Asymptotics of the Solution to the Boundary-value Problems when Limited Equation Has Singular Point

Abstract

The article studies the asymptotic behavior of the solutions of a singularly perturbed three boundary value problems on an interval. The object of the study is a linear inhomogeneous ordinary differential equation of the second order with a small parameter with the highest derivative of the unknown function. The singularities of the problem are that the small parameter is found at the highest derivative of the unknown function and the corresponding unperturbed first-order differential equation has higher order an irregular singular point at the left end of the segment. At the ends of the segment, boundary conditions are imposed. Three problems are considered, in one Dirichlet problem, in two Neumann problem and in the three Roben problem. Asymptotic expansions of problems are constructed by the classical method of Vishik–Lyusternik–Vasilyeva–Imanaliev boundary functions. However, this method cannot be applied directly, since the external solution has a singularity. We first remove this singularity from the external solution, then apply the method of boundary functions. The constructed asymptotic expansions are substantiated using the maximum principle, i.e. estimates for the residual functions are obtained.