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研究生:李景隆
研究生(外文):Ching-LongLi
論文名稱:推導一些演化方程的解公式的讀書心得
論文名稱(外文):A note for derivation of the formulas of solution to some evolution equations
指導教授:方永富
指導教授(外文):Yung-fu Fang
學位類別:碩士
校院名稱:國立成功大學
系所名稱:數學系應用數學碩博士班
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2020
畢業學年度:108
語文別:英文
論文頁數:54
中文關鍵詞:非齊次方程波方程Dirac 方程Klein-Gordon 方程Schördinger 方程擴散方程
外文關鍵詞:Fourier transformInhomogeneous equationwave equationDirac equationKlein-Gordon equationSchrödinger equationDiffusion equation
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在此篇文章中,我學習了 Walter A.Strauss 的偏微分方程,其中介紹了許多常見的偏微分方程,這篇論文會試著推導這些非齊次偏微分方程的解,希望能夠找到他們的通解或是解的表示式,為了推導高階的微分方程,我會由一維空間中的線性偏微分方程解開始討論到更高維度的方程,其中包含了二維空間和三維空間,我們希望在推導的過程中認識更多偏微分方程之間的共通性,在推導的過空間程上我空間們會詳細的描述和介紹。
這篇文章,我們參考了林琦焜教授所著的傅立葉分析與應用,利用傅立葉轉換試著在各種偏微分方程中找出關聯性,了解有關高維度的傅立葉轉換和球座標轉換的應用,傅立葉轉換在多變數的函數中屬於向量微積分,我們採取球座標轉換技巧在一些相異的偏微分方程,對於這些常見的偏微分方程,我們可以更容易推導。在文章開始前,我們需要先介紹一些相關的公式和推導,其中包含: 傅立葉變換、球座標變換、Gamma 函數、Beta 函數、和高斯函數。在文章中我們詳細介紹了波方程、狄拉克方程、Klein-Gordon 方程、薛丁格方程、擴散方程的齊次解和非齊次解。
In this paper, I studied the partial differential equations of Walter A. Strauss, which introduced many common partial differential equations. This paper will try to derive the solutions of these non-homogeneous partial differential equations, hoping to find their general solution or the expression of the solution. In order to derive the higher order differential equations, we will start from the solution of the one-dimensional linear partial differential equation to the higher order, which includes the second and third orders. We hope to recognize the commonality between more partial differential equations in the derivation process. We will describe and introduce in detail in the derivation process.
In this article, we refer to the Fourier analysis and application of Professor C.K.Lin, and use Fourier transform to try to find correlations in various partial differential equations and to understand about the application of high-dimensional Fourier transformation and
spherical coordinate transformation. Fourier transform belongs to vector calculus in the function of multivariables. We adopt the skill of spherical coordinate conversion in some different partial differential equations. For these common partial differential equations, we can more easily derive. Before the article starts, we need to introduce some related formulas and derivations, including: Fourier transform, spherical coordinate transform, Gamma function, Beta function, and Gaussian function. In the article we introduced in detail the homogeneous and non-homogeneous solutions of the wave equation, Dirac equation, Klein-Gordon equation, Schrodinger equation, diffusion equation
1 Introduction 1
2 Preliminaries and Notations 5
3 Klein-Gordan equation 13
3.1 Klein-Gordon equation in 1-D . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Klein-Gordon equation in 2-D . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Klein-Gordon equation in 3-D . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Dirac equation 28
5 wave equation 33
5.1 Fourier inverse transform in one dimension . . . . . . . . . . . . . . . . . . 35
5.2 Fourier inverse transform in two dimension . . . . . . . . . . . . . . . . . . 36
5.3 Fourier inverse transform in three dimension . . . . . . . . . . . . . . . . . 38
6 Diusion equation and Schrodinger equation 40
6.1 Diusion equation(Heat equation) . . . . . . . . . . . . . . . . . . . . . . . 40
6.2 Schrodinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Laplace equation and Helmholtz equation 47
7.1 Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
7.2 Helmholtz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8 Fourth order Schrodinger equation and Fourth order wave equation 51
Bibliography 54
[1] B. A. Matania, K. Herbert, S. Jean-Claude, Dispersion estimates for fourth order Schrödinger equations. C. R. Acad. Sci. Paris, t. 330, Serie I, Pages 87-92, 2000
[2] T. Cazenave, Semilinear Schrödinger equations. American Mathematical Soc., Vol. 10, 2003.
[3] Mehdi Dehghan, Ali Shokri, Journal of Computational and Applied Mathematics. Volume 230, Issue 2, 15, August 2009, Pages 400-410
[4] S. M. El-Sayed The decomposition method for studying the Klein-Gordon equation. Chaos Solitons Fractals, 18 (2003), Pages 1025-1030
[5] R. Finkelstein, R. LeLevier, M. Ruderman, Nonlinear spinor elds. Phys. Rev. Pages 83 (2), 326-332.
[6] V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations. Phys. Rev. E 53 (2), (1996)
[7] C. K. Lin, Fourier Analysis and Applications, Tsang Hai, R. O. C, 470-472, 489, 2010
[8] Shuji Machihara, Kenji Nakanishi, Tohru Ozawa, Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation. Rev. Mat. Iberoamericana Volume 19, Number 1 (2003), 179-194.
[9] Walter A. Strauss, Partial di erential equations. ew York, NY, USA: John Wiley & Sons, 1992.
[10] A. M. Wazwaz, The tanh and the sine-cosine methods for compact and noncompact solutions of the nonlinear Klein-Gordon equation. Appl. Math. Comput., 167 (2005), Pages 1179-1195
[11] V. E. Zakharov, Collapse of Langmuir waves. Sov. Phys. JETP 35, 908-914, 1972.
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