樑柱的問題對工程實用上或理論上都是非常重要的課題，本論文研究此類的非線性四階常微分方程，搭配非線性的邊界條件下，其解的存在唯一性。我們並將設計不同的算法，透過微分方程、積分方程或優化，來計算其數值解，而每一類方法還可再細分成差分、譜方法等。最後將比較各種算法得知其優劣。

Beam problem is very important for engineering theoretically and practically. In this thesis we study such kind of nonlinear 4-th order ordiniary differential equations with nonlinear boundary conditions. The well-posedness of these boundary value problems will be discussed. Moreover, we will design different schemes to solve them, through differential equation, integral equation or minimization. Each type can further be discretized by finite difference, finite element or spectral method, etc. In the end we will compare all methods and find the best one.

1. Methods for Differential Equation..........2
1.1 Introduction............................2
1.2 Finite Difference Method................4
1.2.1 Mathematica Solver................7
1.2.2 Fixed Point Method................9
1.2.3 JOR and SOR Methods...............11
1.2.4 Newton's Method...................15
1.3 Weighted Residual Method................21
2. Minimization Method........................25
2.1 Introduction............................25
2.2 Finite Difference Method................27
2.2.1 Mathematica Solver................29
2.2.2 Newton's Method...................30
2.3 Weighted Residual Method................32
3. Methods for Differential Equation..........35
3.1 Introduction............................35
3.2 Existence and Uniqueness................37
3.3 Picard's Iteration......................48
3.4 Finite Difference Method................58
3.5 Collocation Method......................63
3.6 Weighted Residual Method................65
3.7 Conclusion..............................67

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