本文探討熱傳導問題在初始值為未知時，以特徵函數展開法（Eigenfunction expansion method）假設未知的初始條件，在某一特定時間下配合量測點溫度值以辛普生法（Simpson’s rule）求取展開式中各項係數值，進而求得問題之初始值。
本文將探討在量測點數量對於所求取之值的準確性、量測時間長短對於初始值預測的影響、以不同量測時間下所測得之溫度值做初始值預測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，則當量測點到達某一數量之後，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，當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個短時間量測值即可求取良好結果。

The study discusses the heat conduction problem when the initial value is unknown. Using the eigenfuction expansion method to express unknown initial value, at a particular time, we take the measured temperature value into Simpson’s rule to solve the coefficients of the expansion in order to get the initial value.
The study investigates into the accuracy of estimation value analyzed in different measurement point numbers, the effects of how long the time we analysis, estimates the initial value using the measurement value in different time, 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 number of measurement points 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 short-time-measured values to improve our estimation. As a result, the inverse algorithm of the study presents a good accuracy.

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