Authors: Lamacz, AgnesSchweizer, 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 asymptoticswave equationeffective equationdispersion URI: http://hdl.handle.net/2003/38166http://dx.doi.org/10.17877/DE290R-20145 Issue Date: 2019-05-17 Appears in Collections: Preprints der Fakultät für Mathematik

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