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 to check access.

## 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). - 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). - 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). - 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). - 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). - 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). - 10.
V. D. Kupradze, “Method of integral equations in the diffraction theory,”

*Mat. Sb.*,**41**, No. 4, 561–581 (1934). - 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). - 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). - 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). - 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). - 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). - 17.
R. Kress, “Boundary integral equations in time-harmonic acoustic scattering,”

*Math. Comput. Modelling*,**15**, No. 3-5, 229–243 (1991). - 18.
D. Colton and R. Kress,

*Integral Equation Methods in Scattering Theory*[Russian translation], Mir, Moscow (1987). - 19.
V. S. Vladimirov,

*Equations of Mathematical Physics*[in Russian], Nauka, Moscow (1981). - 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). - 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.

## Author information

### Affiliations

### Corresponding author

Correspondence to É. H. Khalilov.

## Additional information

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

## Rights and permissions

## About this article

### 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

Received:

Revised:

Published:

Issue Date: