Geodesic
On some problems for partial differential equations with a small parameter in the principal part
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 183 (2020), pp. 61-72.

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

In this paper, we consider problems for partial differential equations that contain a small parameter in the principal part. We construct exact solutions of the original singularly perturbed problems and, passing to the limit, we state a correspondence between these solutions and solutions of the limit problems obtained from the original problems if the small parameter is set equal to zero. Examples given in this paper show that among singularly perturbed problems for partial differential equations, there are problems that possess characteristic properties of regularly perturbed problems and allow constructing asymptotic solutions by methods of the theory of regular perturbations.
Keywords: small parameter, singularly perturbed equation, limit problem, Dirichlet problem, Cauchy problem. 
@article{INTO_2020_183_a5,
     author = {I. V. Zakharova},
     title = {On some problems for partial differential equations with a small parameter in the principal part},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {61--72},
     publisher = {mathdoc},
     volume = {183},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_183_a5/}
}
TY  - JOUR
AU  - I. V. Zakharova
TI  - On some problems for partial differential equations with a small parameter in the principal part
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 61
EP  - 72
VL  - 183
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_183_a5/
LA  - ru
ID  - INTO_2020_183_a5
ER  - 
%0 Journal Article
%A I. V. Zakharova
%T On some problems for partial differential equations with a small parameter in the principal part
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 61-72
%V 183
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_183_a5/
%G ru
%F INTO_2020_183_a5
I. V. Zakharova. On some problems for partial differential equations with a small parameter in the principal part. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 183 (2020), pp. 61-72. http://geodesic.mathdoc.fr/item/INTO_2020_183_a5/

[1] Vatson T. N., Teoriya besselevykh funktsii, IL, M., 1949

[2] Zakharova I. V., “Odna zadacha dlya uravnenii s chastnymi proizvodnymi ellipticheskogo tipa, soderzhaschego malyi parametr v glavnoi chasti”, Mezhdunar. zh. prikl. fundam. issl., 2017, no. 5, 235–239

[3] Koshlyakov N. S., Uravneniya v chastnykh proizvodnykh matematicheskoi fiziki, Vysshaya shkola, M., 1970

[4] Miranda K., Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa, IL, M., 1957

[5] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, Izd-vo MGU, M., 2004 | MR

[6] Fridman F., Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Mir, M., 1968

[7] Yanushauskas A., “O zavisyaschikh ot malogo parametra uravneniyakh s chastnymi proizvodnymi”, Kraevye zadachi, Irkutsk, 1990, 94–103