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Singular perturbation analysis of boundary value problems for differential-difference equations. IV: A nonlinear example with layer behavior. (English) Zbl 0725.34064

[For parts I-III see SIAM J. Appl. Math. 42, 502-531 (1982; Zbl 0515.34058), ibid. 45, 687-707 (1985; Zbl 0623.34050), ibid. 45, 708-734 (1985; Zbl 0623.34051).]
The authors study the model \(\epsilon y''(x,\epsilon)+[y(x- 1,\epsilon)y(x,\epsilon)]'=f(x)\), \(0<x<1\); \(0<\epsilon \ll 1\); \(y(x,\epsilon)=\phi (x)\) on -1\(\leq x\leq 0\), and \(y(1,\epsilon)=\gamma\). Integrating both sides of the equation and using singular perturbation techniques an exact solution for the constant case is constructed. Also in this case some discussion regarding the existence and uniqueness of the solution is presented. By a combination of singular perturbation analysis and numerical computations the authors give a comprehensive treatment of the full spectrum of solution behavior of the model with \(\phi\neq 0\) on [-1,0].
Reviewer: H.S.Nur (Fresno)

MSC:

34E15 Singular perturbations for ordinary differential equations
65L07 Numerical investigation of stability of solutions to ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
34D15 Singular perturbations of ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34K10 Boundary value problems for functional-differential equations
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References:

[1] Abramowitz, Handbook of Mathematical Functions (1964)
[2] Ascher, Reformulation of boundary value problems in ’standard’ form, SIAM Rev. 23 pp 238– (1981) · Zbl 0461.34021 · doi:10.1137/1023039
[3] Kevorkian, Perturbation Methods in Applied Mathematics (1981) · doi:10.1007/978-1-4757-4213-8
[4] Lagerstrom, Matched Asymptotic Expansions (1989)
[5] Lange, Singular perturbation analysis of boundary-value problems for differential-difference equations, SIAM J. Appl. Math. 42 pp 502– (1982) · Zbl 0515.34058 · doi:10.1137/0142036
[6] Lange, Singular perturbation analysis of boundary-value problems for differential-difference equations. II. Rapid oscillations and resonances, SIAM J. Appl. Math. 45 pp 687– (1985) · Zbl 0623.34050 · doi:10.1137/0145041
[7] Lange, Singular perturbation analysis of boundary-value problems for differential-difference equations. III. Turning point problems, SIAM J. Appl. Math. 45 pp 708– (1985) · Zbl 0623.34051 · doi:10.1137/0145042
[8] Miura, Particular solutions of forced generalized Airy equations, J. Math. Anal. Appl. 109 pp 303– (1985) · Zbl 0584.34043 · doi:10.1016/0022-247X(85)90151-9
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