The generalized boundary function method for bisingular problems in a disk

The aim of the research is the development of the asymptotic method of boundary functions. This work is devoted to constructing complete asymptotic expansions of the solutions of boundary value problems for a bisingular inhomogeneous linear second-order elliptic equation. The equation has the following singularities: there is a small parameter at the Laplace operator and the corresponding unperturbed (limit) equation has singularities both at the boundary of the disk and inside it. Bisingular Dirichlet, Neumann and Robin problems are studied in the disk. Complete asymptotic expansions of the solutions of the bisingular problems are constructed by the generalized method of boundary functions, which differs from the matching method in that the growing singularities of the outer expansion are actually removed from it and included in the inner expansions with the help of auxiliary asymptotic series. The resulting solutions are asymptotic in the sense of Erdelyi, and the asymptotic expansions are Puiseux series. The leading terms of the asymptotic expansions of the solutions are negative fractional powers of the small parameter. The obtained asymptotic expansions of the solutions of the boundary value problems are justified by means of the maximum principle.