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Boundary and Angular Layer Behavior in Singularly Perturbed Semilinear Systems

Published online by Cambridge University Press:  20 November 2018

K. W. Chang
Affiliation:
Department of mathematics and statistics, The university of calgaryCalgary, alberta
G. X. Liu
Affiliation:
Department of mathematics, Nankai universityTiajin, china
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Abstract

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Some authors have employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ∊ → 0+, of the solutions of scalar boundary value problems

∊y" = h(t,y), a < t < b,

y(a,∊) = A, y(b,∊) = B.

In this paper, we extend these results to vector boundary value problems, under analogous stability conditions on the solution u = u(t) of the reduced equation 0 = h(t,u).

Two types of asymptotic behavior are studied, depending on whether the reduced solution u(t) has or does not have a continuous first derivative in (a,b), leading to the phenomena of boundary and angular layers.

Keywords

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Brish, N.I., On Boundary Value Problems for the Equation ∊yn = f(x,y,y')for small ∊, Dokl. Akad. Nauk SSSR 95 (1954), pp. 429432.Google Scholar
2. Hebets, P. and Laloy, M., Étude de problèmes aux limites par la méthod des sur-et sous-solutions, Lecture notes , Catholic University of Louvain, 1974.Google Scholar
3. Bernfeld, S. and Lakshmikantham, V., An Introduction to Nonlinear Boundary Value Problems, Academic Press, New York, 1974.Google Scholar
4. Boglaev, Yu. B., The two-point problem for a class of ordinary differential equations with a small parameter coefficient of the derivative, USSR Comp. Math, and Math. Phys. 10 (1970), 4, pp. 191204.Google Scholar
5. Chang, K.W. and Howes, F.A., Nonlinear Singular Perturbation Phenomena, Theory and Appl. Springer-Verlag Pub. 1984.CrossRefGoogle Scholar
6. O'Donnell, M. A., Boundary and Corner Layer Behavior in Singularly Perturbed Semilinear Systems of Boundary Value Problems, SIAM J. Math. Anal. (To appear).Google Scholar
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