On The Study Of Singular Integral Equation And High Accuracy Numerical Method

In this paper,firstly,we study the numerical solution of Cauchy singular integral equation.The basic idea of this method is to solve the Cauchy singular integral equation by polynomial function approximation method,This paper chooses Bernstein Polynomials as non-orthogonal polynomials,because Bernstein Polynomials as basis function is stable enough in a given interval.In this condition,for the algebraic polynomial in the interval,other non-negative basis function can not produce a smaller number of condition,I will take advantage of this characteristic,using Bernstein polynomial as basis function,the four kinds of kernel functions are approximated respectively,the approximation solution of Cauchy singular integral equation of the first kind can be expressed.Focus on the study of the first kernel function,the other three cases are similar to the first case,it is given the convergence analysis of numerical results,numerical example is presented to demonstrate the effectiveness of this method,so as to achieve numerical solution.Secondly,when we study the numerical solutions of Hadamard hypersingular integral equation,the midpoint formula and collocation method are used,we choose midpoint as the discrete collocation points,and then the discrete system can be derived,we are easy to find that the system is degenerate equations,so by introducing the regularization factor,the system equations become stable,by using this method,superconvergence effect will be obtained,therefore,it is easy to obtain the numerical solution.At the same time,in general,the numerical approximate solution of the hypersingular integral equation is given,finally,the theory of error estimation is established and the theoretical proof is given,the calculation results are checked by numerical experiments,error analysis and numerical results are given.