On semidiscrete models dominated by the heat, wave and Laplace equations

Abstract

We introduce a method to solve linear semidiscrete equations that provides for the first time explicit solutions for some well-known models such as the semidiscrete advection-diffusion equation and the semidiscrete Lighthill-Whitham-Richards equation, among others. We find conditions in the parameters of the model under which it can be dominated by the heat, wave or Laplace equations. We illustrate the fundamental solutions for all the cases based on the newly developed solution formulas. We also study spatial and time regularity in Lebesgue spaces for these models.