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

Kholmat M. Shadimetov¹, Abdullo R. Hayotov^1, and Dilshod M. Akhmedov¹

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

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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References:

[1]	Lifanov I.K. The method of singular equations and numerical experiments. Moscow, TOO "Yanus", 1995. -520p. (in Russian)
[2]	Gakhov F.D. Boundary problems. –Moscow, Nauka, 1977. 640p. (in Russian).
[3]	Muskhelishvili N.I. Singular Integral Equations. -M.: Nauka, 1968. 512p. (in Russian)
[4]	Nikol’skii S.M. Concerning estimation for approximate quadrature formulas, Uspekhi Mat. Nauk 5 (2(36)) (1950) 165-177 (in Russian).
[5]	Blaga P., Coman Gh. Some problems on optimal quadrature, Stud. Univ. Babeş-Bolyai Math. 52 (4) (2007) 21-44.
[6]	Bojanov B. Optimal quadrature formulas, Uspekhi Mat. Nauk. 60 (6(366)) (2005) 33-52 (in Russian).
[7]	Catinaş T., Coman Gh. Optimal quadrature formulas based on the φ-function method, Stud. Univ. Babeş-Bolyai Math. 51 (1) (2006) 49-64.
[8]	Chakhkiev M.A. Linear differential operators with real spectrum, and optimal quadrature formulas, Izv. Akad. Nauk SSSR Ser. Mat. 48 (5) (1984) 1078-1108 (in Russian).
[9]	Nikol’skii S.M. Quadrature Formulas, Nauka, Moscow, 1988 (in Russian).
[10]	Zhensikbaev A.A. Monosplines of minimal norm and the best quadrature formulas, Uspekhi Mat. Nauk. 36 (1981) 107-159 (in Russian).
[11]	Sard A. Best approximate integration formulas; best approximation formulas, Amer. J. Math. 71 (1949) 80-91.
[12]	Lanzara F. On optimal quadrature formulae, J. Inequal. Appl. 5 (2000) 201-225.
[13]	Sobolev S.L. The coefficients of optimal quadrature formulas, in: Selected Works of S.L. Sobolev, Springer, 2006, pp. 561-566.
[14]	Sobolev S.L., VaskevichV.L. The Theory of Cubature Formulas. Kluwer Academic Publishers Group, Dordrecht (1997).
[15]	Shadimetov Kh.M., Hayotov A.R. Optimal quadrature formulas with positive coefficients in L2(m) (0,1) space. Journal of Comp. and Applied Math. 235 (2011) 1114-1128.
[16]	Hayotov A.R, Milovanovic G.V., Shadimetov Kh.M. On an optimal quadrature formula in the sense of Sard. Numerical Algorithms (2011) 57, 487-510.
[17]	Shadimetov Kh.M., Hayotov A.R., Azamov S.S. Optimal quadrature formula in K2P2 space. Applied Numerical Mathematics (2012) 62, 1893-1909.
[18]	Shadimetov Kh.M., Hayotov A.R. Construction of interpolation splines minimizing semi-norm in W2(m,m-1)(0,1) space. BIT Numerical Mathematics (2013) 53, 545-563.
[19]	Shadimetov Kh.M., Hayotov A.R., Nuraliev F.A. Journal of Comp. and Applied Math. 243 (2013) 91-112.
[20]	Hayotov A.R, Milovanovic G.V., Shadimetov Kh.M. Interpolation splines minimizing a semi-norm. Calcolo, 2013.
[21]	Shadimetov Kh.M., Hayotov A.R. Optimal quadrature formulas in the sense of Sard in W2(m,m-1)(0,1) space Calcolo, 2013,
[22]	Belotserkovskii S.M., Lifanov I.K. Numerical Methods in Singular Integral Equations, Nauka, Moscow, 1985 (in Russian).
[23]	Gabdulkhaev B.G. Cubature formulas for many dimensional singular integrals I // Izvestiya Vuzov. Mathematics. 1975. 4. 3-13. (in Russian).
[24]	Israilov M.I., Shadimetov Kh.M. Weight optimal quadrature formulas for singular integrals of the Couchy type. Doklady AN RUz 1991, 8, 10-11.
[25]	Shadimetov Kh.M. Optimal quadrature formulas for singular integrals of the Couchy type. Doklady AN Ruz 1987, 6, Pp.9-11 (in Russian).
[26]	Eshkuvatov Z.K., N.M.A.Nik Long, Abdulkawi M. Numerical evaluation for Caushy type singular integrals on the interval. Journal of Computational and Applied Mathematics. 2009. p. 1995-2001.
[27]	Shadimetov Kh.M. Coefficients of the weight optimal quadrature formulas in the space L2(m) (0,1)// Proceedings of the V-the International Conference “Cubature Formulas and Thier Applications” –Krasnoyarsk, 1999. -Pp.207-214. (in Russian) ].
[28]	Shadimetov Kh.M. The discrete analogue of the operator d2m/dx2m and its properties // Voprosy vichislitelnoy i prikladnoy matematiki. Tashkent, 1985. С. 22-35.