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The existence of the solution of the wave equation on graphs

Yong Lin Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China Yuanyuan Xie School of Mathematics, Renmin University of China, Beijing, 100872, China

Analysis of PDEs mathscidoc:2207.03011

arXiv, 2021.8
Let G=(V,E) be a finite weighted graph, and Ω⊆V be a domain such that Ω^∘≠∅. In this paper, we study the following initial boundary problem for the non-homogenous wave equation ∂^2_t u(t,x) − Δ_Ω u(t,x) = f(t,x), (t,x)∈[0,∞)×Ω^∘ u(0,x)=g(x), x∈Ω^∘, ∂_t u(0,x)=h(x), x∈Ω^∘, u(t,x)=0, (t,x)∈[0,∞)×∂Ω, where Δ_Ω denotes the Dirichlet Laplacian on Ω^∘. Using Rothe's method, we prove that the above wave equation has a unique solution.
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@inproceedings{yong2021the,
  title={The existence of the solution of the wave equation on graphs},
  author={Yong Lin, and Yuanyuan Xie},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707161226414216582},
  booktitle={arXiv},
  year={2021},
}
Yong Lin, and Yuanyuan Xie. The existence of the solution of the wave equation on graphs. 2021. In arXiv. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707161226414216582.
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