This paper addresses the problem of constructing optimal quadrature formulas for the numerical evaluation of Hadamard-type hypersingular integrals with hihg oscillation. Such integrals frequently arise in the analysis of boundary-value problems for partial differential equations, particularly in hyperbolic and wave-type models, as well as in various applied fields such as elasticity theory, fluid dynamics, and electromagnetic scattering. Our primary focus is on deriving the analytical expressions for the coefficients of an optimal quadrature formula tailored to the structure of the singular and oscillatory kernel. The goal is to achieve minimal error in a rigorous sense by constructing formulas that are optimal in the sense of Sard, meaning that they minimize the worst-case error over a specified function class. To this end, we develop an optimal quadrature formula of the form (2) in the Sobolev space, applying the Sobolev method for constructing optimal quadrature formulas. This method leverages variational principles and operator-theoretic tools to systematically minimize the norm of the error functional. The resulting formulas are well-adapted to handle the challenging combination of hypersingularity and oscillation, and they offer both theoretical optimality and practical computational advantages.
| Mualliflar | Akhmedov, D.M., Buvasherov, D.S., Ахмедов, Д.М., Бувашеров, Д.С. |
|---|---|
| Jurnal | Ҳисоблаш ва амалий математика муаммолари |
| Nashriyot | Тошкент ахборот технологиялари университети қошидаги, Ахборот-коммуникация технологиялари илмий-инновацион, маркази |
| ISSN | 2181-8460, 2181-046X |
| Nashr sanasi | 2025-09-20 |
| Son | 4 |
| Betlar | 65-74 |
| Til | Ingliz |
| Ilmiy soha | Fizika-matematika |
| DOI | 10.71310/pcam.4_68.2025.09 |
| Indeksga qo‘shilgan | 2026-08-27 |
DOI: 10.71310/pcam.4_68.2025.09 · Maqolaning asl sahifasi
Sobolev space, Sobolev method, singular integrals, hypersingular integrals, error functional, optimal quadrature formulas, error estimation, пространство Соболева, метод Соболева, сингулярные интегралы, гиперсингулярные интегралы, функционал погрешности, оптимальные квадратурные формулы, оценка погрешности
| APA | Akhmedov, D.M., Buvasherov, D.S., Ахмедов, Д.М., & Бувашеров, Д.С. (2025). An Optimal Quadrature Formula for Hadamard-Type Hypersingular Integrals with High Oscillation in the Sobolev Space. Ҳисоблаш ва амалий математика муаммолари, (4), 65-74. https://doi.org/10.71310/pcam.4_68.2025.09 |
|---|---|
| GOST (OAK) | Akhmedov, D.M. va b. An Optimal Quadrature Formula for Hadamard-Type Hypersingular Integrals with High Oscillation in the Sobolev Space // Ҳисоблаш ва амалий математика муаммолари. – 2025. – № 4. – P. 65-74. – DOI: 10.71310/pcam.4_68.2025.09. |
| MLA | Akhmedov, D.M., Buvasherov, D.S., Ахмедов, Д.М., and Бувашеров, Д.С. “An Optimal Quadrature Formula for Hadamard-Type Hypersingular Integrals with High Oscillation in the Sobolev Space.” Ҳисоблаш ва амалий математика муаммолари, (4), 65-74, 2025. https://doi.org/10.71310/pcam.4_68.2025.09 |
| Chicago | Akhmedov, D.M., Buvasherov, D.S., Ахмедов, Д.М., and Бувашеров, Д.С. “An Optimal Quadrature Formula for Hadamard-Type Hypersingular Integrals with High Oscillation in the Sobolev Space.” Ҳисоблаш ва амалий математика муаммолари, (4), 65-74 (2025). https://doi.org/10.71310/pcam.4_68.2025.09 |
| Harvard | Akhmedov, D.M., Buvasherov, D.S., Ахмедов, Д.М., and Бувашеров, Д.С (2025) ‘An Optimal Quadrature Formula for Hadamard-Type Hypersingular Integrals with High Oscillation in the Sobolev Space’, Ҳисоблаш ва амалий математика муаммолари, (4), 65-74. https://doi.org/10.71310/pcam.4_68.2025.09 |
| BibTeX | @article{akhmedovdm2025,
title = {An Optimal Quadrature Formula for Hadamard-Type Hypersingular Integrals with High Oscillation in the Sobolev Space},
author = {Akhmedov, D.M. and Buvasherov, D.S. and Ахмедов, Д.М. and Бувашеров, Д.С.},
journal = {Ҳисоблаш ва амалий математика муаммолари},
year = {2025},
number = {4},
pages = {65-74},
doi = {10.71310/pcam.4_68.2025.09},
url = {https://doi.org/10.71310/pcam.4_68.2025.09}
} |
Boshqa maqola uchun iqtibos tayyorlash
This article presents a comprehensive mathematical modeling of thermo-electro magneto-elastic processes in anisotropic thin plates of complex shapes using the R function method (RFM). The model considers interrelated…
Many applied problems are described by the Dirichlet problem for the Poisson equa tions. This problem is used in the following practical situations: membrane or plate deflection, electric potential, gravitational and…
This work proposes an expanded 3D Euler model for calculating the spatiotemporal evolution of pollutant concentrations released into the atmosphere under changing meteorological conditions. The model, alongside the…
This article presents an integrated mathematical model of radial filtration in a cylindrical filter, in which the flow movement is carried out from the outer layer to the inner cavity. Free flow areas are described by…
The article considers the problem of constructing self-similar solutions of a system of nonlinear differential equations modeling mutual diffusion processes in multicomponent media. An analysis of a mathematical model…
In this work, the problem of stabilizing the equilibrium state for a hyperbolic equation with negative nonlocal characteristic velocity and measurement error is investigated. The formulation of the mixed control problem…
In the world, special attention is given to developing various optimal computational methods for the approximate calculation of definite integrals. This article focuses on constructing composite optimal quadrature…
The article studies the dynamics of derivatives up to the highest derivative of an ordinary differential equation of the fourth order with a small parameter at the highest derivative. The solution of the differential…
The article investigates optimal quadrature formulas designed for the approximate evaluation of highly oscillatory integrals, which frequently arise in many applied problems of mathematical physics, signal theory, and…
The relationship between class objects is considered for analyzing the cluster structure of the training dataset. The uniqueness of the solution to the problem of minimal coverage of the training set by prototypes is…
Ҳисоблаш ва амалий математика муаммолари — barcha maqolalar · 2025-yil arxivi
Hypersingular integral equations are encountered in a number of fields, such as the dynamics of air and fluid flow, elasticity, and wave propagation theory. Analytical solutions for these integral equations exist, but…
The solution of the Cauchy problem for partial differential equations of hyperbolic type J. Hadamard led to singular integrals of a special form. Later they were called integrals in the sense of Hadamard, or Hadamard…
This article presents the results of 2018-2023, such as biological diversity, inter-landscape distribution characteristics, density,and number of straggly insects identified from natural and anthropogenic areas of…
This article observational data on the species composition, distribution, population, density, and seasonal damage caused causedby herbivorus bugs. Observationes were carried out in April and September 2021. When…
In the article, the research fields of some species belonging to Paeonia, their significance in human life, adaptation features tovarious ecological conditions are highlighted, and the ecologist groups and other…
This article studies the number of species, population density of terrestrial mollusks of the Topalangdarya basin, located in theSurkhandarya region of Uzbekistan, in various altitudinal regions and biotopes and…
This article sheds light on changes in biomass formation and antimicrobial activity against test pathogenic strains under theinfluence of epigenetic modifiers and elicitors of the endophytic fungus Penicillium…
Shahimardonsoy covered issues such as the soil in the river basin and measures to increase its condition, salinity and fertility, theflora and fauna present in the basin landscape and their conservation.