拉普拉斯方程式在力學與工程問題中應用甚廣，本文針對二維拉普拉斯方程式之源型奇性相似解加以探討，並進一步研究發現，在各種拉普拉斯方程式邊界值問題中，若具有適當之幾何形狀與邊界條件，則經由相似變數轉換後，可得到一相同形式之二階常微分方程式及二個類似之邊界條件，其解亦可統一於一個極為簡單優美且相同形式之奇性相似精確解。 本文即根據上述之原理，將楔形區域、矩形區域與半無限長條區域以及扇形區域三大類拉普拉斯方程式邊界值問題之無窮級數解，化為具有相同數學結構之源型奇性相似精確解。 在古典水動力學中，對於二維、穩態、不可壓縮、無旋流場且不考慮其他外力影響之平面射流問題，常用速度矢端圖法將其由物理平面轉換至速度平面，再以複變函數法求得平面射流問題之自由流線參數方程式，但其求解過程甚為繁複。針對平面射流問題，本文直接在實數平面上建構推導出平面自由流線參數方程式之通式，並利用相似法所求得之流線函數相似精確解代入通式中，積分後即可直接得出自由流線之參數方程式。本文利用相似法解法與傳統複變解法做比較，亦可顯現出相似精確解之簡潔性及實用性。此外，在文中試圖尋找出幾何外形與邊界條件皆不同之平面射流問題彼此之關聯性，例如：平面射流沖擊有限長平板問題中，當折射角為九十度時，則其解可直接轉換為無限長平板問題之解；當折射角為零度時，其解可轉換為平面無窮射流沖擊有限長平板問題之解。 本文結合相似轉換法與自由流線理論，提供一較為快速簡單之解法，求得平面射流問題之自由流線參數方程式以及相關之重要參數，並可與傳統上利用複變函數解法及數值法所得之解互相驗證比較。 Laplace''s equation is applied extensively to mechanics and engineering. This thesis is mainly aimed at the source type singular similar solution of two-dimensional Laplace''s equation. If the various boundary value problems of Laplace''s equation have proper geometry and boundary conditions, they can be transformed to an equivalent second-order ordinary differential equation and two similar boundary conditions. Therefore, their solution can integrate to a very simple and graceful source type similar singular solution. According to the above-mentioned principle, we can transform the series solution of the boundary value problem of Laplace''s equation in the wedge, rectangle and semi-infinite strip, sector region into an exact similarity solution with the source type singularity by similarity method. Boundary value problems in classical hydrodynamics are usually transformed from the physical plane to velocity plane by hodograph method. The traditional method to find the equation of free streamlines is by using complex variable analysis. But it is relatively too complicated. We construct a general formula of the parametric equation of free streamlines in real plane in this thesis so we can newly obtain the parametric equation of free streamlines by substituting the exact similarity solution of stream function. This paper provides a simple and fast method to obtain the parametric equations of free jet in the real domain by similarity method. Furthermore, the solutions from the similarity method can be verified with the solutions obtained from the traditional method by using complex variable analysis and numerical calculations.
