Authors: | Lamacz, Agnes Schweizer, Ben |
Title: | Representation of solutions to wave equations with profile functions |
Language (ISO): | en |
Abstract: | Solutions to the wave equation with constant coefficients in $\mathbb{R}^d$ ca be represented explicitly in Fourier space. We investigate a reconstruction formula, which provides an approximation of solutions $u(., t)$ to initial data $u_0(.)$ for large times. The reconstruction consists of three steps: 1) Given $u_0$, initial data for a profile equation are extracted. 2) A profile evolution equation determines the shape of the profile at time $\tau = \varepsilon^2 t$. 3) A shell reconstruction operator transforms the profile to a function on $\mathbb{R}^d$. The sketched construction simplifies the wave equation, since only a one-dimensional problem in an $O(1)$ time span has to be solved. We prove that the construction provides a good approximation to the wave evolution operator for times $t$ of order $\varepsilon^{-2}$. |
Subject Headings: | large time asymptotics wave equation effective equation dispersion |
URI: | http://hdl.handle.net/2003/38166 http://dx.doi.org/10.17877/DE290R-20145 |
Issue Date: | 2019-05-17 |
Appears in Collections: | Preprints der Fakultät für Mathematik |
Files in This Item:
File | Description | Size | Format | |
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Preprint 2019-04.pdf | DNB | 502.45 kB | Adobe PDF | View/Open |
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