In this paper, we discuss discrete versions of the heat equations and the wave equations, which are called the ω-diffusion equations and the ω-elastic equations on graphs. After deriving some basic properties, we solve the ω-diffusion equations under (i) the condition that there is no boundary, (ii) the initial condition and (iii) the Dirichlet boundary condition. We also give some additional interesting properties on the ω-diffusion equations, such as the minimum and maximum principles, Huygens property and uniqueness via energy methods. Analogues of the ω-elastic equations on graphs are also discussed.
Cite this article
Soon-Yeong Chung, Yun-Sung Chung, Jong-Ho Kim, Diffusion and Elastic Equations on Networks. Publ. Res. Inst. Math. Sci. 43 (2007), no. 3, pp. 699–726