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On a Nested Boundary-Layer Problem

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  • 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.

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


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