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.     

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.     

求解非线性Black-Scholes模型的自适应小波精细积分法

针对非线性Black-Scholes方程,基于quasi-Shannon小波函数给出了一种求解非线性偏微分方程的自适应多尺度小波精细积分法.该方法首先利用插值小波理论构造了用于逼近连续函数的多尺度小波插值算子,利用该算子可以将非线性Black-Scholes方程自适应离散为非线性常微分方程组;然后将用于求解常微分方程组的精细积分法和小波变换的动态过程相结合,并利用非线性处理技术(如同伦分析技术)可有效求解非线性Black-Scholes方程.数值结果表明了该方法在数值精度和计算效率方面的优越性.     

二维刚性边值问题的小波-吉尔解法

本文利用微分方程数值解的离散小波表示，讨论了此类方程在满足一定初始条件和边值条件下，在一个方向上利用小波伽辽金法，另一方向上利用吉尔方法进行求解，提出了一种解二维刚性初，边值问题的小波数值算法，计算结果表明，利用该方法所求得的数值解精度高，而且由小波特有的性质，它特别适用于求解带有奇异摄动的刚性问题。     

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.     

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．     

一类具有弱奇异核的偏积分微分方程的Chebyshev小波数值方法（英文）

本文提出一种基于第四类Chebyshev小波配置法,求解了一类具有弱奇异核的偏积分微分方程数值解.利用第四类移位Chebyshev多项式,在Riemann-Liouville分数阶积分意义下,导出Chebyshev的分数次积分公式.通过利用分数次积分公式和二维的第四类Chebyshev小波结合配置法,将具有弱奇异核的偏积分微分方程转化为代数方程组求解.给出了第四类Chebyshev小波的收敛性分析.数值例子证明了本文方法的有效性.     

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.     

用小波—配点法求解一类有奇异性的微分方程

本文用一个含小参数ε的微分方程作为算例，用小波－配点法进行求解，通过不断地对Ｖ空间进行小波的正交分解来提高分辨率，并利用小波的局部化特征，得到了很好的结果。不但看出微分方程的奇异性的存在，而且给出了奇点的位置，也求出了方程的数值解。     

变系数线性微分方程初值问题数值解的小波方法

通过利用小波尺度函数的正交性并结合配点法 ,本文给出了一种求解变系数线性微分方程初值问题数值解的小波算法 .在一定的假设条件下 ,对算法的收敛性进行了理论分析 .最后 ,我们还给出了一个具体的数值计算例子 .     

在有限差分法下一类抛物型方程的离散化

本文对于一类带有狄拉克函数δ０初值的抛物型方程，在有限差分法下进行离散化，证明了其数值解的存在性、唯一性，尤其是它的稳定性．     

On the cardinalities of finite topologies

Following the ideas of Sharp [2,3], we will give a partial answer to the question: “Let k be an integer, k ? 2. What is the smallest integer m for which there is a topology on m points with k open sets.” We state several results in the theory of finite topologies by introducing the idea of generating topologies. Using this concept, it is possible to derive existence theorems and get numerical results in an easy manner.     

An Algebraic Perspective on Multivariate Tight Wavelet Frames

Recent advances in real algebraic geometry and in the theory of polynomial optimization are applied to answer some open questions in the theory of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) are given in terms of Hermitian sums of squares of certain nonnegative Laurent polynomials and in terms of semidefinite programming. These formulations merge recent advances in real algebraic geometry and wavelet frame theory and lead to an affirmative answer to the long-standing open question of the existence of tight wavelet frames in dimension d=2. They also provide, for every d, efficient numerical methods for checking the existence of tight wavelet frames and for their construction. A class of counterexamples in dimension d=3 show that, in general, the so-called sub-QMF condition is not sufficient for the existence of tight wavelet frames. Stronger sufficient conditions for determining the existence of tight wavelet frames in dimension d≥3 are derived. The results are illustrated on several examples.     

最小支集样条小波有限元

本文认真分析研究了最小支集样条小波及其有关性质,用以张量积形式构造的二维小波建立了最小支集样条小波插值函数,讨论了其相关的性质,随后用最小支集样条小波有限元法去解弹性薄板小挠度问题,给出了数值解的误差阶,最后列举了一个数值例子．     

具有紧支撑的非张量积形式二维小波有限元

分析论述了构造非张量积形式二维Daubechies小波的几条定理,在此基础上着重构造了具有紧支撑的非张量积形式二维小波,随后用具有紧支撑的非张量积二维小波有限元去解弹性薄板挠度问题,给出了误差阶,最后列举了一个数值例子．     

美式看跌期权定价中的小波方法

本文采用有限差分格式和 Daubechies正交小波 ,提出了一种求解 Black- Scholes方程数值解新算法 .为美式看跌期定价提供了一条新的途径 .利用小波基的自适应性和消失矩特性 ,使偏微分算子矩阵和小波级数稀疏化 ,大大减少了计算量 .     

一类基于小波基函数插值的有限元方法

在分析具有大的梯度问题中,将具有紧支集的小波基函数引入到传统的有限元插值函数的构造中,对传统的插值方法进行修正。对新的插值模式进行了数值稳定性（解的唯一存在性）分析并通过分片分析讨论了解的收敛性,新的插值模式所引入的附加自由度通过静力凝聚法来消除,最后得到了基于变分原理的小波有限元列式。     

Numerical Method for Singularly Perturbed Third Order Ordinary Differential Equations of Convection-Diffusion Type

In this paper, we have proposed a numerical method for Singularly Perturbed Boundary Value Problems (SPBVPs) of convection-diffusion type of third order Ordinary Differential Equations (ODEs) in which the SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. The numerical method combines boundary value technique, asymptotic expansion approximation, shooting method and finite difference scheme. In order to get a numerical solution for the derivative of the solution, the domain is divided into two regions namely inner region and outer region. The shooting method is applied to the inner region while standard finite difference scheme (FD) is applied for the outer region. Necessary error estimates are derived for the method. Computational efficiency and accuracy are verified through numerical examples. The method is easy to implement and suitable for parallel computing.     

An effective approach for numerical solution of linear and nonlinear singular boundary value problems

In this study, an effective approach is presented to obtain a numerical solution of linear and nonlinear singular boundary value problems. The proposed method is constructed by combining reproducing kernel and Legendre polynomials. Legendre basis functions are used to get the kernel function, and then the approximate solution is obtained as a finite series sum. Comparison of numerical results is made with the results obtained by other methods available in the literature. Furthermore, efficiency and accuracy of the method are demonstrated in tabulated results and plotted graphs. The numerical outcomes demonstrate that our method is very effective, applicable, and convenient.     

A dynamic multiscale lifting computation method using Daubechies wavelet

An important property of wavelet multiresolution analysis is the capability to represent functions in a dynamic multiscale manner, so the solution in the wavelet domain enables a hierarchical approximation to the exact solution. The typical problem that arises when using Daubechies wavelets in numerical analysis, especially in finite element analysis, is how to calculate the connection coefficients, an integral of products of wavelet scaling functions or derivative operators associated with these. The method to calculate multiscale connection coefficients for stiffness matrices and load vectors is presented for the first time. And the algorithm of multiscale lifting computation is developed. The numerical examples are given to verify the effectiveness of such a method.