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
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.
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© 1984 Springer Science+Business Media New York
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Chang, K.W., Howes, F.A. (1984). Examples and Applications. In: Nonlinear Singular Perturbation Phenomena. Applied Mathematical Sciences, vol 56. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1114-3_8
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DOI: https://doi.org/10.1007/978-1-4612-1114-3_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-96066-1
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