Geodesic
The Cauchy problem for a singularly perturbed integro-differential Fredholm equation
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 4, pp. 655-664 
N. N. Nefedov; A. G. Nikitin. The Cauchy problem for a singularly perturbed integro-differential Fredholm equation. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 4, pp. 655-664. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a7/
@article{ZVMMF_2007_47_4_a7,
     author = {N. N. Nefedov and A. G. Nikitin},
     title = {The {Cauchy} problem for a~singularly perturbed integro-differential {Fredholm} equation},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {655--664},
     year = {2007},
     volume = {47},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a7/}
}
TY  - JOUR
AU  - N. N. Nefedov
AU  - A. G. Nikitin
TI  - The Cauchy problem for a singularly perturbed integro-differential Fredholm equation
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2007
SP  - 655
EP  - 664
VL  - 47
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a7/
LA  - ru
ID  - ZVMMF_2007_47_4_a7
ER  - 
%0 Journal Article
%A N. N. Nefedov
%A A. G. Nikitin
%T The Cauchy problem for a singularly perturbed integro-differential Fredholm equation
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2007
%P 655-664
%V 47
%N 4
%U http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_4_a7/
%G ru
%F ZVMMF_2007_47_4_a7

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

An initial problem is considered for an ordinary singularly perturbed integro-differential equation with a nonlinear integral Fredholm operator. The case when the reduced equation has a smooth solution is investigated, and the solution to the reduced equation with a corner point is analyzed. The asymptotics of the solution to the Cauchy problem is constructed by the method of boundary functions. The asymptotics is validated by the asymptotic method of differential inequalities developed for a new class of problems.

[1] Pao C. V., Nonlinear parabolic and elliptic equations, Plenum Press, New York, London, 1992 | MR

[2] Vasileva A. B., Butuzov V. F., Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vyssh. shkola, M., 1990 | MR

[3] Nefedov H. H., “Metod differentsialnykh neravenstv dlya nekotorykh singulyarno vozmuschennykh zadach s vnutrennimi sloyami”, Differents. ur-niya, 31:7 (1995), 1132–1139

[4] Nefedov H. H., Nikitin A. G., “Metod differentsialnykh neravenstv dlya singulyarno vozmuschennykh integrodifferentsialnykh uravnenii”, Differents. ur-niya, 36:10 (2000), 1398–1404 | MR | Zbl

[5] Nefedov H. H., Nikitin A. G., “Metod differentsialnykh neravenstv dlya kontrastnykh struktur tipa stupenki v singulyarno vozmuschennykh integrodifferentsialnykh uravnenii v prostranstvenno dvumernom sluchae”, Differents. ur-niya, 42:5 (2006), 690–700 | MR | Zbl

[6] Nefedov H. H., Nikitin A. G., Urazgildina T. A., “Zadacha Koshi dlya singulyarno vozmuschennogo integrodifferentsialnogo uravneniya Volterra”, Zh. vychisl. matem. i matem. fiz., 46:5 (2006), 805–812 | MR

[7] Zabreiko P. P., Koshelev A. I., Krasnoselskii M. A. i dr., Integralnye uravneniya, Nauka, M., 1968 | MR