# Examples and Applications

• K. W. Chang
• F. A. Howes
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 56)

## Abstract

Consider the Dirichlet problem \eqalign{ & \varepsilon y''{\text{ }} = {\text{ }}{(y - {\text{u}}({\text{t}}))^{2q + 1}},{\text{ }} - 1 < {\text{t}} < 1, \cr & y( - {\text{l}},\varepsilon ){\text{ }} = {\text{A}},{\text{ }}y({\text{l}},\varepsilon ){\text{ B}}, \cr} where q is a nonnegative integer. If the function u(t), defined for $$- 1{\text{ }} < {\text{ t }} < {\text{ }}1$$,is twice continuously differentiable or has a bounded second derivative, then by Theorem 3.1, for sufficiently small $$\varepsilon > 0$$,the Dirichlet problem has a solution $$y{\text{ }} = {\text{ }}y({\text{t}},\varepsilon )$$which satisfies
$$\mathop {\lim }\limits_{\varepsilon \to {0^ + }} {\text{ }}y({\text{t}},\varepsilon ){\text{ }} = {\text{ u}}({\text{t}}){\text{ in }}[ - 1 + \delta ,1 - \delta ],$$
(8.1)
where $$0{\text{ }} < {\text{ }}\delta {\text{ }} < {\text{ }}1$$ Moreover, the behavior of the solution $$y({\text{t}},\varepsilon )$$ in the boundary layers at t = -1 and/or t = 1 (if u(-1)≠A and/or u(1) ≠ B) can be described by means of the layer functions given in the conclusion of Theorem 3.1.

## Keywords

Boundary Layer Sherwood Number Singular Solution Boundary Layer Stability Shock Layer
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