General solutions of a three-level partial difference equation

This paper is concerned with a linear partial difference equation which includes the well-known DuFort Frankel multilevel difference scheme for the heat equation. By introducing Greenʹs functions for this equation, we obtain, via a novel formal approach, an explicit formula for all its solutions. Given exponential initial conditions, separable solutions are also found. As applications, we derive several stability criteria for the solutions of this equation.