Electron. J. Diff. Equ., Vol. 2009(2009), No. 141, pp. 1-15.

Precise asymptotic behavior of solutions to damped simple pendulum equations

Tetsutaro Shibata

Abstract:
We consider the simple pendulum equation
$$\displaylines{ -u''(t) + \epsilon f(u'(t)) = \lambda\sin u(t), \quad t \in I:=(-1, 1),\cr u(t) > 0, \quad t \in I, \quad u(\pm 1) = 0, }$$
where $0 < \epsilon \le 1$, $\lambda > 0$, and the friction term is either $f(y) = \pm|y|$ or $f(y) = -y$. Note that when $f(y) = -y$ and $\epsilon = 1$, we have well known original damped simple pendulum equation. To understand the dependance of solutions, to the damped simple pendulum equation with $\lambda \gg 1$, upon the term $f(u'(t))$, we present asymptotic formulas for the maximum norm of the solutions. Also we present an asymptotic formula for the time at which maximum occurs, for the case $f(u) = -u$.

Submitted April 21, 2009. Published November 7, 2009.
Math Subject Classifications: 34B15.
Key Words: Damped simple pendulum; asymptotic formula.

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Tetsutaro Shibata
Department of Applied Mathematics, Graduate School of Engineering
Hiroshima University, Higashi-Hiroshima, 739-8527, Japan
email: shibata@amath.hiroshima-u.ac.jp

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