這篇論文中，我們將探討熱方程及其相關性質，內容主要取材於Evans 的
Partial Differential Equations 一書。首先，我們會推導出熱方程並且學習如何解決初值問題及非齊次的問題。接著，我們將探討一些熱方程的基本性質，包含均值公式、最大值原理、正則性及唯一性。最後，在對全空間熱方程充分的了解下，對於半線上的問題，我們先介紹古典的方法：反射法，將問題轉換成整條實數線上的初值問題，進而解決在半線上Dirichlet邊界值問題及Neumann邊界值問題。然而，針對在半線上一般的邊界–初值問題，反射法並不適用。因此，我們介紹劉太平院士與尤釋賢教授所提出的方法。他們透過拉普拉斯轉換推導出Dirichlet邊界條件與Neumann邊界條件的關係，進而利用著個關係式及熱方程的基本解構造出解的表達式。

In this thesis, we study the heat equation and its properties, based on Evans' ``Partial Differential Equation'. First, we derive the heat equation and learn to solve the initial-value problem and the nonhomogeneous problem. Next, we talk about some basic properties of the heat equation, including the mean-value formula, maximum principle, regularity and uniqueness. Last but not least, with full knowledge of the heat equation on the whole space, when it comes to half-line problems, we first introduce the standard reflection method, extending the problems to the whole line and solving the Dirichlet boundary value problems and Neumann boundary value problems on the half line. However, the general initial-boundary value problem cannot be solved by the reflection method. Therefore, we introduce Professor Liu, Tai-Ping and Professor Yu, Shih-Hsien's method. They derive a relation between the Dirichlet and the Neumann boundary value through Laplace transform, and furthermore, construct a solution formula using Dirichlet-Neumann relation and the fundamental solution of the heat equation.

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[2] Lawrence C. Evans. Partial differential equations. American Mathematical Society, Providence, R.I., 2010.
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[5] Tai-Ping Liu and Shih-Hsien Yu. On boundary relation for some dissipative systems. Bull. Inst. Math. Acad. Sin.(NS), 6(3):245–267, 2011.
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