In this paper, we employ the boundary integral equation (BIE) to analytically derive the solution of potential flow across a circular cylinder, an elliptical cylinder and a thin airfoil. It is found that the symmetric or antisymmetric potential flow problem can be solved by using the UT or the LM equation alone. Both analytical and numerical approaches were considered. The key tool is using the degenerate kernel instead of the closed-form fundamental solution. A closed-form fundamental solution is expressed in terms of degenerate kernel by using polar and elliptical coordinates for a circular cylinder and an elliptical cylinder, respectively. The role of the dual BEM is also examined by showing the singular vectors of influence matrix. Besides, either the singular or the hypersingular BIE are respectively employed alone to solve the problem of symmetric and anti-symmetric flow field. Analytical derivation as well as the BEM is demonstrated. The orientation of the ellipse, as well as the angle of attack for the thin airfoil, is considered. The result shows that the degenerate kernel can be an alternative tool for solving some BVPs, although the complex variable is always employed. Once the degenerate kernel for the closed-form fundamental solution is available, the BIE is nothing more than the linear algebra and an analytical study is possible. The extension to 3-D object is promising once the degenerate kernel is available.

在本文中，我们采用边界积分方程（BIE）来解析推导穿过圆柱体、椭圆柱体和薄翼型的势流的解。发现单独使用UT或LM方程可以解决对称或反对称势流问题。分析和数值方法都被考虑。关键工具是使用退化内核而不是封闭形式的基本解决方案。通过分别使用圆柱和椭圆柱的极坐标和椭圆坐标，以简并核的形式表示封闭形式的基本解。还通过显示影响矩阵的奇异向量来检验双边界元的作用。除了，分别单独使用奇异或超奇异BIE来解决对称和反对称流场问题。演示了分析推导以及边界元法。椭圆的方向以及薄翼型的攻角都被考虑在内。结果表明，虽然总是使用复变量，但退化核可以作为解决某些 BVP 的替代工具。一旦闭式基本解的退化核可用，BIE 只不过是线性代数，并且可以进行分析研究。一旦退化内核可用，对 3-D 对象的扩展是有希望的。以及薄翼型的攻角，都被考虑在内。结果表明，虽然总是使用复变量，但退化核可以作为解决某些 BVP 的替代工具。一旦闭式基本解的退化核可用，BIE 只不过是线性代数，并且可以进行分析研究。一旦退化内核可用，对 3-D 对象的扩展是有希望的。以及薄翼型的攻角，都被考虑在内。结果表明，虽然总是使用复变量，但退化核可以作为解决某些 BVP 的替代工具。一旦闭式基本解的退化核可用，BIE 只不过是线性代数，并且可以进行分析研究。一旦退化内核可用，对 3-D 对象的扩展是有希望的。