The Cauchy Singular Integral Integral Equations Of Algorithms

A lot of problems in mathematics and physics can be boiled down to the integrals and integral equations with a Cauchy singular kernel. There are many literatures studying this kind of integrals and integral equations and the study becomes more and more important. But it is very difficult to calculate Cauchy integral and integral equations directly. So people switch from the study of these problems to their numerical solution of the integrals and integral equations.Firstly, this paper mainly describes the development background of the methods to get the numerical solutions of the Cauchy integral equations. Secondly, For Cauchy singular integrals, we put forward a new style of integral formula, and Euler-Maclaurin expansion as well as extrapolation formula. At the same time, we give the integral formula of Hilbert singular integral. With these formulas, we give the error result of specific example of Cauchy and Hilbert singular integral quadrate. We show that these formulas are highly accurate by comparing with the numerical results of other algorithms. Thirdly, we discuss the numerical solutions of the integral equations with a Cauchy and Hilbert singular kernel and get the error result of specific examples. These also explain that the integral formula is highly accurate. At the end of this paper, It gives some conclusions and introduces the development direction of the future.