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系統識別號 U0026-1507201915461000
論文名稱(中文) 熱方程的基本性質
論文名稱(英文) The Study on Some Basic Properties of the Heat Equation
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 107
學期 2
出版年 108
研究生(中文) 江云
研究生(英文) Yun Chiang
學號 L16064012
學位類別 碩士
語文別 英文
論文頁數 54頁
口試委員 指導教授-林育竹
口試委員-吳恭儉
口試委員-江鑑聲
中文關鍵字 熱方程  基本解  均值公式  最大值原理  反射法  Dirichlet-Neumann 關係 
英文關鍵字 heat equation  fundamental solution  mean-value formula  maximum principle  reflection method  Dirichlet-Neumann relation 
學科別分類
中文摘要 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。
英文摘要 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.
論文目次 1 Introduction 2
1.1 Derivation 2
1.2 Preliminaries 3
2 Solution of the Heat Equation 6
2.1 Fundamental Solution 6
2.2 Initial-value Problem 11
2.3 Nonhomogeneous Problem 18
3 Basic Properties of the Heat Equation 23
3.1 Mean-value Formula 23
3.2 Maximum Principle 28
3.3 Regularity 35
3.4 Local Estimates for Solutions of the Heat Equation 38
3.5 Energy Methods 39
3.5.1 Uniqueness 39
3.5.2 Backwards Uniqueness 40
4 Solution on the Half Line 43
4.1 Reflection Method 43
4.2 Dirichlet-Neumann Relation 46
Bibliography 54
參考文獻 [1] J.W. Brown and R.V. Churchill. Complex Variables and Applications. Brown-Churchill series. McGraw-Hill Higher Education, 9th edition, 2013.
[2] Lawrence C. Evans. Partial differential equations. American Mathematical Society, Providence, R.I., 2010.
[3] G.B. Folland. Advanced Calculus. Featured Titles for Advanced Calculus Series. Prentice Hall, 2002.
[4] F. John. Partial Differential Equations. Applied Mathematical Sciences. Springer New York, 4th edition, 1995.
[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.
[6] Walter A. Strauss. Partial differential equations: An introduction. Wiley, 2nd edition, 2008.
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