本論文中，利用Laplace Adomian分解法(LADM)討論非線性熱傳和樑的結構之工程問題，藉由LADM分析有時變邊界條件之環形散熱片問題，並求初期非線性之溫度及熱應力分佈情況，其中環形散熱片根部受到週期性溫度變化，並考慮熱輻射效應，且對流係數為與溫度相關之函數。提出的解決方法簡單且有效系統獲得環形散熱片根部的圓周應力及其根部疲勞分析的解決。
本文利用Adomian方法來求取樑在強非線性彈性基底上時之靜態撓曲的解析解，如果外力之函數是可析函數(Analytic function)，則推導所得的樑之非線性撓曲可以馬克勞林級數(Maclaurin series)表示之，同時亦推導出此馬克勞林級數之係數間的遞迴關係式。結果顯示，所提出的方法是精確有效的，有效的成功應用於強非線性問題，結果獲得與擾動法(perturbation)做比較。發現擾動法解的誤差將隨之非線性係數之增加而增加，同時亦將隨著外力的增加而增加。

The LADM in the nonlinear heat transfer and beam structure can be widely used in engineering.
The purpose of this thesis is to propose the Laplace Adomian Decomposition Method (LADM) for studying the nonlinear temperature and thermal stress analysis of annular fins with time dependent boundary condition.
The nonlinear behavior of temperature and thermal stress distribution in an annular fin with rectangular profile subjected to time dependent periodic temperature variations at the root is studied by the LADM. The radiation effect is considered. The convective heat transfer coefficient is considered as a temperature function.
The proposed solution method is helpful in overcoming the computational bottleneck commonly encountered in industry and in academia. The results show that the circumferential stress at the root of the fin will be important in the fatigue analysis.
This study presents an effective solution method to analyze the nonlinear behavior of temperature and thermal stress distribution in an annular fin with rectangular profile subjected to time dependent periodic temperature variations at the root by using LADM.
However, the analytic static deflection solutions of beams resting on nonlinear elastic foundations are developed by the modified Adomian method. If the applied force function is an analytic function, then the deflection function can be derived and expressed in Maclaurin series. A recurrence relation for the coefficients of the Maclaurin series is derived. It is shown that the proposed solution method is accurate and efficient. The solution method can be successfully applied to the problem with strong nonlinearity. The results are also compared with those obtained by the perturbation method. It is found that the error of the perturbation solution will increase not only when the nonlinear parameter is increased but also when the applied load is increased.

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