1. ## the heat equation

A cold steel bar of length
L = 1 is held in mid-air by two posts which can be considered as a heat leak to the lab with a reference temperature Tref = 6. Initially (time t = 0), the temperature distribution across the steel bar is given by
T(t = 0; x) = -3x + 17x^2- 10x^3 0<= x <= 1

The question which i am struggling on is:
Q) Show that
T(x; t) = Tref is a solution of the heat equation including the boundary conditions. This is the state the steel bar will approach for large times. How much deviates the average temperature from Tref at t = 0:5?

Any help will be great.
Thanks

2. This looks ill-posed to me: the initial condition and boundary conditions contradict each other. And therefore, while the solution $u(x,t)=6$ is a solution to the DE and boundary conditions, it is not a $C^2$ solution which satisfies the given IVP. Also, looks better if you write it as:

$u_t=k u_{xx},\quad 0\leq x\leq 1,\quad t>0$

$u(0,0)=6,\quad u(1,0)=6$

$u(x,0)=-3x+17x^2-10x^3$