Asymptotics of the Solution to a Two-Band Two-Point Boundary Value Problem

Abstract

The article investigates the asymptotic behavior of the solution of a two-point boundary value problem on an interval for a linear inhomogeneous ordinary differential equation of the second order with a small parameter at the highest derivative. The essential features of the problem are the presence of a small parameter in front of the second-order derivative of the desired function, the existence of a twodimensional boundary layer at the left end of the segment at x = 0, and the non-smoothness of the solution to the corresponding unperturbed boundary value problem. It is required to construct a uniform asymptotic expansion of the solution to a two-zone two-point boundary value problem on a unit interval, with any degree of accuracy, as the small parameter tends to zero. Due to the second and third features of the problem, it is not easy to construct an asymptotic solution expansion with respect to the small parameter using the known asymptotic methods. When solving the problem, the following methods are used: methods of integration of ordinary differential equations; the method of a small parameter; the classical method of boundary functions; and the generalized method of boundary functions and the maximum principle. The problem is solved in two stages: in the first stage, a formal expansion of the solution to the two-point boundary value problem is constructed, and in the second stage, the justification of this expansion is given, i.e. the remainder term of the expansion is estimated. In the first stage, a formal asymptotic solution is sought in the form of the sum of three solutions: a smooth outer solution on the entire segment; classical boundary layer solution in the vicinity of x = 0, which exponentially decreases outside the boundary layer; and an intermediate boundary layer solution at x = 0, which decreases in power mode outside the boundary layer. The constructed asymptotic expansion of the solution to the two-point boundary value problem is asymptotic in the sense of Erdei.