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I've solved this PDE (I think), and the solution I've reached is $u(x,t)=1$. In the problem as given below, I just wanted to make sure I understand what is happening physically. Since the first space derivative at the boundaries are 0, does this mean that the temperature doesn't vary with space and time? Basically, I'm not very smart and need a basic explanation on what this means. Like I am 5.

The problem reads:

$u_t$=$u_{xx}$, $00$ with boundary conditions $u_x(0,t)=0$ and $u_x(1,t)=0$ and initial condition $u(x,0)=1$ for $0

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    It seems that this bar stays in equilibrium. It is initially uniformly heated and there is no flux through the endpoints (adiabatic condition). – 2017-02-14

1 Answers 1

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If you think of it as a heat conduction problem, the boundary conditions say the ends are perfectly insulated: No heat flowing in or out there. So, with an initial constant temperature, the temperature stays constant forever, since there is no place for it to go.