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| Format: | Preprint |
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2019
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| Online Access: | https://arxiv.org/abs/1912.11559 |
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| _version_ | 1866909211780186112 |
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| author | Nwaigwe, Dwight |
| author_facet | Nwaigwe, Dwight |
| contents | Consider the differential equation ${ m\ddot{x} +γ\dot{x} -xε\cos(ωt) =0}$, $0 \leq t \leq T$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1912_11559 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On the Convergence of WKB Approximations of the Damped Mathieu Equation Nwaigwe, Dwight Classical Analysis and ODEs Mathematical Physics Consider the differential equation ${ m\ddot{x} +γ\dot{x} -xε\cos(ωt) =0}$, $0 \leq t \leq T$. 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. |
| title | On the Convergence of WKB Approximations of the Damped Mathieu Equation |
| topic | Classical Analysis and ODEs Mathematical Physics |
| url | https://arxiv.org/abs/1912.11559 |