Physical Intuition

This equation is a parabolic partial differential equation (PDE). Its solution $T(x,y,z,t)$ models the temperature profile and the heat transfer over a domain in $1-D$, $2-D$ or $3-D$. It is an explicit function of the time, $t$. In other words, the solution of the heat equation (PDE) models the evolution of the temperature profile overtime, starting with a given distribution $T(x,y,z,t=0)$ at $t = 0$. The heat equation assumes independent space and time variables.

Another way to visualize the solution of this equation is though plotting a graph of the temperature profile generated at each $\Delta t$ on a discretized time grid. These graphs will make a movie if displayed in order and at fixed time intervals.