Asymptotic Analysis of a Singularly PerturbedBoundary Value Problem with an Interior Turning Point

Abstract

This paper investigates a two-point boundary value problem for a singularly perturbed linear inhomogeneous second-order ordinary differential equation with a turning point inside the interval of the form:

[epsilon] y''(x) + (x - x0) y'(x) - (x - x0) y(x) = 0, 0 < x < 1, x0 in (0, 1),

subject to the boundary conditions:

y(0) = a, y(1) = b.

Problems of this type arise in various areas of science and engineering, including physics, biology, economics, and applied mathematics. The considered problem possesses two distinctive features: it belongs to the class of singularly perturbed differential equations, and the coefficient of the reduced equation vanishes at an interior point of the interval. The combination of singular perturbation and an interior turning point makes the problem bisingular and leads to a more complicated solution structure. Unlike classical singularly perturbed boundary value problems, boundary layers do not occur near the endpoints of the interval. Instead, a layer is formed only in a neighborhood of the interior singular point x = x0. Consequently, the solution cannot be represented by a single asymptotic expression on the entire interval. To overcome this difficulty, separate asymptotic representations are constructed on the subintervals [0, x0] and [x0, 1], yielding a composite solution. Each local representation consists of the sum of a regular outer solution and an inner solution describing the layer behavior near the singular point. The main objective of the paper is to construct a uniformly valid asymptotic expansion of the solution on the interval [0,1]. A comparison of the constructed asymptotic expansion with A. Nayfeh's result demonstrates their close agreement and confirms the validity of the proposed asymptotic construction.