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An Efficient Way to Figure Spectra and Bifurcation Diagrams of Boundary Value Problems
Author

Jiř Benedikt
Organization: Západočeská univerzita v Plzni
Department: Katedra matematiky
Conference

2006 Wolfram Technology Conference
Conference location

Champaign IL
Description

At the Wolfram Technology Conference 2005, we showed that the so-called Fučk spectrum of the fourth-order Dirichlet boundary value problem

u4(t) = Muu+(t) - Nuu-(t), where t is an element of [0, 1], u (0) = u' (0) = u (1) = u' (1) = 0,

which bears on the stability of suspension bridges, can be figured as the zero-contour of a function of two variables. In the same way, spectra and bifurcation diagrams of many boundary value problems can be depicted. As it showed that evaluation of this function at one point may take even minutes (in the case of the fourth-order problem), it is not appropriate to use the built-in ContourPlot function, which needs values at all n n points.

We present an algorithm that needs values at much fewer points-the dependence of the number of evaluation points on n is linear instead of quadratic (under several assumptions on the zero-contour). Our algorithm is iterative-the user may apply it on the previous state until a satisfactory result is obtained. The algorithm chooses evaluation points in order to both refine the curves and search for another components of the zero-contour, with user-predefined proportion.
Subject

*Mathematics > Calculus and Analysis > Calculus
Keywords

bifurcation, Dirichlet
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BenediktTalk.nb (2.4 MB) - Mathematica Notebook [for Mathematica 5.1]
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SpectraBifurcation.nb (2.5 MB) - Mathematica Notebook [for Mathematica 5.2]