Numerical Solution of a Singular Integral Equation of the First Kind with Hilbert Kernel

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-224
Published: Sep 19, 2025
Cite as:
Kh.M. Shadimetov, D.M. Akhmedov, Kh.Kh. Jabborov, Numerical Solution of a Singular Integral Equation of the First Kind with Hilbert Kernel, Int. J. Anal. Appl., 23 (2025), 224.

Main Article Content

Kh.M. Shadimetov, D.M. Akhmedov, Kh.Kh. Jabborov

Abstract

In solving practical problems in the fields of physics and engineering, singular integral equations are frequently encountered. Among these, singular integral equations with the Hilbert kernel constitute the periodic cases. In this article, we discuss the construction of an optimal quadrature formula for the numerical solution of Fredholm-type singular integral equations of the first kind with Hilbert kernels using the functional approach in the space L2(1)(0,2π). Using the constructed optimal quadrature formula, the error between the exact solution and the approximate solution of the integral equation is demonstrated through examples. Graphs illustrate how the approximate value converges to the exact value as the number of nodes in the optimal quadrature formula increases.

Article Details

References

  1. D. Akhmedov, K. Shadimetov, K. Rustamov, On an Optimal Method for the Approximate Solution of Singular Integral Equations, AIP Conf. Proc. 3045 (2024), 060032. https://doi.org/10.1063/5.0199829.
  2. K. Shadimetov, A. Hayotov, D. Akhmedov, Optimal Quadrature Formulas for Cauchy Type Singular Integrals in Sobolev Space, Appl. Math. Comput. 263 (2015), 302–314. https://doi.org/10.1016/j.amc.2015.04.066.
  3. W. Han, K.E. Atkinson, Theoretical Numerical Analysis, Springer, New York, 2009. https://doi.org/10.1007/978-1-4419-0458-4.
  4. R. Estrada, R.P. Kanwal, Singular Integral Equations, Birkhäuser, Boston, 2000. https://doi.org/10.1007/978-1-4612-1382-6.
  5. F.D. Gakhov, Boundary Value Problems, Nauka, Moscow, 1977.
  6. I.S. Gradshteĭn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 1965.
  7. A. Hayotov, F. Nuraliev, G. Abdullaeva, The Coeffisients of the Spline Minimizing Semi Norm in K2(P3), Int. J. Anal. Appl. 23 (2025), 7. https://doi.org/10.28924/2291-8639-23-2025-7.
  8. N.I. Ioakimidis, A Natural Interpolation Formula for the Numerical Solution of Singular Integral Equations with Hilbert Kernel, BIT 23 (1983), 92–104. https://doi.org/10.1007/bf01937329.
  9. S. KRENK, Numerical Quadrature of Periodic Singular Integral Equations, IMA J. Appl. Math. 21 (1978), 181–187. https://doi.org/10.1093/imamat/21.2.181.
  10. I.K. Lifanov, Method of Singular Equations and Numerical Experiment in Mathematical Physics, Theory of Elasticity and Diffraction, Janus, Moscow, 1995.
  11. S.G. Mikhlin, Integral Equations: And Their Applications to Certain Problems in Mechanics, Mathematical Physics and Technology, Pergamon Press, Oxford, 1964.
  12. N.I. Muskhelishvili, Singular Integral Equations, P. Noordhoff, Groningen, Netherlands, 1953.
  13. S.M. Nikolskii, To Question About Estimation of Approximation by Quadrature Formulas, Uspekhi Mat. Nauk 5 (1950), 165–177.
  14. S.M. Nikolskii, Quadrature Formulas, Nauka, Moscow, 1988.
  15. M. Rahman, Integral Equations and Their Applications, WIT Press, 2007.
  16. A. Sard, Best Approximate Integration Formulas; Best Approximation Formulas, Amer. J. Math. 71 (1949), 80–91. https://doi.org/10.2307/2372095.
  17. A. Sard, Integral Representations of Remainders, Duke Math. J. 15 (1948), 33–45. https://doi.org/10.1215/s0012-7094-48-01532-4.
  18. A. Sard, Linear Approximation, American Mathematical Society, Providence, 1963.
  19. K.M. Shadimetov, The Discrete Analogue of the Differential Operator d2m/dx2m and Its Construction, in: Questions of Computational and Applied Mathematics, pp. 22–35, (1985).
  20. K.M. Shadimetov, Weighted Optimal Quadrature Formulas in a Periodic Sobolev Space, Uzbek Math. J. 2 (1998), 76–86.
  21. D.M. Akhmedov, K.M. Shadimetov, Optimal Quadrature Formulas for Approximate Solution of the Rst Kind Singular Integral Equation with Cauchy Kernel, Stud. Univ. Babes-Bolyai Mat. 67 (2022), 633–651. https://doi.org/10.24193/subbmath.2022.3.15.
  22. K.M. Shadimetov, D.M. Akhmedov, Approximate Solution of a Singular Integral Equation Using the Sobolev Method, Lobachevskii J. Math. 43 (2022), 496–505. https://doi.org/10.1134/s1995080222050249.
  23. K.M. Shadimetov, D.M. Akhmedov, Numerical Integration Formulas for Hypersingular Integrals, Numer. Math.: Theory Methods Appl. 17 (2024), 805–826. https://doi.org/10.4208/nmtma.oa-2024-0028.
  24. K.M. Shadimetov, D.M. Akhmedov, An Optimal Approximate Solution of the I Kind Fredholm Singular Integral Equations, Filomat 38 (2024), 10765–10796. https://doi.org/10.2298/fil2430765s.
  25. K.M. Shadimetov, A.R. Hayotov, G.N. Akhmadaliev, An Algorithm for Construction of Optimal Integration Formulas in Hilbert Spaces and Its Realization, Int. J. Anal. Appl. 22 (2024), 100. https://doi.org/10.28924/2291-8639-22-2024-100.
  26. K. Shadimetov, A. Boltaev, Optimization of the Approximate Integration Formula Using the Quadrature Formula, Int. J. Anal. Appl. 23 (2025), 53. https://doi.org/10.28924/2291-8639-23-2025-53.
  27. K. Shadimetov, F. Nuraliev, S. Kuziev, Optimal Quadrature Formula of Hermite Type in the Space of Differentiable Functions, Int. J. Anal. Appl. 22 (2024), 25. https://doi.org/10.28924/2291-8639-22-2024-25.
  28. K. Shadimetov, A. Hayotov, U. Khayriev, Optimal Quadrature Formulas for Approximating Strongly Oscillating Integrals in the Hilbert Space $widetilde{W}_{2}^{(m,m-1)}$ of Periodic Functions, J. Comput. Appl. Math. 453 (2025), 116133. https://doi.org/10.1016/j.cam.2024.116133.
  29. S.L. Sobolev, Introduction to the Theory of Cubature Formulas, Nauka, 1974.
  30. S.L. Sobolev, The Coefficients of Optimal Quadrature Formulas, in: G.V. Demidenko, V.L. Vaskevich (Eds.), Selected Works of S.L. Sobolev, Springer, 2006: pp. 561–565. https://doi.org/10.1007/978-0-387-34149-1_35.
  31. Z. Chen, Y. Zhou, An Efficient Algorithm for Solving Hilbert Type Singular Integral Equations of the Second Kind, Comput. Math. Appl. 58 (2009), 632–640. https://doi.org/10.1016/j.camwa.2009.01.045.