A new efficient method with error analysis for solving the second kind Fredholm integral equation with Cauchy kernel

The main objective of this paper is to give an efficient numerical method for the solution of the second kind Fredholm integral equation with Cauchy type kernel. Although, numerical treatment of Singular integral equations (SIEs) has been considered by many researchers and many numerical methods have been proposed for the solution of this equation, strong singularity of the Cauchy singular integral operator still remains as a challenge to numerical methods. Most of the previous methods rely on specific quadrature rule or suitable base functions for capturing the singularity. Here we focus on operator transformation and graded mesh. In addition, we study error analysis. In this regard, efficiency of reproducing kernel Hilbert space (RKHS) method using smooth transformation on the graded mesh is improved and compared with some other numerical methods.