Weighted Optimal Formulas for Approximate Integration

Kholmat Shadimetov () and Ikrom Jalolov
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Kholmat Shadimetov: Department of Informatics and Computer Graphics, Tashkent State Transport University, 1 Odilkhodjayev Str., Tashkent 100167, Uzbekistan
Ikrom Jalolov: Department of Informatics and Computer Graphics, Tashkent State Transport University, 1 Odilkhodjayev Str., Tashkent 100167, Uzbekistan

Mathematics, 2024, vol. 12, issue 5, 1-22

Abstract: 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 − 1 2 π 2 d 2 d x 2 m in the Hilbert space H 2 μ R , called D m β . 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 H 2 μ R for the weight functions p x = 1 and p x = e 2 π i ω x when m = 1 .

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: 2024
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