Source identification for the wave equation on graphs
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.
scientific article; zbMATH DE number 6368465