Skip to main content
SpringerLink
Log in

Quadrature Formulas for Integrals with a Weak Singularity in the Kernel and a Weight Function of Jacobi Polynomials with Complex Exponents

  • Chapter
  • First Online:
Solid Mechanics, Theory of Elasticity and Creep

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 185))

Abstract

Quadrature formulas are presented for integrals with a logarithmic singularity and with a sign function, containing the weight function of Jacobi orthogonal polynomials, the exponents of which can be complex numbers with a real part greater than minus one. The latter are remarkable in that they have the same structure as the quadrature formulas for singular and regular integrals and can be used to solve singular integral equations that also contain terms with a weak singularity. Formulas for calculating the integral with a logarithm at an arbitrary point of the complex plane are also presented, and by numerical analysis, the area around the interval is outlined, outside of which this integral, with a certain degree of accuracy, can also be calculated using the quadrature formula for smooth functions.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Belotserkovsky, S.M., Lifanov, I.K.: Method of Discrete Vortices. CRC Press (1992)

    Google Scholar 

  2. Gabdulkhaev, B.G.: Direct Methods for Solving Singular Integral Equations of the First Kind. Kazan University Press, Kazan (1994).(in Russ.)

    Google Scholar 

  3. Ivanov, V.V.: The Theory of Approximate Methods and its Application to the Numerical Solution of Singular Integral Equations. Kyiv, Naukova dumka, (in Russ.) (1968)

    Google Scholar 

  4. Lifanov, I.K.: Method of Singular Integral Equations and Numerical Experiment. Janus LLP, Moscow (1995)

    Google Scholar 

  5. Prössdorf, S.: Einige Klassen singulärer Gleichungen. Birkhäuser (1974)

    Google Scholar 

  6. Pykhteev, G.N.: Exact Methods for Calculating Integrals of the Cauchy Type. Nauka, Siberian Branch, Novosibirsk (1980)

    MATH  Google Scholar 

  7. Sheshko, M.A.: Singular Integral Equations with Cauchy and Hilbert Kernel and their Approximate Solution. Scientific Society of the Catholic University of Lublin, Lublin (2003)

    Google Scholar 

  8. Stroud, A.H., Secrest, D.: Gaussian Quadrature Formulas. Prentice-Hall, N.J. (1966)

    MATH  Google Scholar 

  9. Boykov, I.V.: Approximate methods for calculating singular and hypersingular integrals. Part 1. Singular integrals. Izd-vo PGU, Penza, 360 (in Russ.) (2005)

    Google Scholar 

  10. Boykov, I.V.: Approximate methods for calculating singular and hypersingular integrals. Part 2. Hypersingular integrals. Izd-vo PGU, Penza, 250 (in Russ.) (2009)

    Google Scholar 

  11. Karpenko, L.N.: Approximate solution of a singular integral equation by means of Jacobi polynomials. J. Appl. Math. Mech. 30(3), 668–675 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  12. Erdogan, F., Gupta, G.D., Cook, T.S.: Numerical solution of singular integral equations. In: G.C. Sih (ed.) Mechanics of Fracture, vol. 1, pp. 368–425. Noordoff, Leyden (1973)

    Google Scholar 

  13. Krenk, S.: On the use of interpolation polynomial for solutions of singular integral equations. Quart. Appl. Math. 32(4), 479–484 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lifanov, I.K., Saakian, A.V.: Method of numerical solution of the problem of impressing a moving stamp into an elastic half-plane, taking heat generation into account. J. Appl. Math. Mech. 46(3), 388–394 (1982)

    Article  Google Scholar 

  15. Miller, G.R., Keer, L.M.: A numerical technique for the solution of singular integral equations of the second kind. Quart. Appl. Math. 42(4), 455–465 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Junghanns, P., Rathsfeld, A.: On polynomial collocation for Cauchy singular integral equations with fixed singularities. Integr. Eqn. Oper. Theory 43(2), 155–176 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ma, L.F., Korsunsky, A.M.: An efficient numerical method for the solution of sliding contact problems. Int. J. Numer. Meth. Engng 64, 1236–1255 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Andreev, A.V.: Direct numerical method for solving singular integral equations of the first kind with generalized kernels. Mech. Solids 40(1), 104–119 (2005)

    Google Scholar 

  19. Theocaris, P.S., Ioakimidis, N.I.: A method of numerical solution of Cauchy-type singular integral equations with generalized kernels and arbitrary complex singularities. J. Comp. Phys. 30(3), 309–323 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Andreev, A.V.: Method for numerical solution of complete singular integral equations with complex singularities of power type. Mech. Solids 41(1), 76–87 (2006)

    Google Scholar 

  21. Saakyan, A.V.: Quadrature formulas of maximum algebraic accuracy for Cauchy type integrals with complex exponents of the Jacobi weight function. Mech. Solids 47(6), 695–699 (2012)

    Article  ADS  Google Scholar 

  22. Theocaris, P.S., Chrysakis, A.C., Ioakimidis, N.I.: Cauchy type integrals and integral equations with logarithmic singularities. J. Eng. Math. 13(l), 63–74 (1979)

    Google Scholar 

  23. Yan, Y., Sloan, I.H.: On integral equations of the first kind with logarithmic kernel. Numer. Math. 53, 499–512 (1988)

    Google Scholar 

  24. Saranen, J.: The modified quadrature method for logarithmic-kernel integral equations on closed curves. J. Integr. Eqn. Appl. 3(4), 575–600 (FALL 1991). https://doi.org/10.1216/jiea/1181075650

  25. Celorrio, R., Domınguezand, V., Sayas, F.-J.: An improved quadrature method for integral equations with logarithmic kernel, Monografıas del Semin. Matem. Garcıa de Galdeano 27, 185–192 (2003)

    Google Scholar 

  26. Sahakyan, A.V., Amirjanyan, H.A.: Method of mechanical quadratures for solving singular integral equations of various types. In: IOP Conference Series: Journal of Physics: Conference Series vol. 991, p. 012070 (2018). https://doi.org/10.1088/1742-6596/991/1/012070

  27. Assari, P., Asadi-Mehregan, F., Cuomo, S.: A numerical scheme for solving a class of logarithmic integral equations arisen from two-dimensional Helmholtz equations using local thin plate splines. Appl. Math. Comput. 356, 157–172 (2019). https://doi.org/10.1016/j.amc.2019.03.042

    Article  MathSciNet  MATH  Google Scholar 

  28. Bateman, H., Erdelyi, A.: Higher Transcendental Functions, vol. 2. McGraw-Hill (1953)

    Google Scholar 

  29. Szego, G.: Orthogonal Polynomials. AMS, NY (1959)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanics, National Academy of Sciences of Armenia, 24B Baghramyan Ave., Yerevan, 0019, Republic of Armenia

    Avetik V. Sahakyan & Harutyun A. Amirjanyan

Authors
  1. Avetik V. Sahakyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Harutyun A. Amirjanyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Avetik V. Sahakyan .

Editor information

Editors and Affiliations

  1. Faculty of Mechanaical Engineering, Institute of Mechanics, Otto-von-Guericke-Universität Magdeburg, Magdeburg, Sachsen-Anhalt, Germany

    Holm Altenbach

  2. National Academy of Sciences of Armenia, Yerevan, Armenia

    Suren Manuk Mkhitaryan

  3. Institute of Mechanics, National Academy of Sciences of Armenia, Yerevan, Armenia

    Vahram Hakobyan

  4. Institute of Mechanics, National Academy of Sciences of Armenia, Yerevan, Armenia

    Avetik Varazdat Sahakyan

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Sahakyan, A.V., Amirjanyan, H.A. (2023). Quadrature Formulas for Integrals with a Weak Singularity in the Kernel and a Weight Function of Jacobi Polynomials with Complex Exponents. In: Altenbach, H., Mkhitaryan, S.M., Hakobyan, V., Sahakyan, A.V. (eds) Solid Mechanics, Theory of Elasticity and Creep. Advanced Structured Materials, vol 185. Springer, Cham. https://doi.org/10.1007/978-3-031-18564-9_21

Download citation

Publish with us

Policies and ethics

Search

Navigation