Asymptotic expansion of the solution of the Dirichlet problem for a ring with a singularity on the boundary

D. A. Tursunov ; U. Z. Erkebaev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 1 (2016), pp. 42-52

Résumé

Owing to the large number and variety of applications, the Dirichlet problem for elliptic equations with a small parameter at highest derivatives occupies a unique place in mathematics. The main problem of flow around in hydrodynamics, the problem of torsion and bending in the elasticity theory, determination of temperature inside a plate according to its known values on the contour in physics, the potential of the steady flow of an incompressible fluid, electromagnetic and magnetic potentials, and the determination of the temperature of the thermal field or electrostatic field potential in a certain region at a given temperature or potential on the boundary can be reduced to this problem. It is also closely related to main problems of statistical theory of elasticity and others. The explicit solution of these problems can be constructed in the general case only using different asymptotic and numerical methods. When the corresponding unperturbed equation has a smooth solution, these problems are called bisingular in A.M. Il’in’s terminology. The method of matching was applied before to construct the asymptotic of bisingularly perturbed problems but the method of boundary functions was not used directly. The authors propose to modify the method of boundary functions by use of which it is possible to construct asymptotic solutions of the Dirichlet problem for a bisingularly perturbed second order elliptic equation with two independent variables in a ring domain. The aim of the study is to develop the asymptotic method of boundary functions for bisingularly perturbed problems. The constructed asymptotic series is a series of Puiseux. The principal term of the asymptotic expansion of the solution has a negative fractional power in the small parameter, which is inherent to bisingular perturbed equations or equations with turning points.

Détail

Keywords: asymptotic expansion of a solution, Dirichlet problem, small parameter, generalized method of boundary functions, boundary functions, modification Bessel functions.
Mots-clés : bisingular perturbation, elliptic type equation

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