On a Nested Boundary-Layer Problem

X. Liang; Roderick S. C. Wong

doi:10.3934/cpaa.2009.8.419

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2009, Volume 8, Issue 1: 419-433. Doi: 10.3934/cpaa.2009.8.419

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On a Nested Boundary-Layer Problem

X. Liang^1, and
Roderick S. C. Wong^2,

1.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139
2.
Department of Mathematics, City University of Hong Kong, Kowloon

Received: March 2008

Revised: September 2008

Published: January 2009

Abstract / Introduction Related Papers Cited by

Abstract

Nested boundary layers mean that one boundary layer lies inside another one. In this paper, we consider one such problem, namely,

$\varepsilon^3xy''(x)+x^2y'(x)- (x^3+\varepsilon)y(x)=0$ with $0 < x <1$, $y(0) = 1$ and $y(1) = \sqrt{e}$.

An asymptotic solution, which holds uniformly for $x\in [0,1]$, is constructed rigorously. This result also provides an explicit formula for the exponentially small leading term in the interval where the exact solution exhibits such behavior. This henomenon has never been mentioned in the existing literature.

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Mathematics Subject Classification: Primary: 34E20; Secondary: 34B05.

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