Conventionally, piecewise polynomial basis functions (PBFs) are used in the boundary elements method (BEM) to approximate unknown functions. Since, smooth radial basis functions (RBFs) are more stable and accurate than the PBFs for two and three dimensional domains, the unknown functions are approximated by the RBFs in this paper. Therefore, a new formulation of BEM, called radial BEM, is proposed. There are some singular boundary integrals in BEM which mostly are calculated analytically. Analytical schemes are only applicable for PBFs defined on straight boundary element, and become more complicated for polynomials of higher degree. To overcome this difficulty, this paper proposes a distribution for boundary source points so that the boundary integrals can be calculated by Gaussian quadrature rule (GQR) with high precision. Using advantages of the proposed approach, boundary integrals of the radial BEM are calculated, easily and precisely. Several numerical examples are presented to show efficiency of the radial BEM versus standard BEM for solving partial differential equations (PDEs).

翻译：传统上，边界元法（BEM）采用分段多项式基函数（PBFs）近似未知函数。由于光滑径向基函数（RBFs）在二维和三维区域中比PBFs更稳定且精度更高，本文采用RBFs近似未知函数，进而提出一种新的BEM公式——径向BEM。BEM中存在若干奇异边界积分，通常通过解析方法计算。然而，解析方法仅适用于定义在直线边界元上的PBFs，且对于高阶多项式会变得更为复杂。为克服这一困难，本文提出一种边界源点分布方案，使得边界积分可通过高斯求积法则（GQR）实现高精度计算。利用该方法的优势，径向BEM的边界积分得以简便且精确地求解。通过多个数值算例验证了径向BEM在求解偏微分方程（PDEs）时相较于标准BEM的高效性。