Cauchy核奇异积分的反Gauss求积算法

Anti-Gaussian Quadrature Rules for Singular Integrals with Cauchy Kernel

利用正交多项式的三项循环关系,定义了一新的正交多项式,建立了奇异积分的插值型反Gauss求积公式。用极限方法构造出求积系数和余项积分显式表达式,余项积分表达式表明奇异积分的反Gauss求积算法是收敛的,对奇异积分求积算法进行了模拟与仿真,结果表明,随着求积结点数的增多,通过反Gauss求积公式计算的积分值与积分精确值的误差在缩小,误差曲线也较为平滑,所得积分近似值逐渐逼近积分的精确值。该求积算法可应用到工程技术数值计算中,为应用软件的开发提供了理论依据。

A new orthogonal polynomial was defined by using the three-term recurrence relation for orthogonal polynomials,and the interpolation anti-Gaussian quadrature formulae for singular integrals were established.The explicit expressions of quadrature coefficient and remainder were constructed by the limit method.The expression of remainder shows that the anti-Gaussian quadrature formulae of singular integral are convergent.Finally,the proposed quadrature rules for singular integrals were simulated.The result shows that the error decreases with the number of quadrature nodes increasing,the error curve is relatively smooth,and the approximate value of the integral gradually approaches the exact value of the integral.This quadrature rules can be applied to the numerical calculation of engineering technology,which provides a theoretical basis for the development of some computer application software.