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

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].