值性等优良的数学性质,这均使得小波分析理论在计算数学与计算力学领域具有很大的应用潜力,也进一步为这些领域的突破性发展带来了新的契机.自20世纪90年代以来,大量的研究已经证明,基于小波理论的数值方...     

Wick型随机广义Burgers方程的确切解

借助白噪声分析、Hermite变换和扩展的双曲函数法,研究了Wick型随机广义Burgers'方程,求出了一些精确的Wick型孤立波解和周期波解.由于Wick型函数难以赋值,为此,我们得到一些特殊情形下的Wick型随机广义Burgers'方程的非Wick型解.     

MQ拟插值求解KdV方程

Quasi-interpolation is very useful in the study of approximation theory and its applications,since it can yield solutions directly without the need to solve any linear system of equations.Based on the good performance,Chen and Wu presented a kind of multiquadric （MQ） quasi-interpolation,which is generalized from the L D operator,and used it to solve hyperbolic conservation laws and Burgers’ equation.In this paper,a numerical scheme is presented based on Chen and Wu’s method for solving the Korteweg-de Vries （KdV） equation.The presented scheme is obtained by using the second-order central divided difference of the spatial derivative to approximate the third-order spatial derivative,and the forward divided difference to approximate the temporal derivative,where the spatial derivative is approximated by the derivative of the generalized L D quasi-interpolation operator.The algorithm is very simple and easy to implement and the numerical experiments show that it is feasible and valid.     

MULTILEVEL AUGMENTATION METHODS FOR SOLVING OPERATOR EQUATIONS

We introduce multilevel augmentation methods for solving operator equations based on direct sum decompositions of the range space of the operator and the solution space of the operator equation and a matrix splitting scheme. We establish a general setting for the analysis of these methods, showing that the methods yield approximate solutions of the same convergence order as the best approximation from the subspace. These augmentation methods allow us to develop fast, accurate and stable nonconventional numerical algorithms for solving operator equations. In particular, for second kind equations, special splitting techniques are proposed to develop such algorithms. These algorithms are then applied to solve the linear systems resulting from matrix compression schemes using wavelet-like functions for solving Fredholm integral equations of the second kind. For this special case, a complete analysis for computational complexity and convergence order is presented. Numerical examples are included to demonstra     

Second Kind Chebyshev Wavelet Galerkin Method for Stochastic Itô-Volterra Integral Equations

In this paper, an efficient wavelet Galerkin method based on the stochastic operational matrix of second kind Chebyshev wavelet is proposed for solving stochastic Itô-Volterra integral equations. Convergence and error analysis of the presented wavelets method are investigated. The numerical results are compared with exact solution and those of other existing methods.     

Wavelet method for solving second-order quasilinear parabolic equations with a conservative principal part

A method based on wavelet transforms is proposed for finding classical solutions to initial-boundary value problems for second-order quasilinear parabolic equations. For smooth data, the convergence of the method is proved and the convergence rate of an approximate weak solution to a classical one is estimated in the space of wavelet coefficients. An approximate weak solution of the problem is found by solving a nonlinear system of equations with the help of gradient-type iterative methods with projection onto a fixed subspace of basis wavelet functions.