線性拋物偏微分方程之研究

Abstract

這篇論文中我們討論線性拋物偏微分方程。首先舉出關於此方程的一個實際例子。其次，我們介紹不同方法來解析此類型偏微分方程，包含一維度和多維度的問題。當我們用不同方法解決同一個方程時，將會證明這些方法所得之結果會是一致的。 還有，在數值分析方面，我們用有限差分法來求數值解。我們用數學軟體 Mathematica 6來展示用有限差分法所得的解以及探討實際解之趨近值，最後比較兩者之的差距。

We study the linear parabolic partial differential equations. First, we give a practical example whose mathematical model is a parabolic PDE. Next, we apply some classical methods to solve the linear parabolic PDEs in one and higher dimension. For the same equation, we will identify the solutions if they are derived by different methods. In numerical analysis, we use finite difference method to obtain the numerical solution of the parabolic PDE. We will use Mathematica 6 to find the numerical solutions, to approach the exact solutions, and to compare between those solutions.