In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate th...In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.展开更多
基金This research is supported in part by China NSF under grant 10071096NSF of Guangdong under grant 990228+1 种基金NSF of Hainan under grant 80525the One Hundred Distinguished Young Chinese Scientists Program of the Chinese Academy of Sciences from Yuesheng Xu.
文摘In this paper,we develop Gaussian quadrature formulas for the Hadamard fi- nite part integrals.In our formulas,the classical orthogonal polynomials such as Legendre and Chebyshev polynomials are used to approximate the density function f(x)so that the Gaussian quadrature formulas have degree n-1.The error estimates of the formulas are obtained.It is found from the numerical examples that the convergence rate and the accu- racy of the approximation results are satisfactory.Moreover,the rate and the accuracy can be improved by choosing appropriate weight functions.
文摘微积分学的基本问题是确定曲线的斜率和曲边梯形的面积。如何利用无限小求得解答,已举例说明。下面将概要地介绍如何运用无限小的运算研究微积分理论。一、微分学定义1.若函数 y=f(x)在 x=r 处有定义,且△f/△x=(f(r+△x)-f(r))/△x是有限数,其标准部分与无限小△x 的选取无关,则称 f 在 x=r 处可微,并且 f
