Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev’s Method

Boltaev, A.K.; Hayotov, Abdullo R.; Shadimetov, Kholmat M.

doi:10.1007/s10114-021-9506-6

Cited by 14 publications

(6 citation statements)

References 25 publications

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“…We note that, in the present paper, we study Sard's problem in space W (m,0) 2 for even natural numbers m. For odd m, Sard's problem in this space was solved in [30].…”

Section: Extremal Function Of the Error Functional Of Quadrature Form...mentioning

confidence: 96%

“…It is worth mentioning that the process of constructing the discrete argument function is comparable to the process of constructing discrete analogues of differential operators d 2m dx 2m , and d 2m dx 2m − 1 (for odd m) in the works [27][28][29][30]. The discrete function D m (hβ) plays a significant role in calculating the coefficients of optimal quadrature formulas in the space W (m,0) 2 .…”

Section: A Discrete Analoguementioning

confidence: 99%

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

2023

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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 Dm(hβ) of the differential operator d2mdx2m+2dmdxm+1 in the Hilbert space W2(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 W2(2,0) for the case m=2.

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“…We note that, in the present paper, we study Sard's problem in space W (m,0) 2 for even natural numbers m. For odd m, Sard's problem in this space was solved in [30].…”

Section: Extremal Function Of the Error Functional Of Quadrature Form...mentioning

confidence: 96%

Section: A Discrete Analoguementioning

confidence: 99%

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

2023

Self Cite

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“…We will now find a solution to Equation (25). The theory of periodic generalized functions and the Fourier transform suggest that it is more convenient to search for a harrow-shaped function instead of a discrete function…”

Section: The Parity Of µmentioning

confidence: 99%

“…To complete this, he defined and studied a discrete analogue D dE 2m . In the work [24], it was possible to construct a discrete analogue of [25,26], discrete analogues of differential operators d 2m dx 2m − 1 (for odd m) and d 2m dx 2m + 2 d 2m−2 dx 2m−2 + 1 (if even m) were created, and their properties were studied.…”

Section: Introduction and Problem Statementmentioning

confidence: 99%

Weighted Optimal Formulas for Approximate Integration

Shadimetov,

Jalolov

2024

Mathematics

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Solutions to problems arising from much scientific and applied research conducted at the world level lead to integral and differential equations. They are approximately solved, mainly using quadrature, cubature, and difference formulas. Therefore, in the current work, we consider a discrete analogue of the differential operator 1−12π2d2dx2m in the Hilbert space H2μR, called Dmβ. We modify the Sobolev algorithm to construct optimal quadrature formulas using a discrete operator. We provide a weighted optimal quadrature formula, using this algorithm for the case where m=1. Finally, we construct an optimal quadrature formula in the Hilbert space H2μR for the weight functions px=1 and px=e2πiωx when m=1.

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“…A discrete analogue of the differential operator d 2m dx 2m − 1 in the Hilbert space W (m,0) 2 was constructed in the work [21]. The result which was obtained by using this discrete analogue is:…”

Section: Introductionmentioning

confidence: 99%

The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

Shadimetov,

Davronov

2023

Preprint

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The discrete analogue of the differential operator plays an important role in the construction of interpolation, quadrature and cubature formulas. In this work, we consider a discrete analogue $D_m(h\beta)$ of the differential operator $\frac{d^{2m}}{dx^{2m}}+1$ designed specifically for even natural numbers $m$. The effectiveness of this operator by constructing an optimal quadrature formula in the space $L_2^{(2,0)}(0,1)$ is shown. The errors of the optimal quadrature formula in $W_2^{(2,1)}(0,1)$ space and the optimal quadrature formula in $L_2^{(2,0)}(0,1)$ space are numerically compared. Numerical results showed that the error of the optimal quadrature formula what constructed in this work is smaller than the error of the quadrature formula constructed in $W_2^{(2,1)}(0,1)$ space. MSC: 41A05, 41A15, 65D30, 65D32, 65M60, 65L60.

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Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev’s Method

Cited by 14 publications

References 25 publications

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

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

Weighted Optimal Formulas for Approximate Integration

The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

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