樑柱撓曲行為力學中的邊界值求解問題與初始值與邊界值的交界不連續的熱傳問題長久以來一直是數值分析研究的重要課題。關於樑柱撓曲行為力學中的邊界值求解問題，以往發展出來的數值方法往往由於需要大量繁瑣的計算，或由於過多符號運算變得複雜而冗長。對於求解熱傳問題時，常常因為初始值與邊界值的交界不連續，使得求解相當困難。

　　本論文主要是以微分變換法為基礎，分別對此兩類問題進行求解。微分變換是一種以泰勒級數展開為數學基礎的函數轉換，它能將線性及非線性微分方程轉換成代數方程進行求解，具系統化處理非線性問題，快速收斂的特性，並可將工程問題的解，以k階泰勒級數展開之型式來呈現，在計算的效率與精確度上微分變換法都有相當優越的表現。

　　微分變換法處理樑柱撓曲行為力學中的邊界值問題時，先化為初始值問題再符合邊界條件求出未知數，計算過程相當簡單；對於求解熱傳問題，本文使用近似的連續函數來取代不連續之邊界與初始條件，再以微分變換處理，並深入探討區間連接的問題。

　　The buckling behavior of nonlinear beams and the heat conduction boundary value problems are focal studies in engineering problems.

　　In this dissertation, the differential transformation method with a systematic procedure is applied to solve nonlinear two-point boundary value problems in solid mechanics and heat conduction problem. In solid mechanic problem, three types of nonlinear boundary value problems are considered in this study, i.e., the deflective of a tip-loaded nonlinear cantilever beam, the nonlinear buckling of a nonlinear cantilever beam without and with imperfection, and the nonlinear buckling of a simply supported beam without and with imperfection. The one-dimensional heat conduction problem considered in this dissertation is to study the effect of discontinuity on the resulting solution.

　　In the initial value problems, the differential transformation method can provide a fixed step and adaptive step strategy to improve the computational efficiency. In this study, a boundary value problem can be transformed to an initial value problem with undetermined parameters and it can be solved in a similar manner as an initial value problem. The simulation results show that the solutions obtained by differential transformation are accurate in comparison with exact solutions, and the computational stability and efficiency are also improved with the proposed procedure.

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