On a nested boundary-layer problem. (English) Zbl 1159.34043
Summary: Nested boundary layers mean that one boundary layer lies inside another one. In this paper, we consider one such problem, namely,
\[ \varepsilon^3xy''(x)+x^2y'(x)- (x^3+\varepsilon)y(x)=0, \quad 0 < x <1, \]
\[ y(0)=1 \qquad y(1) = \sqrt{e}. \]
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 phenomenon has never been mentioned in the existing literature.
\[ \varepsilon^3xy''(x)+x^2y'(x)- (x^3+\varepsilon)y(x)=0, \quad 0 < x <1, \]
\[ y(0)=1 \qquad y(1) = \sqrt{e}. \]
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 phenomenon has never been mentioned in the existing literature.
MSC:
| 34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |
| 34B05 | Linear boundary value problems for ordinary differential equations |
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
