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On spurious solutions of singular perturbation problems. (English) Zbl 0539.34046

The straightforward application of classical boundary layer techniques to nonlinear differential equations can sometimes lead to spurious solutions of given boundary value problems.
The aim of the paper is to show that such problems can nevertheless be solved by applying the method of matched asymptotic expansions with some care. Two examples are treated, the main of these being the equation \(\epsilon^ 2y''+y=1, y(-1)=y(1)=0\). The nonlinearity of the problems provides unexpected solutions (which include several free boundary layers).
Despite the complexity of the solutions, the principle of the method does not appear to be more difficult than the classical method for localizing a boundary layer in a linear problem.
Reviewer: P.A.Bois

MSC:

34E15 Singular perturbations for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:

[1] G. F. Carrier C. E. Pearson Ordinary Differential Equations, Ginn-Blaisdell, Waltham, Mass. 1968
[2] O’Malley, Phase-plane solutions to some singular perturbation problems, J. Math. Anal Appl 54 pp 449– (1976) · Zbl 0334.34050 · doi:10.1016/0022-247X(76)90214-6
[3] Rosenblat, Multiple solutions of nonlinear boundary-value problems, Stud. Appl.Math 63 pp 99– (1980) · Zbl 0439.34022 · doi:10.1002/sapm198063299
[4] Chance, Biological and Biochemical Oscillators (1973)
[5] Bender, Advanced Mathematical Methods for Scientists and Engineers (1978)
[6] Ascher, Reformulation of boundary value problems in ”standard” form, SIAM Rev 23 pp 338– (1981) · Zbl 0461.34021 · doi:10.1137/1023039
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