Asymptotics of solution of the singularly perturbed Dirichlet problem with a weak critical point

The Dirichlet problem for a singularly perturbed linear homogeneous ordinary differential equation of second order with a nonsmooth coefficient in real axis is considered. Such problems can be seen in physics, engineering, continuum mechanics, hydrodynamics, etc. Object of the research is to develop the asymptotic technique of boundary functions of Vishik–Lusternik–Vasilyeva–Imanaliev for singularly perturbed differential equations in case when the corresponding non-perturbed equation has nonsmooth solution in the considered area. According to terminology of A. M. Ilyin, such problems are called bisingular. The possibility to use a generalized method of boundary functions for constructing a complete proportional asymptotic expansion of the boundary problem solution for a singularly perturbed linear ordinary differential equation of second order with a weak critical point or an integrable critical point is proved in the article. The constructed expansion of solution is asymptotic in the sense of Erdey. When constructing the proportional asymptotic expansion of the Dirichlet problem, the following methods were used: small parameter method, method of mathematical induction, classical method of boundary functions, and the principle of maximum. Using the principle of maximum, an assessment for the asymptotic expansion's remainder term is obtained, i.e. the proportional complete asymptotic expansion of the solution by small parameter is proved. A specific example is given.