The flow approach for waves in networks (Q2885139)

The linear transport equation NEWLINE\[NEWLINE \dot{y}_j(t,x)=c_jy_j'(t,x), \quad x\in[0,1], \quad t\geq 0 ,NEWLINE\]NEWLINE describes the transport of material along the edges of a directed graph. Here, \(y_j(t,\cdot)\) is the distribution of material on the \(j\)-th edge at time \(t\), \(c_j>0\) is the speed of propagation, and \(\dot{y_j}\) and \(y_j'\) are the time and spatial derivatives, respectively. The flow behavior in the vertices is described by a weighted adjacency matrix \(\mathbb{B}\), the entries of which give the proportion of the total incoming material flowing from the \(j\)-th into the \(i\)-th edge. The outflows are determined by NEWLINE\[NEWLINE y(t,1)=\mathbb{B}\,y(t,0), \quad t\geq0. NEWLINE\]NEWLINENEWLINENEWLINEThe abstract operator is a difference operator NEWLINE\[NEWLINE A:=\text{diag} \left(c_j\frac{d}{dx}\right)_{j=1}^m, \quad D(A):=\{g\in W^{1,p}([0,1], \mathbb{C}^m): g(1)=\mathbb{B}g(0)\}. NEWLINE\]NEWLINE Essential properties for the generated \(C_0\)-semigroup such as type, spectrum and asymptotics can be characterized in terms of the matrix \(\mathbb{B}\). The authors show how the wave equation on a network can be interpreted as a flow and consequently be modeled by means of a difference operator. To do so, the authors present an ``unconventional'' reduction method to treat waves as flows in networks.