On the Convergence of WKB Approximations of the Damped Mathieu Equation

Abstract

Consider the differential equation $m\ddot{x}+\gamma \dot{x}-x\epsilon \cos (\omega t)=0$, $0\le t\le T$. The form of the fundamental set of solutions are determined by Floquet theory. In the limit as $m\to 0$ we can apply WKB theory to get first order approximations of this fundamental set. WKB theory states that this approximation gets better as $m\to 0$ in the sense that the difference in sup norm is bounded as function of $m$ for a given $T$. However, convergence of the periodic parts and exponential parts are not addressed. We show that there is convergence to these components. The asymptotic error for the characteristic exponents are $O(m^2)$ and $O(m)$ for the periodic parts.

arXiv comment10 pages. version 2

References in corpus (1)

• Overdamped dynamics of a Brownian particle levitated in a Paul trap