以Galerkin法解一維偏微分方程式

Abstract

在這篇論文中我們將介紹伽遼金方法去解一些方程式，它是有限元素法常會用到的一種方法，其中有三個物理問題分別是線方程式，樑方程式，熱方程式。第一個線方程式是在描述一條拉緊的細線，是在討論在細線上施力後的行變問題；第二個梁方程式是在研究樑因為自身重量或外力而改變的行狀；第三個是熱方程式是在分析一根桿子或細線上的熱傳導問題。我們將會使用伽遼金方法去解這些方程式，然後更進一步得我們會推廣去解非線性的線方程式、熱方程式、樑方程式。

In this thesis, we introduce the Galerkin finite element method to solve partial differential equations. We solve three types of problems such as string equation, beam equation and heat equation which is commonly seen in physics. String equation describes the tension acting on a taut string, resulting in a rise in deflection. Beam equation is the study of the beam and how it is deflected by its own weight or an external force. Heat equation is the analysis of the heat conduction through a rod or thin wire. We will use the Galerkin method to solve these equations, and futhermore we extend the methods to solve nonlinear string equation, nonlinear heat equation and nonlinear beam equation.