• 论文 •

### 牛顿科茨公式计算超奇异积分的误差估计

1. 1. 山东建筑大学理学院, 济南 250101;
2. LSEC, 中国科学院, 数学与系统科学研究院, 计算数学研究所, 北京 100080
• 收稿日期:2009-12-11 出版日期:2011-02-15 发布日期:2011-03-08
• 基金资助:

国家重点基础研究发展规划项目(No.2005CB321701)资助项目.

Li Jin, Yu Dehao. THE ERROR ESTIMATE OF NEWTON-COTES METHODS TO COMPUTE HYPERSINGULAR INTEGRAL[J]. Mathematica Numerica Sinica, 2011, 33(1): 77-86.

### THE ERROR ESTIMATE OF NEWTON-COTES METHODS TO COMPUTE HYPERSINGULAR INTEGRAL

Li Jin1,2, Yu Dehao2

1. 1. School of Science, Shandong Jianzhu University, Jinan 250101, China;
2. LSEC, Institute of Computational Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China
• Received:2009-12-11 Online:2011-02-15 Published:2011-03-08

The composite Newton-Cotes rules for the computation of hypersingular integral on interval is studied. The emphasis is placed on certain function, denoted by Sk(p)(τ), in the error functional, where τ is the local coordinate of the singular point. When Sk(p)(τ)=0 the so-called point wise superconvergence phenomenon occurs. Besides, the property of Sk(p)(τ) is presented. At last, numerical examples are provided to validate the theoretical analysis.

MR(2010)主题分类:

()
 [1] Andrews L C. Special Functions of Mathematics for Engineers, second ed., McGraw-Hill, Inc, 1992. [2] Du Q K. Evaluations of certain hypersingular integrals on interval[J]. Int. J. Numer. Meth. Eng., 2001, 51: 1195-1210. [3] Elliott D, Venturino E. Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals[J]. Numer. Math., 1997, 77: 453-465. [4] Hasegawa T. Uniform approximations to finite Hilbert transform and its derivative[J]. J. Comput. Appl. Math., 2004, 163: 127-138. [5] Hui C Y, Shia D. Evaluations of hypersingular integrals using Gaussian quadrature[J]. Int. J. Numer. Methods Eng., 1999, 44: 205-214. [6] Ioakimidis N I. On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives[J]. Math. Comp., 1985, 44: 191-198. [7] Linz P. On the approximate computation of certain strongly singular integrals[J]. Computing, 1985, 35: 345-353. [8] Monegato G. Numerical evaluation of hypersingular integrals[J]. J. Comput. Appl. Math., 1994, 50: 9-31, [9] Wu J M, Sun W. The superconvergence of the composite trapezoidal rule for Hadamard finite part integrals[J]. Numer. Math., 2005, 102: 343-363. [10] Zhang X P, Wu J M, Yu D H. The superconvergence of the composite Newton-Cotes rules for Hadamard finite-part integral on a circle[J]. Computing, 2009, 85: 219-244. [11] Li J, Wu J M, Yu D H. Generalized extrapolation for computation of hypersingular integrals in boundary element methods[J]. Comp. Model. Engng. Sci., 2009, 42: 151-175. [12] Wu J M and SunW W. The superconvergence of Newton-Cotes rules for the Hadamard finite-part integral on an interval[J]. Numer. Math., 2008, 109: 143-165. [13] 邬吉明, 余德浩. 区间上超奇异积分的一种近似计算方法[J]. 数值计算与计算机应用, 1998, 19(2): 118-126. [14] Yu D H. The approximate computation of hypersingular integrals on interval[J]. Numer. Math. J. Chinese Univ., 1992, 1(1): 114-127. [15] 余德浩. 自然边界元方法的数学理论[M]. 北京: 科学出版社, 1993. [16] Yu D H. The numerical computation of hypersingular integrals and its application in BEM, Adv. Engng. Software, 1993, 18: 103-109. [17] 余德浩. 圆周上超奇异积分计算及其误差估计[J]. 高等学校计算数学学报, 1994, 16(4): 332-339. [18] Yu D H. Natural Boundary Integrals Method and its Applications. Kluwer Academic Publishers, 2002.
 [1] 徐玉民, 李宣, 陈一鸣, 付小红. 用正交函数求超奇异积分的近似值及其误差估计[J]. 计算数学, 2013, 35(2): 215-224. [2] 陈一鸣, 仪明旭, 魏金侠, 陈娟. Legendre小波求解超奇异积分[J]. 计算数学, 2012, 34(2): 195-202.