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We are interested in the so-called Fučík spectrum of the fourth-order Dirichlet boundary value problem. The Fučík spectrum is defined as the set of all couples (μ,ν) ∈a such that the problem has a nontrivial solution. Existence of a stable solution of an initial-boundary value problem that describes the oscillation of a suspension bridge is related to the shape of the Fučík spectrum of our problem.
The Fučík spectrum of our problem can be figured by Mathematica as the zero-contour of a function f=f(u,v). Evaluation of this function at one point involves a number of computations of solutions of the initial value problems corresponding to our problem. Since the problem is of the fourth order, the solution is very sensitive to the initial conditions and, of course, to the errors in the numerical integration. Hence we must set NDSolve to compute with very high precision.
Moreover, if u≠v the right-hand side is not differentiable at u=0, which causes problems as the solution changes its sign, and NDSolve need not give a correct solution. Consequently, we must stop the integration whenever the solution attains the zero value, and restart the integration. This is easily accomplished with the EventLocator method, new in Mathematica 5.1.
Since we compute with very high precision, evaluation of the function f at one point can take a few minutes. For this reason, we develop an algorithm that allows us to figure the zero-contour of the function f without the need to evaluate it at all n×n points (as it would be done by ContourPlot), omitting the “non-perspective” points.
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