The Numerical Algorithm For Solving Singular Integral Equation Based On The Reproducing Kernel Collocation Method

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 collocation method to present a new high accuarcy method for solving singular integral equation,and the numerical examples are demonstrated to show the validity and applicability of the reproducing kernel collocation method.Firstly,this paper introduces the principles and basic properties of collocation method and reproducing space and gives the expressions of reproducing kernel functions and applications of collocation method in the singular integral equation;Then the singularity of singular integral equation is overcame.Analytical solution can be given by using the reproducing kernel method,and the collocation method is applied to transform the integral equation into a matrix equation via the collocation points,and get the approximate solution.This paper dicusses linear and nonlinear singular Volterra integral equations respectively.The method of this paper overcomes the singularity of singular integral equation and gets numerical results with high accuracy by using reproducing kernel collocation method.Finally,the numerical examples show the advantages of reproducing kernel collocation method are as followed: low computational complexity,high rate of convergence,high accuracy.Moverover,the method of this paper can be used to solve other weakly singular integral equations.