小波分析用于求解微分方程数值解的研究

【作者】 周涛；

【导师】 袁修贵；

【作者基本信息】 中南大学 ， 计算数学， 2007， 硕士

【摘要】 小波分析是当前应用数学中一个迅速发展的新领域，由于小波兼有光滑性和局部紧支撑性质，和传统的有限元、有限差分方法比较，能够更好的处理局部存在奇异性的问题，目前越来越多的应用在偏微分方程数值求解中。本文主要研究了微分算子的小波多分辨表示，并以热传导方程为模型，研究了偏微分方程数值求解。小波求解微分方程的实质就是将方程由原来的坐标系转化到小波系下求解，Leland Jameson给出了一阶导数算子在尺度j=1时的多分辨展开，本文在此基础上，进一步研究了一阶和二阶导数算子的在尺度j=1，2时的小波展开，证明了一阶、二阶导数算子小波域的表示形式，并给出了具体的展开式以及系数的计算。由导数算子的小波展开理论，本文对具有奇异性的热传导方程的求解建立了两种格式。第一种，对于尺度函数空间V_j，利用具有显式表达式的拟Shannon尺度函数，构造基函数，建立热传导方程数值求解格式；第二种，在多分辨空间V_j+1=V_j⊕W_j（j∈z）上，利用本文推导的导数算子小波展开的结论，建立小波数值求解的离散格式，证明了偏微分方程小波域的等价形式。最后，通过数值算例分析，表明小波解不仅精度比传统的有限差分方法要高，而且在间断点附近没有发生解的振荡现象，能更好的逼近真解。更多还原

【Abstract】 Wavelet analysis is a new realm in the applied mathematics with rapid development, because the wavelets have the smooth and local compact property, compared with traditional finite element method and finite difference method, it is a more useful method for the question with local singularity, so, wavelet analysis now more and more applied into the numerical solution of partial deference equations. In this paper, the derivative operator’s expression by wavelets and numerical solution of partial difference equations that based on heat-conduction equation were investigated.Wavelet analysis on difference equations’ substance is to put the equation into wavelet domain. Leland Jameson has given one-order derivative operator’s expression on the wavelets at scale j=1, based on that, one-order and two-order derivative operator’s expression at scale j = 1,2 was given and one-order and two-order derivative operator’s expression on the wavelet domain was proved in this paper.By the theory of derivative operator’s expression on wavelets domain, this paper discuss two kinds of computer formats to heat-conduction equation. The first, on the scaling function space V_j, using like-Shannon wavelet to construct basis functions , the numerical algorithm to solve heat-conduction equation was established; The second, on the multi-resolution space V_j+1 = V_j+ W_j（j∈z）, using the theory of derivative operator’s expression on wavelets domain deduced by this paper, the numerical algorithm to solve heat-conduction equation was established. Finally , compared with finite difference method, the computation results show that wavelet method’s accuracy is more higher, and do not have the oscillation phenomenon.更多还原