Hermann, Martin; Kaiser, Dieter Numerical methods for parametrized two-point boundary value problems – a survey. (English) Zbl 1065.65093 Alt, Walter (ed.) et al., Berichte des IZWR 1, 2003. Jena: Friedrich Schiller Universität. Jenaer Schriften zur Mathematik und Informatik 2003, 6, 23-38 (2003). Summary: This paper has a dual purpose. First, we describe numerical methods for continuation and bifurcation analysis of parametrized nonlinear two-point boundary value problems in ordinary differential equations. Numerical results are given for the Euler-Bernoulli problem of the buckling of a rod and for the so-called Archimedian spiral. Second, we demonstrate how these methods can be used to obtain insight into the buckling behavior of a thin-walled spherical shell under a uniform external static pressure: for this, we consider two different models. The first model is weakly nonlinear, i.e. the governing equations contain only quadratically nonlinear terms, whereas the second one is strongly nonlinear. Although the degree of nonlinearity differs in the two problems under consideration, the structure of the associated solution manifold looks very similar.For the entire collection see [Zbl 1054.65002]. MSC: 65L10 Numerical solution of boundary value problems involving ordinary differential equations 74K35 Thin films 34B15 Nonlinear boundary value problems for ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 65L15 Numerical solution of eigenvalue problems involving ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations Keywords:buckling of rod; Runge-Kutta method; survey paper; continuation; bifurcation; nonlinear two-point boundary value problems; Numerical results; Euler-Bernoulli problem; Archimedian spiral; spherical shell PDFBibTeX XMLCite \textit{M. Hermann} and \textit{D. Kaiser}, Jena. Schr. Math. Inform. 2003, 23--38 (2003; Zbl 1065.65093)