【作者】 邢佳；

【导师】 邓彩霞；

【作者基本信息】 哈尔滨理工大学 ， 应用数学， 2005， 硕士

【摘要】 小波分析理论和再生核理论都是数学的重要分支。在自然界中许多物理现象都可以用微分方程来描述,一般微分方程没有解析解,所以讨论方程的数值解就显得尤为重要。本文分别应用小波分析理论和再生核理论对微分方程的求解和解空间的问题进行了研究。主要做了以下工作: 一方面,本文应用小波分析理论结合Galerkin 方法讨论了某一类二阶变系数微分方程的求解问题。首先,将Littlewood-paley 小波基引入到Galerkin 方法中。对于L 2 (R)上的Littlewood-paley 正交小波基进行“折叠”映射, 使得折叠小波基为L 2[0,1]的标准正交基。其次, 证明Littlewood-paley 折叠小波基满足该微分方程的边值条件。最后,运用Galerkin 方法在小波子空间中得到微分方程的数值解。本文利用Wavelet-Galerkin 方法求解微分方程,这为讨论微分方程的数值解问题提出了新的研究思路。另一方面,本文又应用再生核空间理论的特殊技巧,讨论了波动方程解空间的问题。首先,针对二阶波动方程的解,构造再生核。其次,证明了二阶波动方程的解空间构成再生核空间H 1 [0,+∞)。然后,证明这个二阶波动方程的解具有反演公式及等距恒等式。最后,讨论了二阶波动方程在更一般的边值条件下的解空间,证明了该方程的解空间也为一个再生核空间。有趣的是,这两个再生核空间的再生核具有相关性。本文的研究拓宽了再生核理论的适用范围,为波动方程的求解问题提供了新的框架。更多还原

【Abstract】 The wavelet analysis theory and reproducing kernel theory are important branch of mathematics. In nature, many physical phenomena can be described by differential equations, but the general differential equations have no exact solutions, therefore it is very important to find the numerical solution of the equation. In this thesis, it is the main problem to investigate that the numerical solution and solution space of the different differential equations by making use of the theory of the Wavelet-Galerkin method and the reproducing kernel Hilbert space, respectly. The main works are as following: On the one hand, using the wavelet analysis theory and the Galerkin methods to solve a kind of second-order variable coefficient differential equation is discussed. Firstly, the Littlewood-paley wavelet basis, that is an orthonormal basis of the space L2 (R), is selected as a basis in the Galerkin methods. Through the “folded”transform, the “folded”orthonormal wavelet basis for the space L2 [0,1] is got. Secondly, it is proved that the “folded”orthonormal wavelet basis satisfies the boundary condition of the equation. In the end, it is given that the Galerkin numerical solution of the equation in the wavelet subspace, and the numerical solution of the equation is also got. Wavelet-Galerkin method is applied to solve the differential equations, and this provides not only the theoretic bases for solving the differential equations, but also a new method to extand applications for the wavelets analysis theory . On the other hand,the reproducing kernel space of the wave equation is investigated in this thesis. Firstly, the reproducing kernel of the solution space of the second-order wave equation is gotten. Secondly, it is proved that the solution space of second-order wave equations can form the reproducing kernel Hilbert space H 1 [0,+∞). Then we prove the isometrical identity and inversion formula of the solution of the wave equation. Besides, the solution of another wave equation is discussed by the same way. The solution space is another RKHS H (C). It is very beautiful that the difference of two kernels is constant. It not only explores a novel application of the reproducing kernel Hilbert space theory, but also provides a new view on solving the differential equation.更多还原