Optimization of the Approximate Integration Formula Using the Quadrature Formula

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DOI: https://doi.org/10.28924/2291-8639-23-2025-53
Published: Mar 3, 2025
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Kh.M. Shadimetov, A.K. Boltaev, Optimization of the Approximate Integration Formula Using the Quadrature Formula, Int. J. Anal. Appl., 23 (2025), 53.

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Kh.M. Shadimetov, A.K. Boltaev

Abstract

The article explores Sard’s problem of constructing optimal quadrature formulas in the space W(4,0)2(0,1) by using Sobolev’s method. This problem involves two steps: first, calculating the norm of the error functional, and then finding the minimum of this norm by the coefficients of the quadrature formulas. The norm of the error functional is computed using the extremal function. Then, by using the method of Lagrange multipliers, a system of linear equations for the coefficients of the optimal quadrature formulas in the space W(4,0)2(0,1) is derived, and the existence and uniqueness of the solution to this system are discussed. The paper then proceeds to construct the optimal quadrature formula using the discrete analogue D4(hβ) of the high-order differential operator. Finally, the optimal quadrature formulas that are exact for exponential-trigonometric functions are obtained.

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