Source identification for the wave equation on graphs
type: Item
instance of: scholarly article
title: Source identification for the wave equation on graphs
zbMATH Open document ID: 1322.35171
author: Sergei Avdonin
author: Serge Nicaise
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2014-11-11T00:00:00Z
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35R02
zbMATH DE Number: 6368465
zbMATH Keywords: integral Volterra equation
zbMATH Keywords: source identification
zbMATH Keywords: star graph
reviewed by: Kunihiko Kajitani
MaRDI profile type: Publication
full work available at URL: j.crma.2014.09.008
OpenAlex ID: W1999456430
cites work: Identifiability, stability and reconstruction results of point sources by boundary measurements in heterogeneous trees
cites work: Well-posedness of an inverse hyperbolic problem by the Hilbert uniqueness method
cites work: Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by a control method
DOI: 10.1016/J.CRMA.2014.09.008
review text: The authors consider source identification problems for the wave equation on an interval and on trees. The main advantage of their approach is its locality: to recover unknown functions on a part of the graph, a piece of information is used which is only relevant to this subgraph. This feature of the method allows the authors to propose a very efficient identification algorithm, which is new, even in the one-dimensional case, and much simpler than similar algorithms proposed in the past.NEWLINENEWLINE In the case of an interval, consider the initial boundary value problem NEWLINE\[NEWLINE\begin{aligned} u_{tt}- u_{xx} & = f(t,g(x),\quad 0< x< l,\quad 0<t<T,\tag{1}\\ u(0,t) &= u(l,t)= 0,\quad 0< t< T,\tag{2}\\ u(x,0) &= u_t(x,0)= 0,\quad 0<x<1,\tag{3}\end{aligned}NEWLINE\]NEWLINE Here, \(f\) is a known function, while \(g\) is an unknown function. The following cases are considered:NEWLINENEWLINE (a) \(g\in L^2(0,T)\),NEWLINENEWLINE (b) \(g(x)= \sum^N_{j=1} \alpha_j\delta(x- \xi_j)\), \(0< \xi_1<\cdots< \xi_N<l\).NEWLINENEWLINE Then, the authors prove the following theorem.NEWLINENEWLINE (a) If \(f\in L^2(0,T)\) and \(g\in L^2(0,l)\) in the IBVP (1)--(3), then \(u_x(0,\cdot)\in C(0,T)\) and it can be represented by NEWLINE\[NEWLINEu(0,t)= \int^t_0 g(x)w(x,t)\,dx.NEWLINE\]NEWLINE (b) If \(f\in C[0,T]\) and \(g\) is a linear combination of \(\delta\)-functions, then \(u_x(0,\cdot)\in C[0,T]\) and it can be represented by NEWLINE\[NEWLINEu_x(0,T)= \sum_{j;\xi_j\leq t}\alpha_j w(\xi_, t),\quad 0\leq l,NEWLINE\]NEWLINE where \(w\) is a solution of the problem NEWLINE\[NEWLINE\begin{aligned} w_{tt}-w_{xx} & =0,\quad 0<x<l,\, 0<t<T,\\ u(0, t) &= f(t),\;0<t< T,\\ w(x, 0) A &= w_t(x,0)= 0,\;0<x< l.\end{aligned}NEWLINE\]NEWLINE It should be remarked that the above approach can be extended to equations on graphs.
description: scientific article; zbMATH DE number 6368465