The Numerical Algorithm For Solving The System Of Singular Integral Equations Based On The Reproducing Kernel Method

The system of singular integral equations has a wide application in solving practical problems,such as fracture mechanics,elastic mechanics,contact mechanics,aerodynamicist and so on.At present,more and more scholars have a strong interest and engage in the study of it.Due to the singularity and complexity of the system of singular integral equations,it is difficult to solve.Those numerical results that ones have obtained are not very ideal.Therefore,if we can find a new method for the solutions of the system of singular integral equations,it will have an important practical significance.The paper solves two kinds of the system of singular integral equations mainly with the reproducing kernel method and gives the exact and numerical solutions.Firstly,the paper briefly introduces the basic properties and the basic theorems of the reproducing kernel in the reproducing kernel space,and gives the expression of reproducing kernel function in the reproducing kernel space.Secondly,in order to solve the equations in the Hilbert space by using reproducing kernel method,we have to overcome the singularity of the equations and transform it into equivalent equations.Finally,based on this theory,the Hilbert space suited for the equations is built,and the reproducing kernel method is applied to give the solutions for two kinds of the system of singular integral equations.In one hand,according to the system of singular integral equations,the orthogonal system is obtained in the Hilbert space by the Gram-Schmidt process,and then get the exact solution of the system of singular integral equations with the infinite series and the approximate solution by truncating the infinite series.In the other hand,a similar method is used to discuss the system of singular integral equations with another form.Through transforming and overcoming the singularity of this kernel function,the exact solution and approximate solution of the system of singular integral equations are given.At the same time,the approximate solution converges uniformly to the exact solution,which verifies the reliability of the method again.The reproducing kernel method used in this paper overcomes such kernel singularity of the system of singular integral equations and obtains numerical results with high accuracy.Lastly,the numerical examples show that the reproducing kernel method has the following advantages: less amount of calculation,fast convergence speed and high precision.In addition,the method can also be used to solve other similar system of singular integral equations.