一阶双曲方程的Legendre—Tau方法

文章采用Legendre—tau方法对一阶双曲方程进行数值求解,此方法可以被有效实施,且可以得到L^2模意义下的最优误差估计,将以往对此类问题的收敛阶估计由O（N^1-τ）提高到O（N^-τ）,改进了原有的理论分析结果,数值算例证实了此方法的有效性．     

非线性发展型方程的Legendre拟谱逼近

本文考虑非稳态Burgers方程的拟谱逼近，构造了一类Legendre拟谱计算格式并证明了其收敛性，数值结果显示了格式的有效性。     

Legendre小波函数求解非线性Volterra积分微分方程组的数值解

Legendre小波函数被用于逼近非线性Volterra积分微分方程组的解,方法是基于Legendre小波的性质构建相应的积分算子矩阵,进而将原问题转化为关于未知解系数的线性方程组,通过求解该方程组,即得原问题的数值解.数值结果表明所述方法对于求解此类问题是行之有效的.     

基于Legrndre小波的第一类Fredholm积分方程的数值解法研究

提出利用Legendre小波函数去获得第一类Fredholm积分方程的数值解,函数定义在区间[0,1)上,然后结合Garlerkin方法将原问题转化为线性代数方程组.而且还对算法的收敛性和误差进行了分析,最后通过两个数值算例验证了所提算法的可行性及有效性.     

Legendre小波求解超奇异积分

超奇异积分的数值算法一直是近些年来研究的重要课题. 基于超奇异积分的 Hadamard 有限部分积分定义, 本文给出了利用 Legendre 小波计算超奇异积分的方法. 当奇异点位于区间内时, 由于 Legendre 小波具有很好的正交性、显式表达式以及小波函数的可计算性, 将区间内的奇异点变换到区间端点处, 再利用区间端点处 Hadamard 有限部分积分的定义,进而可以计算 p+1(p∈N⁺) 阶超奇异积分. 文中最后给出的算例表明了该方法的可行性和有效性.     

调和方程自然边界元Shannon 小波方法

1 引言 调和方程无论是在数学上还是在物理学中都占有重要地位,它有很多不同的物理背景,在力学和物理学中研究的许多问题都可归结为调和方程的边值问题,所以对调和方程进行深入研究有重要意义.余德浩教授在[3]中主要对调和方程在典型域(即单位圆,上半平面)上的情形进行了考虑.特别地,对单位圆的情形给出了刚度矩阵系数的计算公式和调和方程解的存在唯一性.本文采用由冯康教授[1]开创的自然边界元方法和Galerkin小波方法相耦合,对上半平面的调和方程Neumann问题进行了研究,得到十分有效的计算结果.     

一类非线性Schrodinger方程的差分／Legendre谱元法

考察一类带幂次非线性项的Schrodinger方程的Dirichlet初边值问题，提出了一个有效的计算格式，其中时间方向上应用了一种守恒的二阶差分隐格式，空间方向上采用Legendre谱元法．对于时间半离散格式，证职了该格式具有能量守恒性质，并给出了L^2误差估计，对于全离散格式，应用不动点原理证明了数值解的存在唯一性，并给出了L^2误差估计．最后，通过数值试验验证了结果的可信性．     

Cahn—Hilliard方程的Legendre谱逼近

1.引 言本文我们将考虑非线性Cahn—Hilliard方程的初边值问题     

基于自适应小波神经网络的第二类Fredholm积分方程数值解法

该文构造了一类三层前馈自适应小波神经网络,将小波分析中平移因子和伸缩因子的拟合设置为输入层到隐层的权值与阈值,采用小波基函数作为隐层激活函数,并根据梯度下降算法自适应地调整参数.应用自适应小波神经网络数值求解第二类Fredholm积分方程,通过数值算例验证了该方法的可行性和有效性.     

基于Legendre函数的Landau''''s型不等式

首先研究了L2w(-1,1)上关于Legendre多项式Xn(x)的Landau's型不等式.利用Xn(x)的正交性,建立了代数多项式pn(x)的Landau's型不等式,并且指出其不等式的系数在某种意义上是最好可能的.     

A Method for Solving Fredholm Integral Equations of the First Kind Based on Chebyshev Wavelets

In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.     

Numerical solution for the weakly singular Fredholm integro-differential equations using Legendre multiwavelets

An effective method based upon Legendre multiwavelets is proposed for the solution of Fredholm weakly singular integro-differential equations. The properties of Legendre multiwavelets are first given and their operational matrices of integral are constructed. These wavelets are utilized to reduce the solution of the given integro-differential equation to the solution of a sparse linear system of algebraic equations. In order to save memory requirement and computational time, a threshold procedure is applied to obtain the solution to this system of algebraic equations. Through numerical examples, performance of the present method is investigated concerning the convergence and the sparseness of the resulted matrix equation.     

Legendre polynomial solutions of high-order linear Fredholm integro-differential equations

In this study, a Legendre collocation matrix method is presented to solve high-order Linear Fredholm integro-differential equations under the mixed conditions in terms of Legendre polynomials. The proposed method converts the equation and conditions to matrix equations, by means of collocation points on the interval [−1, 1], which corresponding to systems of linear algebraic equations with Legendre coefficients. Thus, by solving the matrix equation, Legendre coefficients and polynomial approach are obtained. Also examples that illustrate the pertinent features of the method are presented and by using the error analysis, the results are discussed.     

A new wavelet method for solving the Helmholtz equation with complex solution

The Helmholtz equation which is very important in a variety of applications, such as acoustic cavity and radiation wave, has been greatly considered in recent years. In this article, we propose a new efficient computational method based on the Legendre wavelets (LWs) expansion together with their operational matrices of integration and differentiation to solve this equation with complex solution. Because of the fact that both of the operational matrices of integration and differentiation are used in the proposed method, the boundary conditions are taken into account automatically. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifies the problems. As an applied example, “propagation of plane waves” is investigated to demonstrate the validity and applicability of the presented method. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 741–756, 2016     

一般Volterra-Stieltjes微积分方程的比较定理

In this paper, A. B, Mingarelli‘s result is generalized to General Volterra-Stieltjes Integro-differential Equations. Comparison theorem and equivalence condition of non-oscillation are obtained.Clasical Sturm comparison theorem and some conclusions are generalized.     

基于切比雪夫小波基与年龄相关种群模型的数值解

基于切比雪夫小波基给出与年龄相关种群模型的数值解.利用切比雪夫小波基的性质使得所求偏微分方程转化为矩阵方程,从而简化了数值解的求解过程.最后通过数值例子验证其理论结果.     

Error Analysis of Discrete Legendre Multi-Projection Methods for Nonlinear Fredholm Integral Equations

In this paper, polynomially-based discrete M-Galerkin and M-collocation methods are proposed to solve nonlinear Fredholm integral equation with a smooth kernel. Using su?ciently accurate numerical quadrature rule, we establish superconvergence results for the approximate and iterated approximate solutions of discrete Legendre M-Galerkin and M-collocation methods in both infinity and L²-norm. Numerical examples are presented to illustrate the theoretical results.     

二阶脉冲积微分方程初值问题

用不动点理论研究 Banach空间中定义在无穷区间上的二阶脉冲积微分方程初值问题 ,建立了方程(1 )的解的存在唯一性定理 .     

二阶非线性脉冲积分-微分方程的边值问题

本文利用单调迭代技术给出了Banach空间中含有非线性一阶微分项x′的二阶脉冲积分-微分方程边值问题存在最大最小解的充分条件;作为主要结论的应用,我们给出了一个无限系统的例子.     

Wavelets collocation methods for the numerical solution of elliptic BV problems

Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems.