Diffusion and Elastic Equations on Networks
Soon‐Yeong Chung, Yun-Sung Chung, Jong-Ho Kim · Publications of the Research Institute for Mathematical Sciences · 2007
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.