Numerical Quadratures for Hadamard Hypersingular Integrals †

Youjian Shen 1*

1 Department of Mathematics, Hainan Normal University, Haikou 571158, China/Department of Mathematics, Zhongshan University, Guangzhou 510275, China.

Received October 13, 2003; Accepted (in revised version) August 23, 2004
Abstract

quadrature formulas for the Hadamard finite 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 accuracy of the approximation results are satisfactory. Moreover, the rate and the accuracy can be improved by choosing appropriate weight functions.

Key words: Gaussian quadrature; finite part; hypersingular integral; orthonormal polynomial.

AMS subject classifications: 65D30,65D32

Correspondence to: Youjian Shen , Department of Mathematics, Hainan Normal University, Haikou 571158, China/Department of Mathematics, Zhongshan University, Guangzhou 510275, China. Email: syjian@hainnu.edu.cn
† This research is supported in part by China NSF under grant 10071096, NSF of Guangdong under grant 990228, NSF of Hainan under grant 80525 and the One Hundred Distinguished Young Chinese Scientists Program of the Chinese Academy of Sciences from Yuesheng Xu.