以數值方法估計熱傳導反問題的解

Abstract

反問題是從測量的結果反推回起始原因，而熱傳導方程是描述熱量藉由傳導方式傳遞能量的過程，本篇探討的問題分為兩種，從某個特定位置的邊界條件反推回原點位置的邊界條件，以及從某個時間點的溫度分佈反推回初始條件。為了方便研究，先給定起始原因，找到某個時間的結果，主要目標就是從此結果反推回起始原因，得到的解可以和真正的起始原因作比較。 使用數值方法將熱傳導方程的解析解離散化，得到的線性系統是非適定性問題，由截斷奇異值分解方法可將其正規化，進行逆運算之後會得到解，若起始原因的狀態具有足夠的平滑性，則這個解的趨勢和真正的起始原因會很接近。

Inverse problems recover the causes from the effects. And heat conduction equations describe the process of energy transfer by heat conduction. In this thesis, we focus on calculating the boundary condition at the origin from the other boundary condition and recovering the initial condition from the temperature distribution at a certain time. For the convenience of research, the causes are given at first and the effects are found directly. The goal is to recover the causes from the effects. The results can be compared with the real causes. The analytical solutions of the heat conduction equations are discretized with some numerical methods. Then, the linear system obtained is an ill-posed problem. It can be regularized by the truncated singular value decomposition. If the state of the causes is smooth enough, the trend of the solution of the regularized problem with inverse operation is approaching to the real causes.