本文係利用二維拉普拉斯算子之廣義相似轉換群，首先應用到拉普拉斯方程式不同邊界值問題，將
其源型奇異性相似解統合於一個簡單優美的形式之中，再將其應用於泊松方程邊界值問題而設計出適
當 的源項分佈以求得其上半平面區域與圓形區域具有相同形式之相似精確解以供數值計算之參考和結
果之比 對.最後再將其應用於振動薄膜膜態之亥姆霍茲方程特徵值問題並巧妙的利用相似律，則偏心圓
區域變密度薄膜膜態之特徵值可直接由已知之同心圓區域等密度薄膜膜態特徵值求出. The similarity transfonnation groups of Two-dimensional Laplacian operator are exhibited. The singular
source type similarity solution are applied to some boundary-value problems of Laplace's equation.
Designing the appropriate source function of Poisson equation under the similarity analysis , some exact
similarity solutions of Poisson equation are constructed to two different boundary-value problems with the
same similarity solutions .
Designing the appropriate weighting function and using the similarity rule , the eigen values of eccentric
circle with membrane of variable density are easily obtained from the eigen values of concentric circles with
membrane of constant density.
