| Summary: | 博士 === 國立成功大學 === 化學工程研究所 === 83 === Daubechies小波函數為一種新創的連續函數，其具有正交化、緊緻支撐及
良好的頻域與時域的局部定位解析能力，在應用數學及工程分析有極大的
應用潛力深受重視。在工程分析上常將之與Galerkin數值法聯用，以其當
作Galerkin 數值法的基底函數，藉以求得常微分與偏微分方程式的數值
解。然而，在應用wavelet-Galerkin解析法時，其必須先求得其聯結係數
之值，由於Daubechies小波缺乏明確的數學表示式，且函數波形起伏分佈
，傳統的數值積分法並不易求得精確且穩定的聯結係數值。雖然文獻上已
探討如何求得精確的聯結係數值，但其所求的聯結係數的積分範圍均為無
限的，此舉對於有限區間的微分方程式或是邊界條件為有限且非週期性的
問題並不適用。本論文針對此一問題提出解決之法，該法乃應用建造小波
的尺度函數之二尺度關係式與慣量關係式，可推導出積分區間為有限的聯
結係數之精確值的演算法。因Daubechies小波函數皆可以尺度函數的線性
組合來表示，而且，藉由參數變換法的應用，我們可將不同解析度的聯結
係數轉換成零階聯結係數的線性組合。故在本論文中，我們僅推導相關於
尺度函數的零階聯結係數。利用這些聯結係數不僅可求解定義域為無限的
微分方程式，而且可以求解定義域為有限且非週期性邊界條件的問題，在
本論文中我們也舉出實際應用的範例，由這些範例中驗證小波解析法的優
越性。在這些範例中，包含求解帶有尺度因子的群數平衡方程式、具可移
動邊界的Stefan問題，及具有急劇變化的Burgers方程式。上述的範例皆
為一般數值法不易求得精確解的微分方程式，而應用小波-Galerkin數值
法不僅可獲得較準確的數值解，而且在精確度相近時，其所需的展開係數
項數較其他的正交展開係數法來的少。在應用上，小波解析法可輕易的與
其他數學程式庫的副程式相鏈結使用，更增加其使用上的方便性。 (略)
The class of Daubechies'' wavelets is a class of continuous
function with the elegant properties of orthonormal, compact
support and time--frequency localization. It has been widely
used as the bases of Galerkin method to the solution of
ordinary and partial differential equations. In the application
of wavelet--Galerkin method, it is essential to calculate the
connection coefficients of the wavelet bases, which involve
integrals of products of wavelet functions and their
derivatives and/or integrals. Since there are no closed form
for expressing Daubechies wavelets and the derivatives of
wavelets vary severely, it is hardly to find the precise and
stable values for the wavelet connection coefficients by
traditional integration methods. In the literature, efforts
have been devoted to the computation of connection coefficients
of the wavelet bases over the infinite domain. The main purpose
of this dissertation is to extend the application scope of the
wavelet--Galerkin method to the finite--domain problems. The
extension of the wavelet--Galerkin method to solving a finite--
domain problem relies on the evaluation of wavelet connection
coefficients over a bounded interval. Aiming directly at this
requirement, this dissertation applies the two-scale relation
and the moment relation of the Daubechies wavelets to derive
algorithms for computing the exact values of the connection
coefficients. Having obtained finite-domain wavelet connection
coefficients, this dissertation applies the wavelet-Galerkin
method to solve the population balance equations with a scaled
factor of 2, the Stefan problem with moving boundary condition,
the shock wave of Burgers equation. The results of wavelet-
Galerkin solution are also compared with those obtained by
other numerical methods. The elegant localization property of
the wavelet analysis is clearly demonstrated from the
comparison of the results. (omit)
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