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Slowly Varying Phase Planes and Boundary‐Layer Theory
Author(s) -
Kath William L.
Publication year - 1985
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1002/sapm1985723221
Subject(s) - mathematics , nonlinear system , phase plane , plane (geometry) , method of matched asymptotic expansions , mathematical analysis , boundary (topology) , scale (ratio) , phase (matter) , singular perturbation , asymptotic analysis , dynamical systems theory , boundary value problem , geometry , physics , quantum mechanics
A method is presented that combines phase‐plane techniques with the ideas of multiple scale and matched asymptotic expansions to explain the behavior of solutions to second‐order, nonlinear, nonautonomous, singular boundary‐value problems. It is shown that if one is willing to give up the detailed information provided by a procedure such as matched asymptotic expansions, then complete qualitative information can be obtained by the much simpler method given here. (“Complete” here means that the method provides a way of categorizing all possible solutions of such problems.) In addition, the similarities and differences between the present method and that of Melnikov, which has been useful in the study of dynamical systems, are noted.