Abstract
For an inverse-monotone boundary value problem with the nonlinear ODE –ɛu" + sinh(u)=1,ɛ>0, u(0)=u(1)=0, applications of the following enclosure methods are presented and discussed:
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(i)
on the basis of a piecewise replacement of sinh(u) by polynomials, the construction of monotone sequences of upper and lower bounds foru;
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(ii)
on the basis of Lohner’s enclosure algorithms for solutions of ODEs, simple and multiple shooting methods.
Existence of a classical solution follows from literature and (independently) from the execution of (ii). Whereas (i) requires the inverse-monotonicity of the problem, this is not so for (ii). For small ε, the unique solution of the BVP is strongly repellent.
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Dedicated to Professor William F. Ames on the occasion of his 70th birthday
Supported by grant SM 093 of Kuwait University
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Adams, E., Baumann, K. & Grossmann, C. Comparison of direct to shooting enclosures for an inverse-monotone boundary value problem with locally steep solution. Computing 59, 63–83 (1997). https://doi.org/10.1007/BF02684404
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DOI: https://doi.org/10.1007/BF02684404