Integral representation of the solution of the initial value problem for the wave equation on a geometric graph without boundary vertices

We study the initial value problem $u(x,0)=\varphi(x),$   $u_t(x,0)=0$ for the wave equation $u_{xx}(x,t)=u_{tt}(x,t)$ for $x\in\Gamma\setminus J$ and $t>0,$   where $\Gamma$ is a geometric graph (according to Yu. V. Pokornyi) with straight-line edges and without boundary vertices ($\partial\Gamma=\varnothing$), $J$ is the set of all internal vertices of $\Gamma,$   and the function $\varphi$ is given; the transmission conditions that close the problem are, in addition to the continuity of the function $u(\,\cdot\,,t)$ at the interior vertices, the smoothness conditions for it, the essence of which is that for each $t\geqslant0$ at each interior vertex $a\in J$ the sum of the right derivatives of the function $u(\,\cdot\,,t)$ in all admissible directions is 0. It is proved that if $G^\ast$ is a generalized Green's function (according to M. G. Zavgorodniy, 2019) for the boundary value problem $-y''(x)=f(x),$   $x\in\Gamma\setminus J,$   under smooth transmission conditions (here $y$ is the desired function, continuous at the points of $J,$   and $f$ is a given function, uniformly continuous on each edge of $\Gamma$), then the classical solution $u$ of the initial value problem is representable in form:
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u(x,t)=\langle\varphi\rangle-\int\limits_\Gamma g^\ast(x,t,s)\varphi''(s)\,ds,

where $\langle\varphi\rangle$ is the average of $\varphi$ over $\Gamma,$   and $g^\ast(x,t,s)=[\mathcal C(t)G^\ast(\,\cdot\,,s)](x),$   where, in turn, $\mathcal C$ is an operator function finitely described only through the metric and topological characteristics of $\Gamma.$    The approach to obtaining this representation of $u$ is similar to the approach implemented by the author earlier (2006) in the case where $\partial\Gamma\ne\varnothing$ and Dirichlet conditions are imposed at the points of $\partial\Gamma.$