臺灣博碩士論文加值系統

摘 要
本文將利用五點有限差分法、切線預測法、牛頓迭代法、隱函數定理、虛擬弧長延拓法及有限維度 Liapunov-Schmidt 降階法等數值方法，來獲得非線性橢圓方程組之解路徑。其中，虛擬弧長延拓法能順利通過正則點、轉彎點及分歧點。本文之主要目的在於探討兩個非線性橢圓方程組多重解的存在，並對其解路徑之分歧加以分析。

Abstract
In this paper, we use the 5-point finite difference method, tangent predictor, Newton iterative method, implicit function theorem, pseudo-arclength continuation method and finite dimensional Liapunov-Schmidt reduction method to obtain the solution path of the nonlinear system of elliptic equations. The pseudo-arclength continuation can pass through regular points, turning points and bifurcation points. We investigate the existence of multiple solutions of two nonlinear systems of elliptic equations and analyze the behavior of branches of the solution path.

Contents
1. Introduction 1
2. Bifuraction theory and continuation method 5
2.1 Basic definitions and theorems ……………………………………………… 5
2.2 Local continuation methed ………………………………………………… 11
2.2.1 Predictor ……………………………………………………………… 11
2.2.2 Solver ………………………………………………………………… 12
2.3 Parameterization of solution arcs ………………………………………… 13
2.4 Bifurcation at simple eigenvalue ………………………………………… 15
3. Numerical computation for bifurcation problem 22
3.1 Finite difference method …………………………………………………… 22
3.2 Pseudo-arclength continuation method …………………………………… 25
3.2.1 Tangent predictor ……………………………………………………… 25
3.2.2 Newton’s iterative solver ……………………………………………… 28
3.3 Switching branch at bifurcation point……………………………………… 29
3.3.1 Liapunov-Schmidt reduction method ………………………………… 29
3.3.2 Direction of branches and selecting starting points…………………… 31
3.4 Algorithm ………………………………………………………………… 35
4. Numerical result 39
4.1 Numerical Experiments for Model A ………………………………………39
4.2 Numerical Experiments for Model B ………………………………………58
5. Conclusion 77
Reference 78

Reference
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