January  2009, 8(1): 419-433. doi: 10.3934/cpaa.2009.8.419

On a Nested Boundary-Layer Problem

1. 

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, United States

2. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  March 2008 Revised  September 2008 Published  October 2008

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.

Citation: X. Liang, Roderick S. C. Wong. On a Nested Boundary-Layer Problem. Communications on Pure and Applied Analysis, 2009, 8 (1) : 419-433. doi: 10.3934/cpaa.2009.8.419
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