Heat Equation identity with dirichlet boundary condition Show an energy identity for the heat equation with convection and Dirichlet boundary condition.
$$u_t -ku_{xx}+Vu_x=0 \qquad 0<x<1, t>0$$
$$u(0,t) = u(1,t)=0 \qquad t>0$$
$$u(x,0) = \phi(x) \qquad 0<x<1$$
Attempt: I think I can apply maximum principle, but dont know how to approach it. 
Thanks!!

You can use the separation of variables techniques. Basically, we assume the solution to have the form 

$$ u(x,t)= X(x)F(t), $$

and subs back in the pde to get the following ode's 

$$ X'' \left( x \right) = \frac{V }{k} X'\left( x \right)+\frac{\lambda}{k} X(x),\quad F'(t)=\lambda F(t).  $$

Now, you can follow the technique in this problem.
