Asymptotic expansion of the solution of a perturbed elliptic equation when the limit equation has singular points

D. A. Tursunov   ; U. Z. Erkebaev

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 3 (2015), pp. 26-34

Résumé

The classical method of boundary functions is used to construct asymptotic expansions of solutions of perturbed Prandtl–Tikhonov type equation in the case of the exponential asymptotic stability of solutions of the equation in the fast variable, i.e. when the condition of A. N. Tikhonov’s theorem is satisfied. When this condition is not satisfied, the boundary functions method cannot be applied directly. For this reason, in such cases, many researchers previously used the Van Dyke matching principle. But the disadvantage of the method of matching is that the formal asymptotic expansion of the solution constructed by matching cannot be justified is all cases. We have proved the possibility of applying the method of boundary functions for constructing a uniform asymptotic expansion of the solution of the Dirichlet problem for the bisingular perturbed elliptic equation when the limit equation is the first order differential equation with singular points, and the condition of A. N. Tikhonov's theorem is not satisfied at these points. An estimate of the remainder term has been obtained, i.e., the formal asymptotic expansion solution of the problem has been justified. The uniform asymptotic expansion of the solution of the problem we have constructed consists of four solutions: the regular (smooth) external solution, the classical boundary layer solution, and two generalized boundary layer solution. The regular external part of the solution satisfies the boundary condition, and this solution has no singularities, i.e. is an everywhere smooth function. The classical boundary layer solution satisfies the second part of the boundary condition, and tends exponentially to zero outside the border inside the area. The generalized boundary functions satisfy the boundary condition at the singular points, and have the power damping property outside the singular points inside the region.

Détail

Keywords: asymptotic, bisingular perturbed, singular point, Dirichlet problem, generalized method of boundary functions, boundary function, small parameter.
Mots-clés : solution, elliptic type equation

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