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

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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. 
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     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/}
}
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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/