本文探討熱傳導問題在初始值為未知時，以多項式函數展開假設未知的初始條件，並在某一特定量測點搭配數個不同時間求取展開式中各項係數值，進而求得問題之初始值。
本文將探討在圓柱座標系統中，量測點位置對於所求取之值的準確性、量測時間間距對於初始值預測的影響、並以中央差分法( Central difference method )計算結果模擬不同量測時間下所測得之溫度值，以及當具有量測誤差時所對應求取的初始值影響。結果顯示，當量測時間較短時，且所選取量測點的溫度隨時間變化較劇烈時，不論是否有量測誤差，其皆可獲得好的預測結果；而當量測時間較長時，但初始值為一平滑曲線時，可獲得良好結果，但若初始值為不連續函數或是在空間域內變化較劇烈時，所求得之值相對較差，故本文將此一情形配合數段多項式曲線即可求取良好結果。

The study discusses the heat conduction problem when the initial value is unknown. Using the polynomial expansion method to express unknown initial value, at a particular point, we take several measured temperature value at different times to solve the coefficients of the expansion in order to get the initial value.
The study investigates into the accuracy of estimation value analyzed at different measurement point in cylindrical coordinate, the effects of how long the time we analysis, estimates the initial value using the measurement value at different time, the results of the central difference method as a simulation temperature value and the impact of estimation of initial value with measurement error. The results show that, in short time, we can get a good estimation in a particular measurement points that the temperature change larger with time whether or not the measurement error; on the other hand, in long time, we can also get good results for the initial value is a smooth curve. If the initial value is not a smooth curve or changes rapidly, we would not get great results, and we use several polynomial expansions to improve our estimation. As a result, the inverse algorithm of the study presents a good accuracy.

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