Optimal Quadrature Formulas with Odd-Order Derivative Corrections in the Sobolev Space \(L_2^{(m)}(0,1)\)

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DOI: https://doi.org/10.28924/2291-8639-24-2026-295
Published: Sep 21, 2026
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Farxod Nuraliev, Shaxobiddin Kuziev, Saidakhon Toshboeva, Zilola Khursanova, Makhliyo Dekhqonova, Umidakxon Akxmedova, Optimal Quadrature Formulas with Odd-Order Derivative Corrections in the Sobolev Space \(L_2^{(m)}(0,1)\), Int. J. Anal. Appl., 24 (2026), 295.

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Farxod Nuraliev, Shaxobiddin Kuziev, Saidakhon Toshboeva, Zilola Khursanova, Makhliyo Dekhqonova, Umidakxon Akxmedova

Abstract

This paper presents the construction of optimal quadrature formulas with derivative corrections in the Sobolev space \(L_2^{(m)}(0,1)\). The proposed approach is based on Sobolev's method together with the discrete analogue of the differential operator of order \(2m\). A complete analytical derivation of the optimal coefficients is obtained by reducing the problem to a finite system of linear equations for the unknown parameters. Closed-form representations of these coefficients are established, and an explicit expression for the squared norm of the corresponding error functional is derived. The resulting quadrature formulas extend several previously known optimal formulas and provide an effective framework for the accurate numerical evaluation of definite integrals of sufficiently smooth functions.

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