An Elementary Approach to a Model Problem of Lagerstrom

TL;DR

This paper presents an elementary method to solve a classical model equation of Lagerstrom, providing a uniformly convergent series solution, asymptotic expansions, and a simple proof of existence and uniqueness for all parameter values.

Contribution

It introduces a straightforward approach to derive a convergent series solution and asymptotic expansions without matching, along with an elementary proof of existence and uniqueness for the model equation.

Findings

Infinite series solution converges uniformly on the domain.

Inner and outer asymptotic expansions are derived without matching.

Existence and uniqueness are established for all parameter values.

Abstract

The equation studied is u"+((n-1)/r)u'+epsilon u u'+ku'^{2}=0, with boundary conditions u(1)=0, u(infinity)=1. This model equation has been studied by many authors since it was introduced in the 1950s by P. A. Lagerstrom. We use an elementary approach to show that there is an infinite series solution which is uniformly convergent on 1<=r<infinity. The first few terms are easily derived, from which one quickly deduces the inner and outer asymptotic expansions, with no matching procedure or a priori assumptions about the nature of the expansion. We also give a short and elementary existence and uniqueness proof which covers all epsilon > 0, k >= 0, and n >= 1.