臺灣博碩士論文加值系統

本論文主要在探討一個非線性分布參數系統穩定解的分歧問題.在分歧理論及隱函數定理的基礎下,我們利用切線猜測法,割線猜測法,牛頓迭代法及虛擬弧長延拓法…等數值方法,計算分歧點與轉彎點,並延拓出所有通過分歧點或轉彎點的解分支.
最後,我們藉由改變參數,觀察分歧點及轉彎點的變化情形,並探討解路徑圖上的分歧現象,以了解分布參數系統多重穩定解的定性問題.

The main purpose of this thesis is to investigate bifurcation problems of steady-state solutions in a distributed-parameter system. With the bifurcation theory and implicit function theorem, we use the tangent predictor, secant predictor, Newton’s iterative method, and pseudo- arclength continuation method etc. We calculate the bifurcation points or turning points, and get solution paths which pass through bifurcation points or turning points.
Finally, we observe the variation of bifurcation points and turning points by changing parameters. We also discuss bifurcation phenomenons on solution diagrams to investigate the qualitative problems of multiple steady-state solutions in the distributed-parameter system.

第一章 緒論 …………………………………………………………… 1.
第二章 基礎理論 ……………………………………………………… 3.
2.1 分布參數系統…………………………………………… 3.
2.2 分歧問題………………………………………………… 4.
2.3 隱函數定理與分歧理論………………………………… 7.
2.4 延拓法…………………………………………………… 8.
第三章 分布參數系統平衡解的數值方法 ………………………… 13.
3.1 分布參數系統穩定解的分歧問題 …………………… 13.
3.2 分歧點解分支方向的選取 …………………………… 26.
3.2.1 Liapunov-schmidt降階法 ……………………… 26.
3.2.2 選取解分支延拓方向 …………………………… 30.
3.3 解分支延拓之數值計算 ……………………………… 35.
3.4 演算法 ………………………………………………… 37.
3.4.1 演算法：分布參數系統穩定解之初始值和分歧點
與轉彎點 ………………………………………… 37.
3.4.2 演算法：選取過分歧點的解分支延拓方向與該方向
之起始點 ………………………………………… 38.
3.4.3 演算法：解分支之延拓法 ……………………… 40.
第四章 數值實驗 …………………………………………………… 42.
4.1 實驗（一）……………………………………………… 44.
4.2 實驗（二）……………………………………………… 60.
4.3 實驗（三）……………………………………………… 68.
4.4 實驗（四）……………………………………………… 76.
第五章 結論 ………………………………………………………… 97.
參考文獻 ……………………………………………………………… 99.

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