Tursunov D.A. Asymptotics of the Solution of the Dirichlet Problem With Singularity Inside the Ring

Details

Hits: 578

https://doi.org/10.15688/mpcm.jvolsu.2018.1.5

Dilmurat Abdillazhanovich Tursunov
Doctor of Physical and Mathematical Sciences, Professor, Department of Informatics,
Osh State University
This email address is being protected from spambots. You need JavaScript enabled to view it.
Lenina St., 331, 723500 Osh, Republic of Kyrgyzstan

Abstract. 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.

Key words: elliptic equation, Bessel modified functions, asymptotic, bisingularly problem, Dirichlet problem, boundary layer function, small parameter.

Asymptotics of the Solution of the Dirichlet Problem With Singularity Inside the Ring by Tursunov D.A. is licensed under a Creative Commons Attribution 4.0 International License.

Citation in English: Mathematical Physics and Computer Simulation. Vol. 21 No. 1 2018, pp. 44-52

Attachments:

5_Tursunov.pdf URL: https://mp.jvolsu.com/index.php/en/component/attachments/download/781

707 Downloads