A Variational Approach to Nonlinear Singularly Perturbed Boundary-Value Problems

William L Kath*, C. Knessl, Bernard J Matkowsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


Carrier and Pearson introduced a nonlinear singularly perturbed boundary value problem that has served as a paradigm for problems where the method of matched asymptotic expansions (MAE) apparently fails. The "failure" of MAE is its inability to select the location of possible internal layers, though their structure is determined. Thus, a straightforward application of MAE leaves the positions of any internal layers arbitrary, though the asymptotic expansion of the exact solution to the problem exhibits internal layers only at specific locations. For this reason the solutions produced by MAE have been referred to as spurious solutions. We resolve the question of finding the positions of the interior layers by employing the variational approach of Grasman and Matkowsky. In addition, we show that this method tells how solutions bifurcate as the boundary values are varied, and give an alternative motivation for the variational approach via Newton"s method.

Original languageEnglish (US)
Pages (from-to)61-88
Number of pages28
JournalStudies in Applied Mathematics
Issue number1
StatePublished - Jan 1 1987

ASJC Scopus subject areas

  • Applied Mathematics

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