Skip to main content
SpringerLink
Log in

Substantiation of the Collocation Method for One Class of Systems of Integral Equations

  • Published:
Ukrainian Mathematical Journal Aims and scope

We present the substantiation of the collocation method for a system of surface integral equations of the boundary-value problem of conjugation for the Helmholtz equations. Moreover, we construct a sequence convergent to the exact solution of the boundary-value problem of conjugation and estimate its error.

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. G. A. Mikhailov and A. F. Cheshkova, “Solution of the difference Dirichlet problem for the many-dimensional Helmholtz equation,” Zh. Vychisl. Mat. Mat. Fiz., 38, No. 1, 99–106 (1998).

    MathSciNet  Google Scholar 

  2. A. M. Radin and V. P. Shestopalov, “Diffraction of plane waves on a sphere with circular hole,” Zh. Vychisl. Mat. Mat. Fiz., 14, No. 5, 1232–1243 (1974).

    Google Scholar 

  3. A. G. Sveshnikov, “Numerical methods in the diffraction theory,” in: Proc. of the Internat. Congr. of Mathematicians, Vancouver (1974), pp. 437–442.

  4. Yu. K. Sirenko and V. P. Shestopalov, “Free oscillations of electromagnetic fields in one-dimensional periodic lattices,” Zh. Vychisl. Mat. Mat. Fiz., 24, No. 2, 262–271 (1987).

    Google Scholar 

  5. F. A. Abdullaev and É. G. Khalilov, “Substantiation of the collocation method for one class of boundary integral equations,” Differents. Uravn., 40, No. 1, 82–86 (2004).

    MATH  Google Scholar 

  6. V. S. Bulygin, “Scalar third boundary-value problem of the mathematical diffraction theory on a plane screen and its discrete mathematical model,” Vestn. Kharkov. Nats. Univ., Mat. Model. Inform. Tekhnolog. Avtomat. Sist. Upravl., Issue 7, No. 775, 62–72 (2007).

  7. Yu. V. Gandel’, “Boundary-value problems for the Helmholtz equation and their discrete mathematical models,” Sovr. Mat. Fundam. Naprav., 36, 36–49 (2010).

    MathSciNet  MATH  Google Scholar 

  8. A. A. Kashirin, “On the conditionally well-posed integral equations and numerical solution of stationary problems of diffraction of acoustic waves,” Vestn. TOGU, No. 3(26), 33–40 (2012).

  9. A. A. Kashirin and S. I. Smagin, “On the numerical solution of Dirichlet problems for the Helmholtz equation by the method of potentials,” Zh. Vychisl. Mat. Mat. Fiz., 52, No. 8, 1492–1505 (2012).

    MathSciNet  MATH  Google Scholar 

  10. V. D. Kupradze, “Method of integral equations in the diffraction theory,” Mat. Sb., 41, No. 4, 561–581 (1934).

    MATH  Google Scholar 

  11. M. Yu. Medvedik, Yu. G. Smirnov, and A. A. Tsupak, “Scalar problem of diffraction of plane waves on a system of disjoint screens and inhomogeneous bodies,” Zh. Vychisl. Mat. Mat. Fiz., 54, No. 8, 1319–1331 (2014).

    MATH  Google Scholar 

  12. I. N. Pleshchinskii and N. B. Pleshchinskii, “Integral equations of the problem of conjugation of semiopen dielectric waveguides,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 5(540), 63–80 (2007).

  13. É. G. Khalilov, “Substantiation of the collocation method for the integral equation of the mixed boundary-value problem for the Helmholtz equation,” Zh. Vychisl. Mat. Mat. Fiz., 56, No. 7, 1340–1348 (2016).

    MathSciNet  Google Scholar 

  14. É. G. Khalilov, “On the approximate solution of one class of boundary integral equation of the first kind,” Differents. Uravn., 52, No. 9, 1277–1283 (2016).

    Google Scholar 

  15. P. J. Harris and K. Chen, “On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the threedimensional exterior Helmholtz problem,” J. Comput. Appl. Math., 156, 303–318 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  16. E. H. Khalilov, “On approximate solution of external Dirichlet boundary-value problem for Laplace equation by collocation method,” Azerb. J. Math., 5, No. 2, 13–20 (2015).

    MathSciNet  MATH  Google Scholar 

  17. R. Kress, “Boundary integral equations in time-harmonic acoustic scattering,” Math. Comput. Modelling, 15, No. 3-5, 229–243 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Colton and R. Kress, Integral Equation Methods in Scattering Theory [Russian translation], Mir, Moscow (1987).

    MATH  Google Scholar 

  19. V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1981).

    Google Scholar 

  20. Yu. A. Kustov and B. I. Musaev, Cubature Formula for a Two-Dimensional Singular Integral and Its Applications [in Russian], Deposited at VINITI, No. 4281-8, Moscow (1981).

  21. E. H. Khalilov, “Cubic formula for a class of weakly singular surface integrals,” Proc. Inst. Math. Mech. Nat. Acad. Sci. Azerbaijan, 39(47), 69–76 (2013).

    MathSciNet  MATH  Google Scholar 

  22. G. M. Vainikko, “Regular convergence of operators and the approximate solution of equations,” in: VINITI Series, Mathematical Analysis [in Russian], Vol. 16, VINITI, Moscow (1979), pp. 5–53.

Download references

Author information

Authors and Affiliations

  1. Azerbaijan State University of Oil and Industry, Baku, Azerbaijan

    É. H. Khalilov

Authors
  1. É. H. Khalilov
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 6, pp. 823–835, June, 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khalilov, É.H. Substantiation of the Collocation Method for One Class of Systems of Integral Equations. Ukr Math J 69, 955–969 (2017). https://doi.org/10.1007/s11253-017-1406-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-017-1406-7

Search

Navigation