Now consider the 1-dimensional heat equation u_t = u_{xx} over the x-interval [0,Pi] with boundary temperatures u(0,t)=1, u(Pi,t)=2 and the initial temperature u(x,0)=0.   
The solution u(x,t) given by separation of variables is an infinite series whose first few terms are  

Below is the plot of this solution as a function of x in [0,Pi] for time t in [0,3]. Note that the steady-state temperature is the linear function v(x)=1+x/Pi.