This article describes a technique for numerically solving logarithmic integral equations resulted from boundary value problems of two-dimensional Helmholtz equations. The method uses radial basis functions (RBFs) constructed on scattered points as a basis in the discrete Galerkin method. The proposed scheme applies a special accurate quadrature formula utilizing the the nonuniform Gauss-Legendre integration rule to compute logarithm-like singular integrals appeared in the scheme. The method is meshless, since it does not require any background cells. The scheme is effective and its algorithm can be easily implemented. The validity and efficiency of the new technique are demonstrated through a numerical example.
