Coordination Quadrature Formula Of Singular Integral Equation On Cross Curve

In this paper,we discuss the formula for the Quadrature formula of singular integral on cross curves.For simplicity,we only consider the cross curve in two cases:bifurcated intersection curve and the trigeminal intersection curve.The study of this problem is roughly divided into the following two parts to complete.The first step is to construct the Quadrature formula on this two types of curves respectively,and discuss convergence.The second step is to apply the quadrature formula on the bifurcated integral curve to an example in the direction of elasticity and perform numerical calculations.In terms of structure,this paper firstly gives the necessary basic knowledge,including the related theoretical knowledge of generalized difference quotient function,the construction forms of interpolation type quadrature formula,The construction form of Hunter-Gauss and Paget-Elliott-Gauss type quadrature formulas,Cauchy type integral and its properties near the endpoints,Riemann boundary value problems on curves with nodes and the problem of the solution of characteristic equations.Then the problem of singular integration on the bifurcated intersection curve and the trigeminal intersection curve is discussed.The singular integration formula on the corresponding curve is established,and the error expression is given,and the algebraic accuracy of the singular integration formula is obtained.In addition,the convergence of the quadrature formula is discussed,and it is concluded that the quadrature formula is convergent on the corresponding integral curve.Finally,a numerical example of the quadrature formula on the bifurcated intersection curve in elastic mechanics is given,and MATLAB is used for numerical calculation.In this paper,the construction method of singular integral quadrature formula is modeled on the corresponding general Hunter-Gauss type quadrature formula and Paget-Elliott-Gauss type quadrature formula established in the literature^[1].We are trying to establish the corresponding cross-curve Hunter quasi Gauss type and Paget-Elliott quasi Gauss type quadrature formula.Considering the particularity of the integral curve,we need to discretize and interpolate the integrals on each curve separately.In this article,for the singular integrals with singular parts of the integrand,the node group to be considered selects the zero point of the approximation polynomial and the singular point of the integrand on the corresponding interval.For the singular integral without the singular part of the integrand,it is a normal integral.The node group to be considered only selects the zero point of the approximation polynomial on the corresponding interval.Finally,the corresponding quadrature formula is established.