Asymptotic Expansion of the Solution to One Problem of Ali Hasan Nayfeh

Abstract

This article studies a two-point boundary-value problem for a linear inhomogeneous second-order ordinary differential equation with a small parameter ε under the sign of the highest derivative of the unknown function. The equation under consideration is given in the form εy′′(x) + (x - 1/2)y′(x) - (x - 1/2)y(x) = 0, 0 < x < 1, and the boundary conditions y(0) = a, y(1) = b. Similar problems are encountered in physics, engineering, biology, economics, and other fields of science. The peculiarities of the problem under consideration are that it belongs to the class of singularly excited equations, and that the coefficient of the corresponding unexcited equation vanishes on the interval under consideration. These two factors (singular perturbation and internal singular point) make the problem bisingular and complicate the structure of the solution. However, the difference between the problem under consideration and other singularly perturbed problems is that boundary layers do not appear on the boundaries of the boundary points. The layer is formed only in the neighborhood of the internal singular point x = 0. As a result, we cannot write the solution to the problem under consideration using a single function, i.e. We construct individual solutions on the intervals [0, 12] and [1 2, 1], i.e., a composite solution is constructed. Each solution consists of the sum of two functions: a regular external part and an internal solution describing the layer at the singular point. The goal of the study is to construct a uniform asymptotic expansion of the solution to a two-point boundary value problem with a bisingular perturbation on a given interval. The accuracy of the resulting asymptotic formulas is compared with numerical calculations performed in Maple based on a specific example, and their asymptotic closeness is demonstrated.