Radial boundary elements method and removing singularity of integrals by an optimal selection of boundary source points

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).