Skip to main content
SpringerLink
Log in

Approximate Methods for Calculating Hypersingular Integrals

  • Published:
Technical Physics Aims and scope Submit manuscript

Abstract

At present, hypersingular integrals find an increasing number of fields of application—aerodynamics, elasticity theory, electrodynamics, and geophysics. Moreover, their calculation in an analytical form is possible only in particular special cases. Therefore, approximate methods for calculating hypersingular integrals are an urgent problem in computational mathematics. Many works are devoted to this problem. An even greater number of works are devoted to approximate methods for calculating singular integrals. Studies of approximate methods for calculating singular integrals began much earlier than similar studies of hypersingular integrals. Results have been obtained in this area that have no analogues for hypersingular integrals. It is of considerable interest to extend the methods for calculating singular integrals to hypersingular integrals, based on the connection between some classes of singular and hypersingular integrals. The work is devoted to this task. The construction of quadrature formulas for calculating hypersingular integrals is based on the methods of the constructive theory of functions and the theory of singular and hypersingular integrals. A method for constructing quadrature formulas for calculating hypersingular integrals is proposed that is based on the transformation of quadrature formulas for calculating singular integrals. Quadrature formulas for calculating several classes of hypersingular and polyhypersingular integrals are constructed. Estimates of the error of the constructed quadrature formulas are obtained. The constructed methods make it possible to efficiently calculate hypersingular integrals when solving applied problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. V. V. Ivanov, Theory of Approximate Methods and Its Application to the Numerical Solution of Singular Integral Equations (Naukova Dumka, Kiev, 1968) [in Russian].

    Google Scholar 

  2. I. V. Boikov, Approximate Methods for Computing Singular and Hypersingular Integrals, Part 1: Singular Integrals (Penzensk. Gos. Univ., Penza, 2005) [in Russian].

  3. I. V. Boikov, Approximate Methods for Computing Singular and Hypersingular Integrals, Part 2: Hypersingular Integrals (Penzensk. Gos. Univ., Penza, 2009) [in Russian].

  4. Sh. S. Khubezhty, Quadrature Formulas for Singular Integrals and Some of Their Applications (Yuzhn. Mat. Inst., Vladikavkaz, 2011) [in Russian].

    Google Scholar 

  5. I. K. Lifanov, Method of Singular Integral Equations and Numerical Experiment (Yanus, Moscow, 1995) [in Russian].

    MATH  Google Scholar 

  6. G. M. Vainikko, I. K. Lifanov, and I. N. Poltavskii, Numerical Methods in Hypersingular Equations and Their Applications (Yanus-K, Moscow, 2001) [in Russian].

    MATH  Google Scholar 

  7. H. R. Kutt, Numer. Math. 24, 205 (1975).

    Article  MathSciNet  Google Scholar 

  8. D. F. Paget, Numer. Math. 36, 447 (1981).

    Article  Google Scholar 

  9. M. I. Ioakimidis, Anal. Numer. Theor. Approx. 12, 131 (1983).

    MathSciNet  Google Scholar 

  10. B. Bialecki, Numer. Math. 57, 263 (1990).

    Article  MathSciNet  Google Scholar 

  11. E. Lutz, L. J. Gray, and A. R. Ingraffea, Boundary Elem. 13, 913 (1991).

    Google Scholar 

  12. G. Monegato, Numer. Math. 50, 273 (1987).

    Article  MathSciNet  Google Scholar 

  13. G. Monegato, J. Comput. Appl. Math. 50, 9 (1994).

    Article  MathSciNet  Google Scholar 

  14. G. Monegato, Math. Comput. 64, 765 (1994).

    Article  ADS  MathSciNet  Google Scholar 

  15. K. Diethelm, J. Comput. Appl. Math. 65, 97 (1995).

    Article  MathSciNet  Google Scholar 

  16. G. Crisculo, J. Comput. Appl. Math. 78, 255 (1997).

    Article  MathSciNet  Google Scholar 

  17. A. M. Korsunsky, Q. Appl. Math. 56, 461 (1998).

    Article  MathSciNet  Google Scholar 

  18. I. V. Boykov, J. Math. Math. Sci. 28, 127 (2001).

    Article  Google Scholar 

  19. I. V. Boykov, A. I. Boykova, and E. S. Ventsel, Eng. Anal. Boundary Elem. 28, 1437 (2004).

    Article  Google Scholar 

  20. A. V. Saakyan, Izv. Nats. Akad. Naik Armenii, Mekh. 73 (2), 44 (2020).

    Google Scholar 

  21. A. V. Saakyan, Izv. Vyssh. Uchebn. Zaved., Sev.-Kavkaz. Reg., Estestv. Nauki, No. 2, 94 (2020).

  22. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Yale Univ. Press, New Haven, 1923).

    MATH  Google Scholar 

  23. J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (Dover, New York, 1952).

    MATH  Google Scholar 

  24. L. A. Chikin, Uch. Zap. Kazansk. Gos. Univ. 113 (10), 57 (1953).

    Google Scholar 

  25. I. V. Boikov, N. F. Dobrynina, and L. N. Domnin, Approximate Methods for Calculating Hadamard Integrals and Solving Hypersingular Integral Equations (Penzensk. Gos. Tekhnol. Univ., Penza, 1996) [in Russian].

    Google Scholar 

  26. P. Kolm and V. Rokhlin, Comput. Math. Appl. 41, 327 (2001).

    Article  MathSciNet  Google Scholar 

  27. I. V. Boykov, E. S. Ventsel, and A. I. Boykova, Appl. Numer. 59 (6), 1366 (2009).

    Article  Google Scholar 

  28. A. C. Kaya and E. Erdogan, Q. Appl. Math. 45 (1), 105 (1987).

    Article  Google Scholar 

  29. A. M. Linkov and S. G. Mogilevskaya, Prikl. Mat. Mekh. 50, 844 (1986).

    Google Scholar 

  30. C. Y. Hui and D. Shia, Int. J. Numer. Methods Eng. 44, 205 (1999).

    Article  Google Scholar 

  31. Q. K. Du, Int. J. Numer. Methods Eng. 51, 1195 (2001).

    Article  Google Scholar 

  32. U. J. Choi, S. W. Kim, and B. Yun, Int. J. Numer. Methods Eng. 61, 496 (2004).

    Article  Google Scholar 

  33. I. V. Boykov, E. S. Ventsel, and A. I. Boykova, Eng. Anal. Boundary Elem. 30, 799 (2006).

    Article  Google Scholar 

  34. J. Hildenbrand and G. Kuhn, Anal. Boundary Elem. 10, 209 (1992).

    Article  Google Scholar 

  35. Y.-S. Chan, A. C. Fannjiang, and G. H. Paulino, Int. J. Eng. Sci. 41, 683 (2003).

    Article  Google Scholar 

  36. A. M. Korsunsky, Proc. R. Soc. London, Ser. A 458 (2027), 2721 (2002). https://www.jstor.org/stable/3560029

    Article  ADS  MathSciNet  Google Scholar 

  37. O. Kazuhiro and N. Nao-Aki, Key Eng. Mater. 385–387, 793 (2008).

    Google Scholar 

  38. S. J. Obaiys, R. W. Ibrahim, and A. F. Ahmad, “Hypersingular integrals in integral equations and inequalities: Fundamental review study,” in Differential and Integral Inequalities, Ed. by D. Andrica and T. M. Rassias (Springer, 2019), pp. 687–717.

    MATH  Google Scholar 

  39. L. Yu. Plieva, Numer. Anal. Appl. 9, 326 (2016). https://doi.org/10.1134/S1995423916040066

    Article  MathSciNet  Google Scholar 

  40. I. V. Boikov, Yu. F. Zakharova, G. I. Grinchenkov, and M. A. Semov, Izv. Vyssh. Uchebn. Zaved., Povolzhsk. Reg., Fiz.-Mat. Nauki, No. 3, 5 (2013).

  41. I. V. Boikov and M. A. Semov, Izv. Vyssh. Uchebn. Zaved., Mat., No. 3, 3 (2016).

  42. Yu. S. Soliev, Proc. 19th Int. Saratov Winter School, Dedicated to the 90th Anniversary of Acad. P. L. Ul’yanov “Modern Problems of Functions Theory and Their Applications” (Nauchn. Kniga, Saratov, 2018), pp. 297–299 [in Russian].

    Google Scholar 

  43. I. V. Boikov, P. V. Aikashev, and A. I. Boikova, Zh. Srednevolzh. Mat. O-va 22 (4), 405 (2020).

    Google Scholar 

  44. J. M. Wu and D. H. Yu, J. Numer. Math. Appl. 21, 25 (1999).

    Google Scholar 

  45. X. Zhang, J. Wu, and D. Yu, J. Comput. Appl. Math. 223 (2), 598 (2009).

    Article  ADS  MathSciNet  Google Scholar 

  46. C. Hu, X. He, and T. Lu, Discrete Contin. Dyn. Syst. 20 (5), 1355 (2015).

    Google Scholar 

  47. I. K. Lifanov, in Optimal Calculation Methods and Their Application (Penzensk. Politekh. Inst., Penza, 1985), No. 7, pp. 38–45.

  48. Yu. V. Gandel’, S. V. Eremenko, and T. S. Polyakova, Mathematical Issues of the Discrete Current Method (Khar’kovsk. Gos. Univ., Khar’kov, 1992), Chap. 2.

    Google Scholar 

  49. A. Zygmund, Trigonometric Series, 2nd ed. (Cambridge Univ. Press, Cambridge, 1959), Vol. 1.

    MATH  Google Scholar 

  50. D. Gaier, Konstruktive Methoden der Konformen Abbildung (Springer, Berlin–Heidelberd, 1964).

Download references

Funding

This work was supported financially by the Russian Foundation for Basic Research (grant no. 16-01-00594) and a Rector’s Grant of Penza State University (agreement no. 1/RG of August 4, 2020).

Author information

Authors and Affiliations

  1. Penza State University, 440026, Penza, Russia

    I. V. Boykov & P. V. Aykashev

Authors
  1. I. V. Boykov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. V. Aykashev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to I. V. Boykov or P. V. Aykashev.

Additional information

Translated by M. Drozdova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boykov, I.V., Aykashev, P.V. Approximate Methods for Calculating Hypersingular Integrals. Tech. Phys. 67, 448–455 (2022). https://doi.org/10.1134/S1063784222070039

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1063784222070039

Keywords:

Search

Navigation