Geodesic
The generalized boundary function method for bisingular problems in a disk
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 239-249 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The aim of the research is the development of the asymptotic method of boundary functions. This work is devoted to constructing complete asymptotic expansions of the solutions of boundary value problems for a bisingular inhomogeneous linear second-order elliptic equation. The equation has the following singularities: there is a small parameter at the Laplace operator and the corresponding unperturbed (limit) equation has singularities both at the boundary of the disk and inside it. Bisingular Dirichlet, Neumann and Robin problems are studied in the disk. Complete asymptotic expansions of the solutions of the bisingular problems are constructed by the generalized method of boundary functions, which differs from the matching method in that the growing singularities of the outer expansion are actually removed from it and included in the inner expansions with the help of auxiliary asymptotic series. The resulting solutions are asymptotic in the sense of Erdelyi, and the asymptotic expansions are Puiseux series. The leading terms of the asymptotic expansions of the solutions are negative fractional powers of the small parameter. The obtained asymptotic expansions of the solutions of the boundary value problems are justified by means of the maximum principle.
Keywords: asymptotic expansion of the solution, bisingular problem, second-order elliptic equation, modified Bessel functions, Dirichlet problem, Neumann problem, Robin problem, generalized boundary function method, small parameter, maximum principle. 
@article{TIMM_2017_23_2_a19,
     author = {D. A. Tursunov},
     title = {The generalized boundary function method for bisingular problems in a disk},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {239--249},
     year = {2017},
     volume = {23},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a19/}
}
TY  - JOUR
AU  - D. A. Tursunov
TI  - The generalized boundary function method for bisingular problems in a disk
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2017
SP  - 239
EP  - 249
VL  - 23
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a19/
LA  - ru
ID  - TIMM_2017_23_2_a19
ER  - 
%0 Journal Article
%A D. A. Tursunov
%T The generalized boundary function method for bisingular problems in a disk
%J Trudy Instituta matematiki i mehaniki
%D 2017
%P 239-249
%V 23
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a19/
%G ru
%F TIMM_2017_23_2_a19
D. A. Tursunov. The generalized boundary function method for bisingular problems in a disk. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 239-249. http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a19/

[1] Ilin A.M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka. Gl. red. fiz.-mat. lit., M., 1989, 336 pp. | MR

[2] Ilin A.M., Danilin A.R., Asimptoticheskie metody v analize, Fizmatlit, M., 2009, 248 pp.

[3] Rozov N.Kh., “Nekotorye zamechaniya o bisingulyarnykh kraevykh zadachakh”, Sovremennaya matematika i ee prilozheniya, 35, no. 2, 2005, 44–47

[4] Danilin A.R., “Asimptotika reshenii sistemy singulyarnykh ellipticheskikh uravnenii v pryamougolnike”, Mat. sb., 194:1 (2003), 31–60 | DOI | MR | Zbl

[5] Lelikova E.F., “Ob asimptotike resheniya ellipticheskogo uravneniya s malym parametrom v okrestnosti tochki peregiba granitsy”, Tr. Instituta matematiki i mekhaniki UrO RAN, 22:1 (2016), 197–211 | MR

[6] Khachai O.Yu., “Asimptoticheskoe razlozhenie resheniya odnoi bisingulyarnoi zadachi Koshi dlya nelineinogo obyknovennogo uravneniya pervogo poryadka”, Dep. v VINITI, 16 (2005), 174-V2005, 1–46

[7] Tursunov D.A., Erkebaev U.Z., “Asimptoticheskoe razlozhenie resheniya zadachi Dirikhle dlya ellipticheskogo uravneniya s osobennostyami”, Ufim. mat. zhurn., 8:1 (2016), 102–112 | Zbl

[8] Tursunov D.A., Erkebaev U.Z., “Asimptotika resheniya zadachi Dirikhle dlya bisingulyarno vozmuschennogo uravneniya v koltse”, Vest. Udmurt. un-ta. Matematika. Mekhanika. Kompyut. nauki, 25:4 (2015), 517–525 | Zbl

[9] Oleinik O.A., “O svoistvakh reshenii nekotorykh kraevykh zadach dlya uravnenii ellipticheskogo tipa”, Mat. sb., 30(72):3 (1952), 695–702 | Zbl

[10] Fedoryuk M.V., Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983, 352 pp. | MR