Optimal Hermite-Type Quadrature with Weakly Singular Weight Via Phi-Functions
DOI:
https://doi.org/10.71310/pcam.4-1_75.2026.13Keywords:
optimal quadrature formula, ????-function method, weakly singular weight, Hermite-type formula, Sobolev space, error functionalAbstract
Background: Many practical problems in physics and engineering involve calculating integrals with singularities. Traditional calculation methods lose accuracy near these singular points, so new methods are needed that can handle such difficult integrals.
Purpose: This paper develops an improved calculation method for such integrals. The new method uses both function values and their slopes at selected points and is optimized to give the best possible accuracy.
Method: We develop the mathematical framework by creating a special function that captures the behavior of the singular integral. By optimizing the parameters of this special function, we determine the best weights and coefficients for the calculation.
Results: We find all the coefficients needed for the new calculation method and prove that the method works correctly. The method is accurate for polynomial functions, and numerical tests show it gives results that are three to five times more accurate than existing standard methods.
Implication: The new method is practical and efficient for solving real problems that involve difficult integrals with singularities, and the approach can be extended to other similar problems.
References
Nikolsky S. M. 1950. On the issue of estimates of approximations by quadrature formulas. Advances in Mathematical Sciences. 5(3): 165–177. (In Russian)
Cătinaş T., Coman Gh. 2006. Optimal quadrature formulas based on the ????-function method. Studia Universitatis Babeş-Bolyai Mathematica. A51(1): 49–64.
Sard A. 1963. Linear approximation. 2nd ed. Providence, Rhode Island: American Mathematical Society.
Sobolev S. L. 1974. Introduction to the theory of cubature formulas. Moscow: Nauka. (In Russian)
Sobolev S. L. 1977. Coefficients of optimal quadrature formulas. Doklady Akademii Nauk SSSR. 235(1): 34–37. (In Russian)
Lanzara F. 2000. On optimal quadrature formulae. Journal of Inequalities and Applications. 5: 201–225.
Shadimetov Kh. M., Hayotov A. R. 2022. Optimal approximation of error functionals of quadrature and interpolation formulas in spaces of differentiable functions. Tashkent: Muhr Press.
Shadimetov Kh. M., Azamov S. S. 2024. On the construction of optimal quadrature formulas with equally spaced nodes. Filomat. 38(29): 10279–10295.
Hayotov A. R., Jeon S., Lee C.-O. 2020. On an optimal quadrature formula for approximation of Fourier integrals in the space ????(1)2 . Journal of Computational and Applied Mathematics. 372: Art. 112713.
Hayotov A. R., Jeon S., Shadimetov Kh. M. 2021. Application of optimal quadrature formulas for reconstruction of CT images. Journal of Computational and Applied Mathematics. 388: Art. 113313.
Hayotov A. R., Shomalikova M. Sh. 2025. An optimal quadrature formula exact to the exponential function. Problems of Computational and Applied Mathematics. 5(69): 74–85.
Hayotov A. R., Berdimuradova U. A. 2025. An optimal quadrature formula with derivatives for arbitrarily fixed nodes in the Sobolev space. Problems of Computational and Applied Mathematics. 2(64): 64–73.
Hayotov A. R., Babaev S., Abduakhadov A., Davronov J. 2024. An optimal quadrature formula exact to the exponential function by the phi function method. Studia Universitatis Babeş-Bolyai Mathematica. 69(3): 651–663. doi: https://doi.org/10.24193/subbmath.2024.3.11.
Abduakhadov A. A. 2023. Construction of an optimal quadrature formula by the ????-function method. Problems of Computational and Applied Mathematics. 3/1(50): 100–110.
Abduakhadov A. A. 2024. Optimal approximation of Fourier integrals by the phi-function method. Problems of Computational and Applied Mathematics. 4/2(60): 109–118.
Abduakhadov A. A. 2026. Construction of an optimal quadrature formula for the approximate evaluation of Fourier integrals using the ????-function method. Scientific Reports of Bukhara State University. 3: 26–30.
Abduakhadov A. A. 2025. Coefficients of optimal quadrature formula with parameter. Acta NUUz Exact Sciences. 2/2.1: 4–10.
Hayotov A. R., Babaev S. S., Abduaxadov A. A. 2026. Construction of an optimal quadrature formula for the approximate calculation of Fourier integrals using the ????-function method. Russian Mathematics. 70(1): 73–85.
Hayotov A. R., Ravshanova M. Q. 2025. Construction of optimal quadrature formulas exact for exponential functions using the ????-function method. Bulletin of the Institute of Mathematics. 8(4): 44–50.
Hayotov A. R., Khaitov T. O. 2025. Algebraic-trigonometric optimal formulas for numerical integration. Problems of Computational and Applied Mathematics. 5(69): 102–113.
Hayotov A. R., Khaitov T. O., Hayotov M. A. 2025. Optimal quadrature formulas with derivative exact for trigonometric functions. Uzbek Mathematical Journal. 69(1): 71–78.
Hayotov A. R., Khaitov T. O. 2025. On optimality and error estimation of a quadrature formula with derivative that is exact for trigonometric functions. International Journal of Analysis and Applications. 23(270): 1–11.
Downloads
Published
Issue
Section
License
Copyright (c) 2026 D.M. Akhmedov
This work is licensed under a Creative Commons Attribution 4.0 International License.
