The Least Squares Reproducing Kernel Method For Solving Singular Integral Equation With Cauchy Kernel

Singular integral equation theory is of great significance for many practical problems.The problems such as boundary value problems of analytic function,tidal theory,elastic theory and fluid mechanics can sum up to singular integral equation.With the rapid development of computer science,the numerical algorithm and applications of singular integral equation become a hot topic once again.Because of the particularity of singular integral equation,we can't use normal method to solve it.The purpose of this paper is to combine reproducing kernel with least square method to present a new method for solving the?-approximate solution of singular integral equation with Cauchy kernel.and the numerical examples are demonstrated to show the validity and applicability of the reproducing kernel least square method.Firstly,this paper introduces the principles and basic properties of reproducing space and least square method and gives the expressions of reproducing kernel functions.Then An equivalent transformation is constructed,the strong singularity of singular integral equations with Cauchy type singularities is overcome,and use a set of bases in a dense subspace in a squared integrable space,the definition mapping,a subspace is formatted which is dense in the reproducing kernel space.Thus,a basic framework for solving the solution space of the transformed operator equation is constructed,an ?-approximate solution to the problem with the least square method.The method of this paper overcomes the singularity of singular integral equation and the approximate solution of the Cauchy type singular integral equation is given by using the least square reproducing kernel method.Finally,the numerical examples show the advantages of least square reproducing kernel method are as followed: low computational complexity,high rate of convergence.