An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space

Hayotov, A.R., Berdimuradova, U.A., Хаётов, А.Р., Бердимурадовa, У.А.

Ҳисоблаш ва амалий математика муаммолари · 2025-yil

Annotatsiya

The numerical computation of definite integrals plays a crucial role in various applied and theoretical disciplines. In many cases, exact analytical evaluation of integrals is in feasible due to the complexity of the integrand or the nature of the integration limits. Quadrature formulas provide an efficient approach for approximating definite integrals, relying on weighted sums of function values at selected nodes. Traditional quadrature formulas, such as those of Newton-Cotes, Gauss, aim to optimize accuracy by care fully choosing nodes and weights. However, optimal quadrature formulas can also be constructed in the sense of Sard, minimizing the error functional within a given func tion space. In this paper, we focus on constructing an optimal quadrature formula in the Sobolev space with arbitrarily fixed nodes. Unlike conventional approaches where coefficients are determined sequentially, we simultaneously optimize both function and derivative coefficients, improving overall accuracy and stability. The derivation employs the method of- functions, which allows us to express the quadrature formula’s co efficients explicitly and analyze its error properties. The obtained quadrature formula minimizes the error norm in the chosen function space, ensuring an improved approxi mation of definite integrals. Furthermore, when the nodes are equally spaced, our results generalize the well-known Euler-Maclaurin formula, demonstrating the effectiveness of our approach.

Maqola ma’lumotlari
MualliflarHayotov, A.R., Berdimuradova, U.A., Хаётов, А.Р., Бердимурадовa, У.А.
JurnalҲисоблаш ва амалий математика муаммолари
NashriyotТошкент ахборот технологиялари университети қошидаги, Ахборот-коммуникация технологиялари илмий-инновацион, маркази
ISSN2181-8460, 2181-046X
Nashr sanasi2025-05-15
Son2
Betlar64-73
TilIngliz
DOI10.71310/pcam.2_64.2025.06
Indeksga qo‘shilgan2026-08-27

DOI: 10.71310/pcam.2_64.2025.06 · Maqolaning asl sahifasi

Kalit so‘zlar

quadrature formula, Sobolev space, optimal approximation, Euler-Maclaurin formula, квадратурная формула, пространство Соболева, оптимальная аппроксимация, формула Эйлера-Маклорена

Iqtibos keltirish

APAHayotov, A.R., Berdimuradova, U.A., Хаётов, А.Р., & Бердимурадовa, У.А. (2025). An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space. Ҳисоблаш ва амалий математика муаммолари, (2), 64-73. https://doi.org/10.71310/pcam.2_64.2025.06
GOST (OAK)Hayotov, A.R. va b. An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space // Ҳисоблаш ва амалий математика муаммолари. – 2025. – № 2. – P. 64-73. – DOI: 10.71310/pcam.2_64.2025.06.
MLAHayotov, A.R., Berdimuradova, U.A., Хаётов, А.Р., and Бердимурадовa, У.А. “An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space.” Ҳисоблаш ва амалий математика муаммолари, (2), 64-73, 2025. https://doi.org/10.71310/pcam.2_64.2025.06
ChicagoHayotov, A.R., Berdimuradova, U.A., Хаётов, А.Р., and Бердимурадовa, У.А. “An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space.” Ҳисоблаш ва амалий математика муаммолари, (2), 64-73 (2025). https://doi.org/10.71310/pcam.2_64.2025.06
HarvardHayotov, A.R., Berdimuradova, U.A., Хаётов, А.Р., and Бердимурадовa, У.А (2025) ‘An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space’, Ҳисоблаш ва амалий математика муаммолари, (2), 64-73. https://doi.org/10.71310/pcam.2_64.2025.06
BibTeX
@article{hayotovar2025,
  title = {An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space},
  author = {Hayotov, A.R. and Berdimuradova, U.A. and Хаётов, А.Р. and Бердимурадовa, У.А.},
  journal = {Ҳисоблаш ва амалий математика муаммолари},
  year = {2025},
  number = {2},
  pages = {64-73},
  doi = {10.71310/pcam.2_64.2025.06},
  url = {https://doi.org/10.71310/pcam.2_64.2025.06}
}

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