Heat Transport and Dissipation

The Heat transport equation in one dimension x reads

D[u[x, t], t]  ==  D[u[x, t], {x, 2}]

where u is the temperature (or internal energy density). 
Or more fancy:

(∂_t) u[x, t]  ==  (∂_(x, x))u[x, t]

1)Use NDSolve to solve this equation for different boundary conditions in a finite domain (a bar of finite length)
2)Produce an animated plot of the heat distribution along the bar as a function of a time

Fixed temperature at the edges

As boundary conditions take a temperature profile u(x,t=0) 
at t=0 (for example a gaussian curve. This can be centered in the centre of the bar or at one of the edges.
In addition, you need boundary conditions at the edges of the rod for all times, describing the reservoirs at fixed temperatures:
u(a,t)=(Subscript[T, a])  , u(b,t) = (Subscript[T, b]) , a and b being the edges side of the rods

Isolated bar

In this case the boundary conditions correspond to having no heat flow at the ends of the bar, i. e.
D[u[x,t],x]/.x->a==0   and D[u[x,t],x]/.x->b==0

Further

You can now play e.g. with 
1) mixed conditions
2) 2+1 or 3+1 dimensions
....