Asymptotic solutions of the bisingular perturbed elliptic equation. Case of a singular point on the boundary - Dilmurat Tursunov (Известия Томского политехнического университета: Инжиниринг георесурсов, 2019)
For mathematical modeling the convective-diffusive transport, chemical kinetics the boundary value problems occur for elliptic equations of the second order with a small parameter in the highest derivatives. The explicit solution of these problems can be constructed in a general case using different asymptotic methods. The fundamental work in this direction was done by A.N. Tikhonov, A.B. Vasilyeva, S.A. Lomov, V.B. Butuzov, L.I. Lyustemik, M.I. Vishik, A.M. Ilin. When the corresponding unperturbed equation has a smooth solution these problems are called bisingular in A.M. Ilin 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 author has proposed to modify the method of boundary functions that makes possible the construction of the asymptotic solutions of bisingularly perturbed elliptic equation. The aim of the study is to develop the asymptotic method of boundary functions for bisingularly perturbed equations. Applying the generalized method of boundary functions, the author constructed the asymptotic expansion of the solution for bisingularly perturbed elliptic equation in the case when the limit equation has a singularity at the boundary points of the region. The problem is considered in the circle.