The numerical solution of two-dimensional logarithmic integral equations on normal domains using radial basis functions with polynomial precision

Abstract

The main purpose of this article is to investigate the numerical solution of two-dimensional Fredholm integral equations of the second kind on normal domains, whose kernels have logarithmic singularity. Radial basis functions constructed on scattered points are utilized as a basis in the discrete collocation method to solve these types of integral equations. We encounter logarithm-like singular integrals in the process of setting up the presented scheme which cannot be computed by classical quadrature formulae. Therefore, a special numerical integration rule is required to approximate such integrals based on the use of dual non-uniform composite Gauss---Legendre quadratures on normal domains. Since the method proposed in the current paper does not need any background mesh, it is meshless and consequently independent of the geometry of domain. The error estimate and the convergence rate of the approach are studied for the presented method. The convergence accuracy of the new technique is examined over four integral equations on the tear, annular, crescent, and castle domains, and obtained results confirm the theoretical error estimates.