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Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator

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  • Kholmat Shadimetov

    (Department of Informatics and Computer Graphics, Tashkent State Transport University, 1 Odilkhodjayev Str., Tashkent 100167, Uzbekistan
    Computational Mathematics Laboratory, V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b University Str., Tashkent 100174, Uzbekistan)

  • Aziz Boltaev

    (Department of Informatics and Computer Graphics, Tashkent State Transport University, 1 Odilkhodjayev Str., Tashkent 100167, Uzbekistan
    Computational Mathematics Laboratory, V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, 4b University Str., Tashkent 100174, Uzbekistan
    Department of Computational Mathematics and Information Systems, National University of Uzbekistan Named after M. Ulugbek, 4 University Str., Tashkent 100174, Uzbekistan)

  • Roman Parovik

    (International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, 4 Pogranichnaya St., Petropavlovsk-Kamchatskiy 683032, Russia
    Laboratory for Simulation of Physical Processes, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, 7 Mirnaya St., Kamchatka Krai, Yelizovsky District, Paratunka 684034, Russia)

Abstract

It is known that discrete analogs of differential operators play an important role in constructing optimal quadrature, cubature, and difference formulas. Using discrete analogs of differential operators, optimal interpolation, quadrature, and difference formulas exact for algebraic polynomials, trigonometric and exponential functions can be constructed. In this paper, we construct a discrete analogue D m ( h β ) of the differential operator d 2 m d x 2 m + 2 d m d x m + 1 in the Hilbert space W 2 ( m , 0 ) . We develop an algorithm for constructing optimal quadrature formulas exact on exponential-trigonometric functions using a discrete operator. Based on this algorithm, in m = 2 , we give an optimal quadrature formula exact for trigonometric functions. Finally, we present the rate of convergence of the optimal quadrature formula in the Hilbert space W 2 ( 2 , 0 ) for the case m = 2 .

Suggested Citation

  • Kholmat Shadimetov & Aziz Boltaev & Roman Parovik, 2023. "Optimization of the Approximate Integration Formula Using the Discrete Analogue of a High-Order Differential Operator," Mathematics, MDPI, vol. 11(14), pages 1-20, July.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:14:p:3114-:d:1194278
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    References listed on IDEAS

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    1. Hayotov, Abdullo R. & Milovanović, Gradimir V. & Shadimetov, Kholmat M., 2015. "Optimal quadratures in the sense of Sard in a Hilbert space," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 637-653.
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    Cited by:

    1. Kholmat Shadimetov & Ikrom Jalolov, 2024. "Weighted Optimal Formulas for Approximate Integration," Mathematics, MDPI, vol. 12(5), pages 1-22, February.

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