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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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

  • K. W. Chang
    • 1
  • F. A. Howes
    • 2
  1. 1.Department of MathematicsUniversity of CalgaryCalgaryCanada
  2. 2.Department of MathematicsUniversity of CaliforniaDavisUSA

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