Nonlinear Theory and Its Applications, IEICE
Online ISSN : 2185-4106
ISSN-L : 2185-4106
Special Section on Recent Progress in Verified Numerical Computations
Verified solutions of two-point boundary value problems for nonlinear oscillators
Florian Bünger
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2011 Volume 2 Issue 1 Pages 90-110

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Abstract

Using techniques introduced by Nakao [4], Oishi [5, 6] and applied by Takayasu, Oishi, Kubo [11, 12] to certain nonlinear two-point boundary value problems (see also Rump [7], Chapter 15), we provide a numerical method for verifying the existence of weak solutions of two-point boundary value problems of the form

-u″ = a(x, u) + b(x, u)u′, 0 < x < 1,
u(0) = 0 = u(1),

in the vicinity of a given approximate numerical solution, where a and b are functions that fulfill some regularity properties. The numerical approximation is done by cubic spline interpolation. Finally, the method is applied to the Duffing, the van der Pol and the Toda oscillator. The rigorous numerical computations were done with INTLAB [8].

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© 2011 The Institute of Electronics, Information and Communication Engineers
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