具對流與擴散項之非線性Burgers方程式，已有熟知之Hopf-Cole變換解法，而本文採用相似法直接在物理面上探討Burgers方程式，並有系統地以伸縮群及行波相似轉換求得其相似精確解並加以應用在流體力學及氣動力學問題中。本文另以適當的起始條件求出Hopf-Cole變換積分解，並與本文相似解加以連結，使吾人對Burgers方程式之數學結構與物理意義有更進一步之認識和了解。
運用Burgers方程式之一般積分解，有利於在工程數值的模擬分析上驗證相關模型之非線性行為，而採行Burgers方程式之相似解則有利於從應用物理的行為分析上，解釋各種相關工程模型之非線性特性。研究結果顯示，在運用相似轉換法直接對Burgers方程式加以探討，較容易看出本研究問題之本質所在，亦即利用相似法可直接瞭解研究問題中物理量間的關聯性，並可獲得更多在應用物理上的直觀性瞭解。

Abstract:
The prototype equation for nonlinear convection-diffusion processes is Burgers’ equation. An analytical solution of Burgers’ equation was discovered independently by Hopf and Cole. In this paper, a similarity study of Burgers’ equation is exhibited. A systematic approach for obtaining the stretching group and traveling wave similarity solutions are constructed and applied to fluid mechanics, gas-dynamic or stock market dynamics problems. Some exact solutions are also obtained from integral solution of Hopf-Cole transformation, which are related to the stretching group and traveling wave similarity solutions.
Using the general integral solution of Burgers equation is beneficial to efficiently verifying the relevant engineering model in numerical analysis of nonlinear physics. The derived results reveal that one can straightforward find the meaningful solution through similarity transformation of Burgers equation to estimate the key physical variables and obtain more important implications.

目錄
目錄首頁………………………………………………………i
誌謝……………………………………………………………iii
中文論文提要…………………………………………………iv
英文論文提要…………………………………………………v
圖目錄…………………………………………………………vi
符號說明……………………………………………………vii
第一章 緒論…………………………………………………1
1－1 研究動機與目的……………………………………………3
1－2 研究流程與論文架構………………………………………5
第二章 Burgers方程式之非線性波動模型………………7
2－1 Burgers方程式的數理模型特性……………………………7
2－2 Burgers方程式在建構物理模型上的應用…………………8
2－2－1 在電磁波訊號的頻譜分析上的應用……………9
2－2－2 在生物訊號傳遞現象分析上的應用………………11
2－2－3 在建構股市非線性行為模型上的應用……………13
2－3 方程式之一般解形式………………………………………15
第三章 Burgers方程式相似解之探討……………………17
3－1 相似法求解非線性力學問題上的重要性…………………18
3－2 非線性波動問題的相似精確解……………………………19
3-2-1具對流與擴散項之非線性Burgers方程式……………19
3-2-1-1 線性擴散方程式(α=0)…………………………20
3-2-1-2 非線性對流方程式(ν=0,α=1)………………20
3－3 Burgers方程式伸縮群之相似解……………………………21
3-3-1 線性擴散方程式(α=0)………………………………22
3-3-1-1 Stokes第一問題之相似解 (m=0)……………22
3-3-1-2 擴散方程式之基本解 …………………………24
3-3-2 非線性對流方程式 (α=1,ν=0)……………………25
3-3-2-1 膨脹波解 (m=0)…………………………………26
3-3-2-2 非線性對流方程式之基本解 (m=-1/2)………27
3-3-2-3  Burgers方程式點源起始條件之基本解 ……28
3－4 Burgers方程式之行波相似解………………………………30
3-4-1 線性擴散方程式之相似解(α=0)……………………31
3-4-1-1  Stokes第二問題之相似解……………………31
3-4-1-2 分離變數法之相似解……………………………32
3-4-2 非線性對流方程式之震波解 (α=1,ν=0)…………33
3-4-3  Burgers方程式之行波相似解 (α=1,ν≠0)………35
第四章 Burgers方程式的一般解與相似解之關聯性……38
4－1 Burgers方程式的一般解與相似解之異同………………38
4－2 建構一般積分解與相似精確解之關聯性…………………40
4－3 方程式點源起始條件之積分解與相似解的關聯性………41
4-3-1 擴散項為主 (ν→∞,R→0)…………………………42
4-3-2 非線性對流項為主 (ν→0,R→∞)…………………42
4－4 Burgers方程式黏性衝擊波之積分解與相似解的關聯性…45
第五章 結論………………………………………………50
參考文獻……………………………………………………54
論文簡要版…………………………………………………59

圖目錄

圖1－1  論文研究流程………………………………………………6
圖2－1 非線性傳輸線之等效電路模型………………………………9
圖3－1  Stokes第一問題之解………………………………………23
圖3－2  擴散方程式之基本解………………………………………25
圖3－3  對流方程式之膨脹波解……………………………………27
圖3－4  Stokes第二問題之解………………………………………32
圖3－5 非線性對流方程式之壓縮波解………………………………34
圖3－6  Burgers方程式之行波相似轉換－黏性衝擊波解………37
圖4－1 點源衝擊波之形成……………………………………………44

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