Geodesic
On the approximate solution of a class of weakly singular integral equations
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 3, pp. 310-325.

Voir la notice de l'article provenant de la source Math-Net.Ru

The work is devoted to the study of the solution of one class of weakly singular surface integral equations of the second kind. First, a Lyapunov surface is partitioned into “regular” elementary parts, and then a cubature formula for one class of weakly singular surface integrals is constructed at the control points. Using the constructed cubature formula, the considered integral equation is replaced by a system of algebraic equations. As a result, under the additional conditions imposed on the kernel of the integral, it is proved that the considered integral equation and the resulting system of algebraic equations have unique solutions, and the solution of the system of algebraic equations converges to the value of the solution of the integral equation at the control points. Moreover, using these results, we substantiate the collocation method for various integral equations of the external Dirichlet boundary-value problem for the Helmholtz equation. 
@article{ISU_2020_20_3_a3,
     author = {E. H. Khalilov},
     title = {On the approximate solution of a class of weakly singular integral equations},
     journal = {Izvestiya of Saratov University. Mathematics. Mechanics. Informatics},
     pages = {310--325},
     publisher = {mathdoc},
     volume = {20},
     number = {3},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ISU_2020_20_3_a3/}
}
TY  - JOUR
AU  - E. H. Khalilov
TI  - On the approximate solution of a class of weakly singular integral equations
JO  - Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
PY  - 2020
SP  - 310
EP  - 325
VL  - 20
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/ISU_2020_20_3_a3/
LA  - ru
ID  - ISU_2020_20_3_a3
ER  - 
%0 Journal Article
%A E. H. Khalilov
%T On the approximate solution of a class of weakly singular integral equations
%J Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
%D 2020
%P 310-325
%V 20
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/ISU_2020_20_3_a3/
%G ru
%F ISU_2020_20_3_a3
E. H. Khalilov. On the approximate solution of a class of weakly singular integral equations. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 3, pp. 310-325. http://geodesic.mathdoc.fr/item/ISU_2020_20_3_a3/

[1] F. A. Abdullayev, E. H. Khalilov, “Constructive method for solving the external Neumann boundary-value problem for the Helmholtz equation”, Proceedings of IMM of NAS of Azerbaijan, 44:1 (2018), 62–69 | MR | Zbl

[2] J. Bremer, Z. Gimbutas, “A Nystrom method for weakly singular integral operators on surfaces”, J. Comput. Phys., 231:14 (2012), 4885–4903 | DOI | MR | Zbl

[3] O. Gonzalez, J. Li, “A convergence theorem for a class of Nystrom methods for weakly singular integral equations on surfaces in $\mathbb{R}^3$”, Mathematics of Computation, 84:292 (2015), 675–714 | DOI | MR | Zbl

[4] I. G. Graham, I. H. Sloan, “Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in $\mathbb{R}^3$”, Numer. Math., 92:2 (2002), 289–323 | DOI | MR | Zbl

[5] P. J. Harris, K. Chen, “On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem”, J. Comput. Appl. Math., 156:2 (2003), 303–318 | DOI | MR | Zbl

[6] R. Kress, “Boundary integral equations in time-harmonic acoustic scattering”, Math. Comput. Modelling, 15:3-5 (1991), 229–243 | DOI | MR | Zbl

[7] T. Cai, “A fast solver for a hypersingular boundary integral equation”, Appl. Numer. Math., 59 (2009), 1960–1969 | DOI | MR | Zbl

[8] L. Farina, P. A. Martinc, V. Peron, “Hypersingular integral equations over a disc: Convergence of a spectral method and connection with Tranter's method”, J. Comput. Appl. Math., 269 (2014), 118–131 | DOI | MR | Zbl

[9] E. H. Khalilov, “Constructive Method for Solving a Boundary Value Problem with Impedance Boundary Condition for the Helmholtz Equation”, Differ. Equ., 54:4 (2018), 539–550 | DOI | DOI | MR | Zbl

[10] R. Kress, “A collocation method for a hypersingular boundary integral equation via trigonometric differentiation”, J. Integral Equations Applications, 26:2 (2014), 197–213 | DOI | MR | Zbl

[11] I. K. Lifanov, S. L. Stavtsev, “Integral equations and sound propagation in a shallow sea”, Differ. Equ., 40:9 (2004), 1330–1344 | DOI | MR | Zbl

[12] V. S. Vladimirov, Equations of mathematical physics, Marcel Dekker Inc., New York, 1971, 426 pp. | MR | MR | Zbl

[13] G. M. Vainikko, “Regular convergence of operators and approximate solution of equations”, Journal of Soviet Mathematics, 15:6 (1981), 675–705 | DOI | MR | Zbl

[14] D. L. Colton, R. Kress, Integral equation methods in scattering theory, John Wiley and Sons, 1983, 271 pp. | MR | Zbl

[15] E. H. Khalilov, “Some properties of the operators generated by a derivative of the acoustic double layer potential”, Sib. Math. J., 55:3 (2014), 564–573 | DOI | MR | Zbl