Geodesic
Asymptotics of the solution of the Dirichlet problem with singularity inside the ring
Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 21 (2018) no. 1, pp. 44-52.

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

The Dirichlet problem for elliptic equations with a small parameter in the highest derivatives occupies the unique position in different fields of science such as mathematics, physics, mechanics, and fluid dynamics. It is necessary to apply different asymptotic or numerical methods, because an explicit solution to these problems cannot be obtained through analytical methods. An asymptotic expansions design of solutions for singularly perturbed problems is an urgent problem, especially for bisingular problems. These problems have two singularities. The first singularity is associated with a singular dependence of the solution from a small parameter. The second singularity is related to the asymptotic behavior of the nonsmoothness members of the external solution, i.e. the problem corresponding to the original singular problem is the singular problem, too. The aim of the research is to develop the asymptotic method of boundary functions for bisingular perturbed problem. The possibility of using the generalized method of boundary functions for designing a complete asymptotic expansion of the solution of the Dirichlet problem was shown. This method has been applied to find the solutions of bisingular perturbed, linear, non-homogeneous, second-order elliptic equations with two independent variables in the ring with the assumption that the second singularity appears inside the region. The obtained asymptotic series is a Pyuyzo series. The main term of the asymptotic expansion of the solution has a negative fractional power of the small parameter which is typical to bisingular perturbed equations.
Mots-clés : elliptic equation
Keywords: Bessel modified functions, asymptotic, bisingularly problem, Dirichlet problem, boundary layer function, small parameter. 
@article{VVGUM_2018_21_1_a4,
     author = {D. A. Tursunov},
     title = {Asymptotics of the solution of the {Dirichlet} problem with singularity inside the ring},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
     pages = {44--52},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VVGUM_2018_21_1_a4/}
}
TY  - JOUR
AU  - D. A. Tursunov
TI  - Asymptotics of the solution of the Dirichlet problem with singularity inside the ring
JO  - Matematičeskaâ fizika i kompʹûternoe modelirovanie
PY  - 2018
SP  - 44
EP  - 52
VL  - 21
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/VVGUM_2018_21_1_a4/
LA  - ru
ID  - VVGUM_2018_21_1_a4
ER  - 
%0 Journal Article
%A D. A. Tursunov
%T Asymptotics of the solution of the Dirichlet problem with singularity inside the ring
%J Matematičeskaâ fizika i kompʹûternoe modelirovanie
%D 2018
%P 44-52
%V 21
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/VVGUM_2018_21_1_a4/
%G ru
%F VVGUM_2018_21_1_a4
D. A. Tursunov. Asymptotics of the solution of the Dirichlet problem with singularity inside the ring. Matematičeskaâ fizika i kompʹûternoe modelirovanie, Tome 21 (2018) no. 1, pp. 44-52. http://geodesic.mathdoc.fr/item/VVGUM_2018_21_1_a4/

[1] K. Alymkulov, D. A. Tursunov, “A Method for Constructing Asymptotic Expansions of Bisingularly Perturbed Problems”, Russian Mathematics, 2016, no. 12, 3–11 | MR | Zbl

[2] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Nauka Publ., Moscow, 1989, 464 pp.

[3] A. M. Ilyin, Matching of Asymptotic Expansions of Boundary Value Problems, Nauka Publ., Moscow, 1989, 334 pp.

[4] D. A. Tursunov, U. Z. Erkebaev, “Asymptotic of the Solution to the Bisingular Perturbed Dirichlet Problem in the Ring With Quadratic Growth on the Boundary”, Bulletin of the South Ural State University. Series “Mathematics. Mechanics. Physics”, 8:2 (2016), 52–61 | Zbl

[5] D. A. Tursunov, U. Z. Erkebaev, “Asymptotic of the Solution of the Dirichlet Problem for a Bisingularly Perturbed Equation in a Ring”, Bulletin of the Udmurt University. Mathematics. Mechanics. Computer Science, 25:4 (2015), 517–525 | Zbl

[6] D. A. Tursunov, U. Z. Erkebaev, “Asymptotic Expansions of Solutions to Dirichlet problem for Elliptic Equation with Singularities”, Ufa Mathematical Journal, 8:1 (2016), 102–112 | Zbl

[7] M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations, Nauka Publ., Moscow, 1983, 352 pp.

[8] P. P. N. de Groen, Critical points of the degenerate operator in elliptic singular perturbation problem, ZW 28/75, Mathematish Centrum, Amsterdam, 1975, 67 pp.

[9] W. Eckhaus, E. M. de Jager, “Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type”, Arch. Rat. Mech. Anal, 23:1 (1966), 26–86 | DOI | MR | Zbl

[10] W. Eckhaus, “Boundary layers in linear elliptic singular perturbation problems”, SIAM Rev., 14 (1972), 225–270 | DOI | MR | Zbl

[11] N. Levinson, “The first boundary value problem for $\varepsilon \Delta u+Au_{x}+Bu_{y}+Cu = D$ for small $\varepsilon$”, Ann. of Math., 51 (1950), 428–445 | DOI | MR | Zbl