The existence of the solution of the wave equation on graphs

TL;DR

This paper proves the existence and uniqueness of solutions to the wave equation on finite weighted graphs with Dirichlet boundary conditions, extending classical PDE results to graph structures using Rothe's method.

Contribution

The paper establishes the existence and uniqueness of solutions for the wave equation on graphs, a novel extension of PDE theory to discrete graph domains.

Findings

Proves the wave equation on graphs has a unique solution.

Extends classical PDE results to graph structures.

Uses Rothe's method for the proof.

Abstract

Let $G=(V, E)$ be a finite weighted graph, and $\Omega\subseteq V$ be a domain such that $\Omega^\circ\neq\emptyset$. In this paper, we study the following initial boundary problem for the non-homogenous wave equation \begin{equation*} \left\{ \begin{aligned} &\partial_t^2 u(t,x)-\Delta_\Omega u(t,x)=f(t,x),\qquad&&(t,x)\in[0,\infty)\times \Omega^\circ,\\ &u(0,x)=g(x),\qquad&& x\in\Omega^\circ,\\ &\partial_tu(0,x)=h(x),\qquad&& x\in\Omega^\circ,\\ &u(t,x)=0,\qquad&&(t,x)\in[0,\infty)\times\partial \Omega, \end{aligned} \right. \end{equation*} where $\Delta_\Omega$ denotes the Dirichlet Laplacian on $\Omega^\circ$. Using Rothe's method, we prove that the above wave equation has a unique solution.