Conventionally, piecewise polynomials have been used in the boundary elements method (BEM) to approximate unknown boundary values. Since infinitely smooth radial basis functions (RBFs) are more stable and accurate than the polynomials for high dimensional domains, the unknown values are approximated by the RBFs in this paper. Therefore, a new formulation of BEM, called radial BEM, is obtained. To calculate singular boundary integrals of the new method, we propose a new distribution for boundary source points that removes singularity from the integrals. Therefore, the boundary integrals are calculated precisely by the standard Gaussian quadrature rule (GQR) with n = 16 quadrature nodes. Several numerical examples are presented to check the efficiency of the radial BEM versus standard BEM and RBF collocation method for solving partial differential equations (PDEs). Analytical and numerical studies presented in this paper admit the radial BEM as a perfect version of BEM which will enrich the contribution of BEM and RBFs in solving PDEs, impressively.

翻译：传统上，边界元方法采用分段多项式逼近未知边界值。由于无限光滑的径向基函数在高维域中比多项式更稳定、更精确，本文采用径向基函数近似未知值，由此衍生出边界元方法的新形式——径向边界元。为计算新方法中的奇异边界积分，我们提出了一种边界源点新分布，成功消除了积分奇异性。因此，采用标准高斯求积法则（含n=16个求积节点）即可精确计算边界积分。通过多个数值算例，将径向边界元与标准边界元及径向基函数配点法求解偏微分方程的效率进行对比。本文的解析与数值研究证实，径向边界元作为边界元方法的完美改良版本，将显著丰富边界元与径向基函数在偏微分方程求解中的贡献。