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Shooting Method for Nonlinear Singularly Perturbed Boundary‐Value Problems
Author(s) -
Ou C. H.,
Wong R.
Publication year - 2004
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 - 10.1111/j.0022-2526.2004.01509.x
Subject(s) - mathematics , nonlinear system , mathematical analysis , boundary value problem , shooting method , interval (graph theory) , boundary (topology) , zero (linguistics) , term (time) , initial value problem , differential equation , method of matched asymptotic expansions , physics , combinatorics , quantum mechanics , linguistics , philosophy
Asymptotic formulas, as ɛ→ 0 + , are derived for the solutions of the nonlinear differential equation ɛ u″ + Q ( u ) = 0 with boundary conditions u (‐1) = u (1) = 0 or u ′(‐1) = u ′(1) = 0 . The nonlinear term Q ( u ) behaves like a cubic; it vanishes at s ‐ , 0, s + and nowhere else in [ s ‐ , s + ] , where s ‐ < 0 < s + . Furthermore, Q ′( s ± ) < 0, Q ′(0) > 0 and the integral of Q on the interval [ s ‐ , s + ] is zero. Solutions to these boundary‐value problems are shown to exhibit internal shock layers, and the error terms in the asymptotic approximations are demonstrated to be exponentially small. Estimates are obtained for the number of internal shocks that a solution can have, and the total numbers of solutions to these problems are also given. All results here are established rigorously in the mathematical sense.