【作者】 权豫西；

【导师】 石智；

【作者基本信息】 西安建筑科技大学 ， 应用数学， 2008， 硕士

【摘要】 随着计算机技术的发展和完善以及实际应用的需要,科学和工程中的所有计算问题与计算机已经密不可分.弹性力学中的微分方程,由于其阶数较高,而且很多方程还无法找到其精确解,所以关于此类方程的数值解在工程中就显得尤为重要.小波分析作为数值分析的一种强有力的工具是二十世纪七十年代发展起来的一门新兴的数学学科,现在已广泛地应用到求解微分方程和积分方程中.本文对弹性力学中的微分方程的小波方法进行研究,得到了一系列结果.全文共分三章,每章的主要内容如下:第一章简述本文所用小波分析的一些基础知识以及弹性力学的基本理论,对目前小波分析在微分方程中的应用现状进行探讨,提出应用正交小波基对微分方程进行求解的可行性.第二章研究求解弹性地基梁的小波方法,主要分为四个部分进行说明:Winkler地基上梁的微分控制方程;线性Legendre多小波的定义及其性质和积分矩阵;用线性Legendre多小波求解Winkler地基上梁的方程;数值举例.第三章研究求解弹性地基板的小波解法,主要分为五个部分:双参数弹性地基上板的微分控制方程:一维Sine-Cosine小波的定义及其微分矩阵;利用张量积构造二维的Sine-Cosine小波基并推导出其相应的两个微分矩阵;用Sine-Cosine小波求解双参数弹性地基上四边自由矩形板的挠度问题;数值举例.更多还原

【Abstract】 With the development of computer technology and the requirements of practical application, all the problems arising in science and engineering field are closed related with computers. Due to the high rank of the differential equations in elastic mechanics and the difficulties of finding accurate solutions for these equations, the numerical solutions to these equations are especially important in engineering. As a powerful tool of numerical analysis, wavelet analysis was developed from 1970s and has been widely used to solve differential equations and integral equations. This thesis is focused on the research of wavelet methods to solve differential equations in elastic mechanics and a series of results are obtained. It is divided into 3 chapters and the main ideas of each chapter are as follows:In Chapter 1, some general knowledge of wavelet analysis and basic theories of elastic mechanics are briefly discussed, the current applications of wavelet analysis in differential equations are investigated, and the feasibility of using orthogonal wavelets to solve differential equations is also proposed.In Chapter 2, the wavelet method for solving problems of beams on elastic groundsill is investigated, and four aspects are illustrated: differential control equation of a beam on Winkler groundsill; the definition and property and integral operational matrix of the linear Legendre multi-wavelet; the numerical solutions to the differential control equation of a beam on Winkler groundsill solved by linear Legendre multi-wavelet; numerical examples.In Chapter 3. the wavelet solutions to problems of laminas on elastic groundsill is researched and is divided into five sections: differential control equation of a plate on the two-parameter elastic groundsill; the definition and differential operational matrix of the Sine-Cosine wavelet: the construction of the two-dimensional Sine-Cosine wavelet using tensor product and two differential operational matrices; the numerical solutions to the differential equation of a rectangular plate on the two-parameter elastic groundsill solved by two-dimensional Sine-Cosine wavelet; numerical example.更多还原