Stroucken, A. C. J.; Verhulst, F. The Galerkin-averaging method for nonlinear, undamped continuous systems. (English) Zbl 0638.35057 Math. Methods Appl. Sci. 9, 520-549 (1987). This paper is concerned with initial-boundary value problems for nonlinear evolution equations of the form: \[ (1)\quad u_{tt}+Au=\epsilon f(u,u_ t,t),\quad t\geq 0. \] For these problems, the authors set up a first and second order approximation theory, by means of Galerkin approximation and averaging procedures. They, also, study the difference between two cases: the case with nontrivial resonance at first order, illustrated by the nonlinear wave equation \(u_{tt}-u_{xx}=\epsilon u^ 3\), and the case with no resonance at first order, illustrated by the Klein-Gordon equation \(u_{tt}- u_{xx}+u=\epsilon u^ 3.\) Reviewer: D.Huet Cited in 6 Documents MSC: 35L70 Second-order nonlinear hyperbolic equations 35A35 Theoretical approximation in context of PDEs 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs Keywords:small parameter; initial-boundary value; nonlinear evolution equations; Galerkin approximation; averaging; nontrivial resonance at first order; nonlinear wave equation; no resonance at first order; Klein-Gordon equation PDFBibTeX XMLCite \textit{A. C. J. Stroucken} and \textit{F. Verhulst}, Math. Methods Appl. Sci. 9, 520--549 (1987; Zbl 0638.35057) Full Text: DOI References: [1] Chow, Asymptotic solutions of inhomogeneous initial boundary value problems for weakly nonlinear partial differential equations, SIAM J. Appl. Math. 22 pp 629– (1972) · Zbl 0237.35007 · doi:10.1137/0122059 [2] Fink, A convergent two-time method for periodic differential equations, J. Diff. Eqs. 15 pp 459– (1974) · Zbl 0258.34042 · doi:10.1016/0022-0396(74)90068-0 [3] Foias, Asymptotic analysis of the Navier-Stokes equations, Physica 9D pp 157– (1983) [4] Keller, Asymptotic solutions of initial value problems for nonlinear partial differential equations, SIAM J. Appl. Math. 18 pp 748– (1970) · Zbl 0197.37001 · doi:10.1137/0118067 [5] Moore, Resonances introduced by discretization, IMA J. Appl. Math. 31 pp 1– (1983) · Zbl 0563.76002 · doi:10.1093/imamat/31.1.1 [6] Rafel, Asymptotic Analysis II, LNM 985 pp 349– (1983) · Zbl 0541.73067 · doi:10.1007/BFb0062376 [7] Rafel , G. G. An internal resonance problem for a compressed column. Int. J. Nonlinear Mechanics [8] Sanders, Averaging methods in nonlinear dynamical systems. Applied Math. Sciences 59 (1985) · Zbl 0586.34040 · doi:10.1007/978-1-4757-4575-7 [9] Verhulst, Discretization and instability in nonlinear evolution equations, Publ. de l’U. E. R. Math. Pures et Appliquées 5 (1983) [10] Verhulst, Asymptotic Analysis II; LNM 985 (1983b) · Zbl 0515.70014 · doi:10.1007/BFb0062359 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.