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.
