Geodesic
Justification of the collocation method for the integral equation for a mixed boundary value problem for the Helmholtz equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 7, pp. 1340-1348 
E. G. Khalilov. Justification of the collocation method for the integral equation for a mixed boundary value problem for the Helmholtz equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 7, pp. 1340-1348. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_7_a9/
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     title = {Justification of the collocation method for the integral equation for a mixed boundary value problem for the {Helmholtz} equation},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The surface integral equation for a spatial mixed boundary value problem for the Helmholtz equation is considered. At a set of chosen points, the equation is replaced with a system of algebraic equations, and the existence and uniqueness of the solution of this system is established. The convergence of the solutions of this system to the exact solution of the integral equation is proven, and the convergence rate of the method is determined.

[1] Ahner J. F., “The exterior Dirichlet problem for the Helmholtz equation”, J. Math. Anal. Appl., 52 (1975), 415–429 | DOI | MR | Zbl

[2] Kleinman R. E., Wendland W., “On Neumann's method for the exterior Neumann problem for the Helmholtz equation”, J. Math. Anal. Appl., 57 (1977), 170–202 | DOI | MR | Zbl

[3] Ahner J. F., Kleinman R. E., “The exterior Neumann problem for the Helmholtz equation”, Arch. Rational Mech. Anal., 52 (1973), 26–43 | DOI | MR | Zbl

[4] Musaev B. I., Khalilov E. G., “O priblizhennom reshenii odnogo klassa granichnykh integralnykh uravnenii metodom kollokatsii”, Tr. IMM AN Azerb. Resp., 9(17) (1998), 78–84

[5] Abdullaev F. A., Khalilov E. G., “Obosnovanie metoda kollokatsii dlya odnogo klassa granichnykh integralnykh uravnenii”, Differents. ur-niya, 40:1 (2004), 82–86 | MR | Zbl

[6] Kashirin A. A., Smagin S. I., “O chislennom reshenii zadach Dirikhle dlya uravneniya Gelmgoltsa metodom potentsialov”, Zh. vychisl. matem. i matem. fiz., 52:8 (2012), 1492–1505 | MR | Zbl

[7] Panich O. I., “K voprosu o razreshimosti vneshnikh kraevykh zadach dlya volnovogo uravneniya i dlya sistemy uravnenii Maksvella”, Uspekhi matem. nauk, 20:1 (1965), 221–226 | MR | Zbl

[8] Kolton D., Kress R., Metody integralnykh uravnenii v teorii rasseyaniya, Mir, M., 1987 | MR

[9] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1976

[10] Kustov Yu. A., Musaev B. I., Kubaturnaya formula dlya dvumernogo singulyarnogo integrala i ee prilozheniya, Dep. v VINITI No 4281-81, M., 1981

[11] Khalilov E. H., “Cubic formula for class of weakly singular surface integrals”, Proc. IMM of NAS Azerbaijan, 39(47) (2013), 69–76 | MR | Zbl

[12] Vainikko G. M., “Regulyarnaya skhodimost operatorov i priblizhennoe reshenie uravnenii”, Itogi nauki i tekhniki. Ser. Matem. analiz, 16, 1979, 5–53