Geodesic
Solution of a class of integral equations with fixed singularities in the kernel by the mechanical quadrature method
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 156 (2014) no. 2, pp. 5-15 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Computational schemes for the mechanical quadrature method were constructed on the basis of a simplest interval quadrature formula for Fredholm integral equations of the second kind with fixed integrable singularities in the kernel on the internal and external variables. The theoretical and functional justification of these schemes was given. In particular, the convergence of the method in the space of functions square-integrable in the interval of integration with a weight depending on the characteristics of the integral operator's kernel was proved. The results obtained for the one-dimensional case were also extended to the multidimensional case.
Keywords: integral equation, fixed singularities in the kernel, weighted Lebesgue space, method of mechanical quadratures, convergence of a method.
Mots-clés : interval quadrature formula 
@article{UZKU_2014_156_2_a0,
     author = {Yu. R. Agachev and A. I. Leonov and I. P. Semenov},
     title = {Solution of a~class of integral equations with fixed singularities in the kernel by the mechanical quadrature method},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {5--15},
     year = {2014},
     volume = {156},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2014_156_2_a0/}
}
TY  - JOUR
AU  - Yu. R. Agachev
AU  - A. I. Leonov
AU  - I. P. Semenov
TI  - Solution of a class of integral equations with fixed singularities in the kernel by the mechanical quadrature method
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2014
SP  - 5
EP  - 15
VL  - 156
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/UZKU_2014_156_2_a0/
LA  - ru
ID  - UZKU_2014_156_2_a0
ER  - 
%0 Journal Article
%A Yu. R. Agachev
%A A. I. Leonov
%A I. P. Semenov
%T Solution of a class of integral equations with fixed singularities in the kernel by the mechanical quadrature method
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2014
%P 5-15
%V 156
%N 2
%U http://geodesic.mathdoc.fr/item/UZKU_2014_156_2_a0/
%G ru
%F UZKU_2014_156_2_a0
Yu. R. Agachev; A. I. Leonov; I. P. Semenov. Solution of a class of integral equations with fixed singularities in the kernel by the mechanical quadrature method. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 156 (2014) no. 2, pp. 5-15. http://geodesic.mathdoc.fr/item/UZKU_2014_156_2_a0/

[1] Vainikko G. M., “O skhodimosti metoda mekhanicheskikh kvadratur dlya integralnykh uravnenii s razryvnymi yadrami”, Sib. matem. zhurn., 12:1 (1971), 40–53 | MR

[2] Gabdulkhaev B. G., “K chislennomu resheniyu integralnykh uravnenii metodom mekhanicheskikh kvadratur”, Izv. vuzov. Matem., 1972, no. 12, 23–39 | MR | Zbl

[3] Sharipov R. N., “Nailuchshie intervalnye kvadraturnye formuly dlya klassov Lipshitsa”, Konstruktivnaya teoriya funktsii i funktsionalnyi analiz, 4, Izd-vo Kazan. un-ta, Kazan, 1983, 124–132 | MR | Zbl

[4] Milovanović G. V., Cvetković A. S., “Nonstandard Gaussian quadrature formulae based on operator values”, Adv. Comput. Math., 32:4 (2010), 431–486 | DOI | MR | Zbl

[5] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980, 496 pp. | MR | Zbl

[6] Gabdulkhaev B. G., “Splain-metody resheniya odnogo klassa singulyarnykh integro-differentsialnykh uravnenii”, Izv. vuzov. Matem., 1975, no. 6, 14–24 | MR | Zbl

[7] Gabdulkhaev B. G., Optimalnye approksimatsii reshenii lineinykh zadach, Izd-vo Kazan. un-ta, Kazan, 1980, 232 pp. | MR

[8] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1984, 752 pp. | MR | Zbl