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研究生：	王姵蓁 Wang, Pei-Zhen
論文名稱：	拉普拉斯轉換於偏微分方程中初邊值問題上的應用 Application of Laplace Transform in the Initial-Boundary Value Problems for Partial Differential Equations
指導教授：	郭鴻文 Kuo, Hung-Wen
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系應用數學碩博士班 Department of Mathematics
論文出版年：	2020
畢業學年度：	109
語文別：	英文
論文頁數：	40
中文關鍵詞：	拉普拉斯轉換、基本解
外文關鍵詞：	Laplace transform, fundamental solution
相關次數：	點閱：113 下載：5
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查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

在這篇論文中，我們將探討一些基本的偏微分方程初邊值問題，並透過拉普拉斯轉換解決問題。我們藉由拉普拉斯變換建構了全部的邊界值。並且，我們建立了基本解於拉普拉斯轉換前後之關係。

In this thesis, we study initial-boundary value problems of some basic partial differential equations and solve them via the Laplace transform. We construct the full boundary data by means of the Laplace transform. Moreover, we found the relations between fundamental solutions and their Laplace transforms.

1 Introduction                                                             1

2 Examples of some basic partial differential equations                    4

  2.1 Heat equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
    2.1.1 The heat equation on the half line (I) . . . . . . . . . . . . . 4
    2.1.2 The heat equation on the half line (II). . . . . . . . . . . . . 9

  2.2 Transport equations . . . . . . . . . . . . . . . . . . . . . . . . 12
    2.2.1 The transport equation on the half line . . . . . . . . . . . . 12

  2.3 Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . .  16
    2.3.1 The wave equation on the half line (I) . . . . . . . . . . . .  16
    2.3.2 The wave equation on the half line (II). . . . . . . . . . . .  23

Appendices                                                                27
A Detailed calculation                                                    28

  A.1 Heat equations . . . . . . . . . . . . . . . . . . . . . . . . . .  28
    A.1.1 The heat equation on the whole line . . . . . . . . . . . . . . 28
    A.1.2 The heat equation on the half line (I) . . . . . . . . . . . .  30
    A.1.3 The heat equation on the half line (II). . . . . . . . . . . .  32

  A.2 Wave equations . . . . . . . . . . . . . . . . . . . . . . . . . .  33
    A.2.1 The wave equation on the whole line . . . . . . . . . . . . . . 33
    A.2.2 The wave equation on the half line (I) . . . . . . . . . . . .  35
    A.2.3 The wave equation on the half line (II). . . . . . . . . . . .  38

Bibliography                                                              40
                                    

[1] Deng, Shijin; Wang, Weike; Yu, Shih-Hsien, Bifurcation on boundary data for linear Broadwell model with conservative boundary condition. J. Hyperbolic Differ. Equ. 11 (2014), no. 3, 603–619.

[2] Deng, Shijin; Wang, Weike; Yu, Shih-Hsien, Green’s functions of wave equations in Rn+ × R+. Arch. Ration. Mech. Anal. 216 (2015), no. 3, 881–903.

[3] Liu, Tai-Ping; Yu, Shih-Hsien, On boundary relation for some dissipative systems. Bulletin Inst. Math. Acad. Sin. (N.S.) 6 (2011), no. 3, 245-267.

[4] Liu,Tai-Ping; Yu, Shih-Hsien, Dirichlet-Neumann kernel for hyperbolic-dissipative system in half-space. Bull. Inst. Math. Acad. Sin. (N.S.) 7 (2012), no. 4, 477-543

[5] Liu, Tai-Ping; Yu, Shih-Hsien, Boundary wave propagator for compressible Navier- Stokes equations. Found. Comput. Math. 14 (2014), no. 6, 1287–1335.

[6] Wang, Haitao; Yu, Shih-Hsien, Algebraic-complex scheme for Dirichlet-Neumann data for parabolic system. Arch. Ration. Mech. Anal. 211 (2014), no. 3, 1013–1026.

校內：2024-01-01公開
校外：不公開電子論文尚未授權公開，紙本請查館藏目錄

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