A meshless discrete collocation method for the numerical solution of singular-logarithmic boundary integral equations utilizing radial basis functions

The main intention of the current paper is to describe a scheme for the numerical solution of boundary integral equations of the second kind with logarithmic singular kernels. These types of integral equations result from boundary value problems of Laplace's equations with linear Robin boundary conditions. The method approximates the solution using the radial basis function (RBF) expansion with polynomial precision in the discrete collocation method. The collocation method for solving logarithmic boundary integral equations encounters more difficulties for computing the singular integrals which cannot be approximated by the classical quadrature formulae. To overcome this problem, we utilize the non-uniform composite Gauss-Legendre integration rule and employ it to estimate the singular logarithm integrals appeared in the method. Since the scheme is based on the use of scattered points spread on the analyzed domain and does not need any domain elements, we can call it as the meshless discrete collocation method. The new algorithm is successful and easy to solve various types of boundary integral equations with singular kernels. We also provide the error estimate of the proposed method. The efficiency and accuracy of the new approach are illustrated by some numerical examples.