American Journal of Numerical Analysis
ISSN (Print): 2372-2118 ISSN (Online): 2372-2126 Website: http://www.sciepub.com/journal/ajna Editor-in-chief: Emanuele Galligani
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American Journal of Numerical Analysis. 2013, 1(1), 22-31
DOI: 10.12691/ajna-1-1-4
Open AccessArticle

Optimal Quadrature Formulas for the Cauchy Type Singular Integral in the Sobolev Space L2(2)(-1,1)

Kholmat M. Shadimetov1, Abdullo R. Hayotov1, and Dilshod M. Akhmedov1

1Department of Computational Methods, Institute of Mathematics, National University of Uzbekistan, Tashkent, Uzbekistan

Pub. Date: December 02, 2013

Cite this paper:
Kholmat M. Shadimetov, Abdullo R. Hayotov and Dilshod M. Akhmedov. Optimal Quadrature Formulas for the Cauchy Type Singular Integral in the Sobolev Space L2(2)(-1,1). American Journal of Numerical Analysis. 2013; 1(1):22-31. doi: 10.12691/ajna-1-1-4

Abstract

This paper studies the problem of construction of the optimal quadrature formula in the sense of Sard in L2(2)(-1,1) S.L.Sobolev space for approximate calculation of the Cauchy type singular integral. Using the discrete analogue of the operator d4/dx4 we obtain new optimal quadrature formulas. Furthermore, explicit formulas of the optimal coefficients are obtained. Finally, in numerical examples, we give the error bounds obtained for the case h=0.02 by our optimal quadrature formula and compared with the corresponding error bounds of the quadrature formula (15) of the work [26] at different values of singular point t. The numerical results show that our quadrature formula is more accurate than the quadrature formula constructed in the work [26].

Keywords:
optimal quadrature formula singular integral of Cauchy type Sobolev space

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