Solve the following problem (Convection–diffusion equation) in the strip $[0,L]\times [0,+\infty)$ :
$u_t (x,t) -2u_x (x,t)=0\ \ \ 0
$u_x (x,0)=x^2\ \ \ 0\leq x\leq L$
$u_t (L,t)=t^3 + L^2\ \ \ t>0$
I know the following resolutive formula (valid under certain assumptions):
$u(x,t)=e^{-bt}\int_{0}^{t}e^{bs}\ f(x+a(s-t),s) ds + e^{-bt}g(x-at)$
where
$u_t (x,t)+a\ u_x (x,t)+b\ u(x,t)=f(x,t), \ \ \ u(x,0)=g(x)$
My problem is that I do not understand how to use the condition of exercise: $u_t (L,t)=t^3 + L^2\ \ \ t>0$. Any suggestions please?