本文主要探討混合拉氏轉換法求解傅立葉和非傅立葉熱傳導問題以及傅立葉和非傅立葉相變化問題。在求解傅立葉熱傳導問題時，本文使用混合拉氏轉換法和有限差分法求解，並與解析解作比較。由計算結果可知，混合拉氏轉換法之累加誤差相對有限差分法明顯較小，其準確性相對較佳。在非傅立葉熱傳導問題方面，本文主要討論熱波模式(CV wave model)，並使用拉氏轉換法搭配線性形狀函數、有限差分法、控制體積法等三種方法求解不同座標(直角、圓柱、球座標)下之各種熱傳問題，並探討這些問題之物理現象。其中，控制體積法相對於線性形狀函數和有限差分法，可最有效克服數值不穩定現象，其準確性相對最佳。對於非傅立葉相變化熱傳問題，本文主要探討非傅立葉史蒂芬問題，在非傅立葉相變化問題的相關研究方面，目前尚未有學者使用混合拉氏轉換法搭配單區法求解非傅立葉相變化問題，本文則使用拉氏轉換法以及有限差分法搭配求取潛熱的數種單區法求解相變化問題，可成功克服非傅立葉相變化問題會產生震盪現象，並預測其物理趨勢。

The main focus of this thesis is to use the hybrid Laplace transformation method to investigate the Fourier and non-Fourier linear and nonlinear (or phase-change) heat transfer problems. In studying Fourier heat transfer problems, the hybrid Laplace transformation and finite difference methods are utilized and the numerical solutions are compared with the exact ones. From the results, the accumulated errors of the Laplace transformation are significantly smaller than those of the finite difference method. In analyzing non-Fourier problems, the CV wave model is applied and the Laplace transformation combined with linear shape function, finite difference method and control volume scheme is used to solve the heat transfer problems of different coordinate systems (Cartesian, cylindrical and spherical coordinate systems). From the analysis, the control volume scheme could conquer the numerically unstable problem most effectively and would have the most accurate solutions. In investigating the non-Fourier phase-change problem, the non-Fourier Stefan problem is studied primarily. The hybrid Laplace transformation method combined with the finite difference method and the effective specific and enthalpy scheme is employed to solve the problem. From the computing results, it can be found that the proposed method could overcome the oscillating phenomenon presented in other methods.

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