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Please use this identifier to cite or link to this item: http://ntour.ntou.edu.tw:8080/ir/handle/987654321/13668

Title: 李群打靶法在非線性二階邊界值問題多解的計算
The Lie group shooting method on the multiple solutions of nonlinear second order boundary value problem
Authors: Shi-Min Zhuo
Contributors: NTOU:Department of Mechanical and Mechatronic Engineering
Keywords: 一步李群;邊界值問題;打靶法;丟失初始條件的判斷
One-step group preserving scheme;boundary value problem;shooting method;estimation of missing initial condition
Date: 2005
Issue Date: 2011-06-30T07:25:19Z
Abstract: 在工程與科學上,大部分的問題,都能由非線性常微分方程(ODEs)所描述。當要求滿足邊界條件被使用時,所產生的問題被稱為邊界值問題(BVPs)。當然地,邊界值問題的解必須滿足邊界條件,但是在許多例子上,當與邊界值問題的數值積分有關時,這將是一個困難的工作。 在此,本文提供一個李群打靶法,對於二階非線性邊界值問題(BVPs)利用在t=0與t=T的條件之下的數值積分。主要目的是能夠找到未知初始值而且很容易地找到所有的解。我們建構一個緊密空間打靶法去找出未知初始條件。關鍵點是基於一步李群元素 的建構與廣義中點李群元素 的建立。接著利用 = ,我們可以經由權重因子 的疊代解,去找出丟失初始條件。經過數個例子的測試,可看出方法不但有高效率而且準確度高,權重因子 的解,經由半疊代法更可以快速的達到收斂效果。甚至在廣泛的時間邊界座標之下,新的方法也可以適用,而且只須少數的疊代。 關鍵詞: 一步李群 ; 邊界值問題 ; 打靶法 ; 丟失初始條件的判斷。
Abstract There are a large number of problems in engineering and science that can be described by nonlinear ordinary differential equation (ODEs). When boundary condition are imposed, the resulting problems are referred to as boundary value problem (BVP). Naturally, the solution of BVPs have to satisfy the boundary condition, but in many case this can be a difficult task when one is concerned with the numerical integration of BVPs. We study the numerical integrations of second order boundary value problem (BVPs) under imposed conditions at t=0 and t=T. It aims to find all solution as easy as possible. We can construct a compact space shooting method for finding unknown initial conditions. The key point is based on the construction of a one-step Lie group element and the establishment of a generalized mid-point Lie group element . Then, by imposing = we can search the missing initial conditions through an iterative solution of the weighting factor . Numerical examples were examined to convince that the new approach has high efficiency and accuracy with a fast convergence speed by solving r with the half-interval method. Even under a large span of the boundary coordinate, the new method is also applicable by requiring only few iterations. The methods are also extended to the BVPs with general boundary conditions. Keyword: One-step group preserving scheme, boundary value problem, shooting method, estimation of missing initial condition.
URI: http://ethesys.lib.ntou.edu.tw/cdrfb3/record/#G0M93720004
Appears in Collections:[機械與機電工程學系] 博碩士論文

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