Finally, consider the 1-dimensional heat equation u_t = u_{xx} over the x-interval [0,Pi] with insulated ends and the initial temperature u(x,0)=sin(x).    
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 [0,2]. Note that the steady-state temperature is 2/Pi=0.6366, the average value of u(x,0) in [0,Pi].