On the Convergence of WKB Approximations of the Damped Mathieu Equation

TL;DR

This paper analyzes the convergence properties of WKB approximations for the damped Mathieu equation, demonstrating convergence of solution components and quantifying the asymptotic error in characteristic exponents.

Contribution

It provides a rigorous analysis of the convergence of WKB approximations for the damped Mathieu equation, including error bounds for characteristic exponents and solution components.

Findings

WKB approximations converge to the true periodic and exponential parts as m approaches zero.

Asymptotic error for characteristic exponents is of order O(m^2).

Error for periodic parts is of order O(m).

Abstract

Consider the differential equation ${ m\ddot{x} +\gamma \dot{x} -x\epsilon \cos(\omega t) =0}$, $0 \leq t \leq T$. 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.