線性橢圓偏微分方程之研究

Abstract

本文的目標是探討一些常用於解決線性橢圓偏微分方程的古典方法及應用。首先，我們給一個有關於在靜電勢中Laplace方程的實際例子並利用有限元素法解之。再來介紹常用的古典解題技巧，像是在不同定義域中分離變數法的使用以及有限與無限空間的傅立葉轉換。最後我們介紹數值方法中的有限差分法並藉助軟體Mathematica去計算一個擁有Dirichlet 邊界條件的Laplace方程問題 。

The aim of this paper is to investigate several classical methods and applications of the linear elliptic partial differential equations. First, a practical example is given based on the Laplace’s equation for the electrostatic potential, and is solved by Finite element method. Secondly, classical solving techniques are introduced, such as separation of variables in different domains, and Fourier transforms in both finite and infinite domains. At last, numerical Finite difference method is introduced to solve the Laplace’s equation on a square with nonhomogeneous Dirichlet boundary condition, which is computed by Mathematica.