在第一章中,對於一種在矩形定義域上的常係數奇異擾動偏微分方程,我們使用分離變量法來構造出. 在此我們只考慮Dirichlet 的邊界條件,而其他種類的邊界條件可以很容易地類推得到. 根據我們所得到的結果,此解與其偏導數的行為可以很容易地觀察出來. 更進一步地,我們提出了一個可找到其真實解的模型,可用它來研究邊界層的行為以及用來測試數值方法的好壞. 因此, 這些解析解對於研究邊界層問題是很重要的.在第二章中,我們研究在Schiff et al. [20]論文中的模型. 它是一個定義在矩形定義域[-a,a]´ [0,b]的四階重調和問題,其邊界條件為夾擠(clamped)邊界條件.我們使用邊界近似法(BAM)求得其最準解,事實上,此種數值方法是譜方法(spectral method)的一種. 然而這個模型的收斂速度並沒有像一般的譜方法的收斂速度達到指數收斂(exponential convergence). 我們發現它收斂緩慢的原因是在定義域的兩個角點具有弱奇異(mild singularity). 我們也發現了基底函數與其偏導數的化簡公式,使用這些公式可構造出一些有用的模型,可用來測試數值方法的好壞. 藉由將lambda,a,b這三個參數化簡成只剩下一個參數t=b/a ,我們也研究了應力強度係數(stress intensity factor)與這三個參數之間的大小關係.

In the …rst chapter, separation of variables is used to derive the explicit particular solutions for a class of singularly perturbed di¤erential equations with constant coe¢ cients on a rectangular domain. Although only the Dirichlet boundary condition is taken into account; it can be similarly extended to other boundary conditions. Based on these results, the behavior of the solutions and their derivatives can be easily illustrated. Moreover, we have proposed a model with exact solution, which can be used to explore the behavior of layer and to test numerical methods. Hence, these analytic solutions are very important to the study in this …eld. In the second chapter, we study the model of Shi¤ et al. [20]. It is a biharmonic equation on the rectangular domain [¡ a; a]£ [0; b] with clamped boundary condition. We compute its most accurate numerical solution by boundary approximation method (BAM), which is a special version of spectral method or collocation method. Its convergence unfortunately is not as good as the usual spectral method with exponential decay rate. We discover that the slowdown is due to the very mild singularity at two corners not considered by BAM. We further simplify the basis functions and their partial
derivatives. Using these functions we can construct several models useful for testing numerical methods. We also explore how the stress intensity factor depends on the sizes of domain a and b, and the load ¸ by reducing the original problem with three parameters lambda, a, b to that with only one parameter t.

1. Exact Solutions for Singularly Perturbed Di¤erential Equations
with Constant Coe¢ cients in Rectangular Domains
1.1 Introduction
1.2 Basic pproaches
1.2.1 Partial Solutions
1.2.2 Particular Solutions for Satisfying the Zero Corner Conditions
1.2.3 Non-homogeneous Equations
1.3 Computational Model
2. Biharmonic Boundary Value Problems with Singularity
2.1 Introduction
2.2 Boundary Approximation Method
2.3 Testing Models
2.3.1 Singular Models
2.3.2 Analytic Models
2.4 Schi¤’s Model
2.5 Varied Domains

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