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

Kholmat Shadimetov, Aziz Boltaev () and Roman Parovik ()
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Kholmat Shadimetov: Department of Informatics and Computer Graphics, Tashkent State Transport University, 1 Odilkhodjayev Str., Tashkent 100167, Uzbekistan
Aziz Boltaev: Department of Informatics and Computer Graphics, Tashkent State Transport University, 1 Odilkhodjayev Str., Tashkent 100167, Uzbekistan
Roman Parovik: International Integrative Research Laboratory of Extreme Phenomena of Kamchatka, Vitus Bering Kamchatka State University, 4 Pogranichnaya St., Petropavlovsk-Kamchatskiy 683032, Russia

Mathematics, 2023, vol. 11, issue 14, 1-20

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 .

Keywords: differential operator; discrete analogue; Hilbert space; discrete argument functions; optimal quadrature formula (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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