Geodesic
Asymptotic solution of a singularly perturbed Cauchy problem in the presence of a rational ``simple'' turning point
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 1, Tome 190 (2021), pp. 81-87.

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

In this paper, based on S. A. Lomov's regularization method, we construct an asymptotic solution of a singularly perturbed Cauchy problem in the case of violation of the stability conditions for the spectrum of the limit operator. In particular, we consider the problem with a “simple” turning point, i.e., where one eigenvalue vanishes for $t=0$ and has the form $t^{m/n}$ (the limit operator is discretely irreversible).
Keywords: singularly perturbed Cauchy problem, asymptotic solution, regularization method, turning point. 
@article{INTO_2021_190_a6,
     author = {A. G. Eliseev and T. A. Ratnikova},
     title = {Asymptotic solution of a singularly perturbed {Cauchy} problem in the presence of a rational ``simple'' turning point},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {81--87},
     publisher = {mathdoc},
     volume = {190},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_190_a6/}
}
TY  - JOUR
AU  - A. G. Eliseev
AU  - T. A. Ratnikova
TI  - Asymptotic solution of a singularly perturbed Cauchy problem in the presence of a rational ``simple'' turning point
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2021
SP  - 81
EP  - 87
VL  - 190
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2021_190_a6/
LA  - ru
ID  - INTO_2021_190_a6
ER  - 
%0 Journal Article
%A A. G. Eliseev
%A T. A. Ratnikova
%T Asymptotic solution of a singularly perturbed Cauchy problem in the presence of a rational ``simple'' turning point
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2021
%P 81-87
%V 190
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2021_190_a6/
%G ru
%F INTO_2021_190_a6
A. G. Eliseev; T. A. Ratnikova. Asymptotic solution of a singularly perturbed Cauchy problem in the presence of a rational ``simple'' turning point. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 1, Tome 190 (2021), pp. 81-87. http://geodesic.mathdoc.fr/item/INTO_2021_190_a6/

[1] Lomov S. A., Vvedenie v obschuyu teoriyu singulyarnykh vozmuschenii, Nauka, M., 1981

[2] Eliseev A. G., Lomov S. A., “Teoriya singulyarnykh vozmuschenii v sluchae spektralnykh osobennostei predelnogo operatora”, Mat. sb., 131:173 (1986), 544–557 | Zbl

[3] Lioville J., “Second memoire sur le development des fonctions en series dont divers termes sont assujettis, a une meme equation”, J. Math. Pure Appl., 2 (1837), 16–35

[4] Tursunov D. A., Kozhbekov K. G., “Asimptotika resheniya singulyarno vozmuschennykh differentsialnykh uravnenii s drobnoi tochkoi povorota”, Izv. Irkutsk. un-ta., 21 (2017), 108–121 | Zbl