Analysis of approximate solution for a class of systems of integral equations. (English. Russian original) Zbl 1505.65321
Comput. Math. Math. Phys. 62, No. 5, 811-826 (2022); translation from Zh. Vychisl. Mat. Mat. Fiz. 62, No. 5, 838-853 (2022).
Summary: The paper presents a substantiation of the collocation method for a system of integral equations for the field-matching boundary value problem for the Helmholtz equation in two-dimensional space. Quadrature formulas are constructed for the single and double layer potentials and the normal derivative of the single layer potential. At definite points, the system of integral equations is replaced by a system of algebraic equations, and the existence and uniqueness of a solution of the system of algebraic equations is shown. The convergence of the solution of a system of algebraic equations to the exact solution of a system of integral equations is proved, and the convergence rate of the method is found. In addition, a sequence is constructed that converges to the exact solution of the field-matching boundary value problem.
MSC:
| 65R20 | Numerical methods for integral equations |
Keywords:
field-matching boundary value problem; Helmholtz equation; system of integral equations; single and double layer potentials; Hankel function; quadrature formulas; collocation methodReferences:
| [1] | Colton, D.; Kress, R., Integral Equation Methods in Scattering Theory (1984), New York: Wiley, New York · Zbl 1291.35003 |
| [2] | Kress, R.; Roach, G. F., Transmission problems Helmholtz equation, J. Math. Phys., 19, 1433-1437 (1978) · Zbl 0433.35017 · doi:10.1063/1.523808 |
| [3] | Kashirin, A. A.; Smagin, S. I.; Taltykina, M. Yu., Mosaic-skeleton method as applied to the numerical solution of three-dimensional Dirichlet problems for the Helmholtz equation in integral form, Comput. Math. Math. Phys., 56, 612-625 (2016) · Zbl 1353.65130 · doi:10.1134/S0965542516040096 |
| [4] | Khalilov, E. H., Justification of the collocation method for the integral equation for a mixed boundary value problem for the Helmholtz equation, Comput. Math. Math. Phys., 56, 1310-1318 (2016) · Zbl 1367.65178 · doi:10.1134/S0965542516070101 |
| [5] | Harris, P. J.; Chen, K., On efficient preconditioners for iterative solution of a Galerkin boundary element equation for the three-dimensional exterior Helmholtz problem, J. Comput. Appl. Math., 156, 303-318 (2003) · Zbl 1029.65131 · doi:10.1016/S0377-0427(02)00918-4 |
| [6] | Khalilov, E. H.; Aliev, A. R., Justification of a quadrature method for an integral equation to the external Neumann problem for the Helmholtz equation, Math. Methods Appl. Sci., 41, 6921-6933 (2018) · Zbl 1402.45003 · doi:10.1002/mma.5204 |
| [7] | Turc, C.; Boubendir, Y.; Riahi, M. K., Well-conditioned boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz problems with impedance boundary conditions in two-dimensional Lipschitz domains, J. Integral Equations Appl., 29, 441-472 (2017) · Zbl 1376.65146 · doi:10.1216/JIE-2017-29-3-441 |
| [8] | Khalilov, E. H., Substantiation of the collocation method for one class of systems of integral equations, Ukr. Math. J., 69, 955-969 (2017) · Zbl 1499.65751 · doi:10.1007/s11253-017-1406-7 |
| [9] | E. H. Khalilov and M. N. Bakhshaliyeva, “Quadrature formulas for simple and double layer logarithmic potentials,” Proc. IMM Natl. Acad. Sci. Az. 45 (1), 155-162 (2019). · Zbl 1495.65227 |
| [10] | Kress, R., Boundary integral equations in time-harmonic acoustic scattering, Math. Comput. Model., 15, 229-243 (1991) · Zbl 0731.76077 · doi:10.1016/0895-7177(91)90068-I |
| [11] | Muskhelishvili, N. I., Singular Integral Equations (1972), Groningen: Wolters-Noordhoff, Groningen · Zbl 0108.29203 |
| [12] | Vladimirov, V. S., Equations of Mathematical Physics (1971), New York: Marcel Dekker, New York · Zbl 0231.35002 |
| [13] | Khalilov, E. H., Justification of the collocation method for a class of surface integral equations, Math. Notes, 107, 663-678 (2020) · Zbl 1442.35081 · doi:10.1134/S0001434620030335 |
| [14] | Vainikko, G. M., Regular convergence of operators and the approximate solution of equations, J. Sov. Math., 15, 675-705 (1981) · Zbl 0582.65046 · doi:10.1007/BF01377042 |
| [15] | Bakhshaliyeva, M. N.; Khalilov, E. H., Justification of the collocation method for an integral equation of the exterior Dirichlet problem for the Laplace equation, Comput. Math. Math. Phys., 61, 923-937 (2021) · Zbl 1473.65355 · doi:10.1134/S0965542521030039 |
| [16] | Khalilov, E. H.; Bakhshaliyeva, M. N., Justification of the collocation method for an integral equation of the mixed boundary value problem for the Laplace equation, Ufa Math. J., 13, 85-97 (2021) · Zbl 1474.45076 · doi:10.13108/2021-13-1-85 |
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