Find solution of the PDE $\partial_t u - \partial _{xx}u = (x^2-2\pi x)\exp(-t)$ with given boundary conditions

Question

Let $\Omega = (0,\infty)\times (0, \pi)$ be the domain. The task is to find solution of the following equation:

$$\begin{cases} \partial_t u - \partial _{xx}u &= (x^2-2\pi x)\exp (-t) & \text{in } \Omega\\ \partial_x u (t,0)&=2\pi\exp(-t) & \text{for }t>0\\ \partial_x u (t,\pi)&=0 & \text{for }t>0\\ u(0,x)&=2\pi x-x^2 &\text{for } x\in[0,\pi] \end{cases}$$

Hint: use the substitution $u(t,x) = (2\pi x - x^2) \exp(-t) + v(t,x)$.

MY ATTEMPT: after applying the substitution in the hint, I get the following problem:

$$\begin{cases} (\partial_t - \partial_{xx})v &= -2\exp(-t) & \text{in } \Omega\\ \partial_x v(t,0) = \partial_x v(t,\pi) &= 0 \\ v(0,x) &=0 \end{cases}$$

Unfortunately I don't see how to proceed further, since using the fundamental solution doesn't take into account values of $\partial_x v$.

Thank you in advance!

Answer 1

For the latter, try a solution $v(t,x) = w(t)$ which is independent of $x$. Then the first two boundary conditions are automatically satisfied and you've reduced the problem to $$\frac{dw}{dt} = -2e^{-t}, \,\,\,\,\, w(0) = 0.$$ The solution to this is, of course, $w(t) = 2e^{-t} - 2$.