Precise asymptotic behavior of solutions to damped simple pendulum equations
Electronic journal of differential equations, Tome 2009 (2009)
We consider the simple pendulum equation
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 $f(u) = -u$.
| $\displaylines{ -u''(t) + \epsilon f(u'(t)) = \lambda\sin u(t), \quad t \in I:=(-1, 1),\cr u(t) > 0, \quad t \in I, \quad u(\pm 1) = 0, }$ |
@article{EJDE_2009__2009__a22,
author = {Shibata, Tetsutaro},
title = {Precise asymptotic behavior of solutions to damped simple pendulum equations},
journal = {Electronic journal of differential equations},
year = {2009},
volume = {2009},
zbl = {1186.34026},
language = {en},
url = {http://geodesic.mathdoc.fr/item/EJDE_2009__2009__a22/}
}
Shibata, Tetsutaro. Precise asymptotic behavior of solutions to damped simple pendulum equations. Electronic journal of differential equations, Tome 2009 (2009). http://geodesic.mathdoc.fr/item/EJDE_2009__2009__a22/