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An integral representation is obtained for the solution to the initial-boundary-value problem (with boundary conditions of the first kind) for the wave equation u  xx (x, t) = u  tt (x, t) on a one-dimensional spatial network with transmission conditions (at the nodes) of the form  $$\sum {\alpha _h (x)u_h^ + (x,t)} = 0$$ , where the summation is over all admissible spatial directions h at the point x, the α h (x) are positive numbers, and u   h  + (x, t) is the right derivative in the direction h of the function u(, t) at the point x. The kernel of the corresponding integral operator is expressed in terms of the Green function of the induced boundary-value problem for the equation −y″(x) = f(x). The expression is used to give a finite description of this kernel in terms of the solution to a system of nondifferential equations with delays equal to the lengths of the branches of the spatial network.

journal of mathematical sciences

Description of solutions to the initial-boundary-value problem for a wave equation on a one-dimensional spatial network in terms of the green function of the corresponding boundary-value problem for an ordinary differential equation