Consider the heat problem
$$
\begin{array}{ll}
u_t = ku_{xx} & u(x,0) = f(x) \\
u_x(0,t) = 0 & u_x(1,t) = 0,
\end{array}
$$
where the initial temperature distribution $f(x)$ looks like so:
Sketch the steady state solution along with the evolution towards that steady state.
I would love to attach a graph that will satisfy the given conditions but "new users are not allowed'. So I will describe it instead. At t=0 we see that the curve looks like the given graph. For t >0, the derivative with respect to x is held to zero at both ends, giving us horizontal tangent lines at x=0 and x=1. This occurs because x = 0 and x = 1 are insulated. As t approaches infinity, the curve approaches y=.5. I believe that the area underneath the curve should remain constant throughout the transformation.
I think the area would represent the total amount of heat in a bar and we're assuming that no heat is lost to the environment. So the heat is being redistributed along the bar, but the total amount of heat will be unchanged. Though somebody please correct me if I'm wrong.
Keep in mind the first derivative of x is set to 0 at the endpoints. Not the actual value. The area under the curve represents total energy. If the ends are insulated, no energy is allows to escape the bar, it just simply redistributes.
And now that I am more than just a new member, I will add a photo!
