小波分析在常微分方程求解中的应用

【作者】 高静；

【导师】 张书玲；

【作者基本信息】 西北大学 ， 计算数学， 2004， 硕士

【摘要】 小波分析是一门新兴理论，它被广泛地应用于各个领域。作为80年代末期出现的时频分析工具，小波变换在信号与图像处理等领域里已经得到了成功的应用，并凭借其自身的诸多优点成为了JPEG2000的标准，基于多尺度分析的尺度函数和小波函数很好的分析特性和计算特性，充分利用这些特性以小波作为基函数离散微分方程，再来求解所得到的代数方程，这种小波有限元法最近得到人们的很大的关注。 本文在详细论述了M带多小波理论的基础上，研究了小波分析方法在常微分方程求解中的应用，讨论了常微分方程在小波基下离散成代数方程后，提出最佳M 项逼近的思想，并结合小波基的特性，对求解代数方程组的迭代法进行了改进，使得计算量大大减少，特别有利于求解大型方程组，从理论上分析了这一改进算法的可行性，并用数值实验证明了其合理性。更多还原

【Abstract】 Wavelet analysis theory is a arisen science which was applied extensively to every domain. Being a time-frequency analysis tool in 1980s’, wavelet transform has succeeded to be applied to the signal and image processing domain and been the criterion of JPEG 2000 by it’s advantages. Based on Multiresolution Analysis, scale functions and wavelet functions have good analysis and computation characteristic which are been made the best of to discrete the differential equations. Then some algebra equations can be acquired. The method is named of wavelet finite element method has attracted many scholars’ interests.In this paper, after wavelet analysis theory being discussed clearly, the multiresolution analysis of M-band multiwavelet theory is introduced which contributed to multiresolution filter banks. The application of wavelet used to the solution of ordinary differential equations (ODE) and a modified algorithm which resolves ODE is proposed. After ODE being a discrete algebra equation systems by wavelet, the best M term approximation idea is combined with the wavelet characteristics. And the iterate method of equation systems is modified. The computation is reduced, the improved algorithm contributes to the resolution of large equation systems. The feasibility and reasonability are proved by theory analysis and numerical experiments.更多还原