An Optimal Method for the Approximate Solution of the Hypersingular Integral Equations
DOI:
https://doi.org/10.71310/pcam.6_70.2025.10Keywords:
optimal quadrature formulas, extremal function, Sobolev space, optimal coefficients, Hadamard type singular integralAbstract
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 solutions themselves are also expressed through singular integrals. Therefore, it becomes necessary to develop formulas for the approximate calculation of these solutions, and such formulas have been developed. Among these formulas, very few are highly accurate. However, they are not optimal. Our work is devoted to constructing an optimal quadrature formula for the approximate calculation of the analytical solutions of a hypersingular integral equation.
References
Akhmedov D.M., Hayotov A.R., Shadimetov Kh.M. Optimal quadrature formulas with derivatives for Cauchy type singular integrals // Applied Mathematics and Computation. – 2018. – Vol. 317. – P. 150-159.
Boykov I., Roudnev V., Boykova A. Approximate methods for solving linear and nonlinear hypersingular integral equations // Axioms (MDPI). – 2020. – Vol. 9, Issue 3. – Art. 74.
Dagnino C., Santi E. Spline product quadrature rules for Cauchy singular integrals // J. Comput. Appl. Math. – 1990. – Vol. 33. – P. 133-140.
Dagnino C., Lamberti P. Numerical evaluation of Cauchy principal value integrals based on local spline approximation operators // J. Comput. Appl. Math. – 1996. – Vol. 76. – P. 231-238.
Diethelm K. Gaussian quadrature formulas of the first kind for Cauchy principal value integrals: basic properties and error estimates // J. Comput. Appl. Math. – 1995. – Vol. 65. – P. 97-114.
Diethelm K. Error bounds for spline-based quadrature methods for strongly singular integrals // J. Comput. Appl. Math. – 1998. – Vol. 89. – P. 257-261.
Dovgy S.A., Lifanov I.K., Cherniy D.I. Method of singular integral equations and computing technologies. – Kyiv: Yuston, 2016.
Eshkuvatov Z.K., Nik Long N.M.A., Abdulkawi M. Quadrature formula for approximating the singular integral of Cauchy type with unbounded weight function on the edges // J.Comput. Appl. Math. – 2009. – Vol. 233. – P. 334-345.
Eshkuvatov Z.K., Nik Long N.M.A., Abdulkawi M. Numerical evaluation for Cauchy type singular integrals on the interval // J. Comput. Appl. Math. – 2010. – Vol. 233, Issue 8. – P. 1995-2001.
Hasegawa T. Numerical integration of functions with poles near the interval of integration // J. Comput. Appl. Math. – 1997. – Vol. 87. – P. 339-357.
Hasegawa T., Suguira H. Quadrature rule for indefinite integral of algebraic-logarifmic singular integrands // J. Comput. Appl. Math. – 2007. – Vol. 205. – P. 487-496.
Lifanov I.K. The method of singular equations and numerical experiments. – Moscow: TOO “Yanus”, 1995.
Lifanov I.K. Hypesingular integral equations and their applacations. – 2004.
Sobolev S.L. Introduction to the Theory of Cubature Formulas. – Moscow: Nauka, 1974.
Shadimetov Kh.M. On an extremal function of quadrature formulas // Reports of Uzbekistan Academy of Sciences. – 1995. – No. 9-10. – P. 3-5.
Shadimetov Kh.M. Reducing the construction of composite optimal quadrature formulas to the solution of difference schemes // Reports of Uzbekistan Academy of Sciences. – 1998. – No. 7. – P. 3-6.
Shadimetov Kh.M. Optimal lattice quadrature and cubature formulas in Sobolev spaces // Doklady Mathematics. – Vol. 63, No. 1. – P. 92-94.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 D.M. Akhmedov
This work is licensed under a Creative Commons Attribution 4.0 International License.
