$\bullet$ I have the following heat equation with a time-periodic transport term:
$$\kappa u_{xx} - a \sin(\omega t)u_x = cu_t$$
I'm considering a 1D domain over $-l<x<l$.
I'd like to be able to solve this for simple boundary conditions such as $u(-l,t) = u_0 - \Delta u_0$ and $u(l,t) = u_0 + \Delta u_0$ and $u(x,0)=0$, and if possible from there to be able to construct a solution for arbitrary time-dependent boundary conditions $u(-l,t) = f(t)$ and $u(l,t)=g(t)$.
$\bullet$ I already have general solutions to the PDE:
$$u_{\lambda} = A_\lambda e^{-\frac{\kappa}{c}\lambda^2 t} [\sin(D(t,\lambda))\cos(\lambda x)+\cos(D(t,\lambda))\sin(\lambda x) ]$$
$$u_{\lambda} = B_\lambda e^{-\frac{\kappa}{c}\lambda^2 t} [\sin(D(t,\lambda))\sin(\lambda x)-\cos(D(t,\lambda))\cos(\lambda x) ]$$
where $A_\lambda$, $B_\lambda$, $\lambda$ are arbitrary constants and
$$D(t,\lambda) = \frac{a\lambda}{c\omega}\cos(\omega t)$$
$\bullet$ Each of those satisfies the PDE. However, due to the presence of both $\sin(\lambda x)$ and $\cos(\lambda x)$ in each solution, constructing a series using e.g. $\lambda = (2n+1)\pi/2l$ doesn't seem to lead anywhere useful.
Does anyone have any ideas how I could solve the boundary value problem, even in the simplest case $u(-l,t) = u(l,t) = u_0$? Initial conditions are less important, as it is only the long-time behaviour which I am interested in.
$\bullet$ Incidentally, I know from numerical simulations that the long time solution is oscillatory in the case $u(-l,t) = u_0 - \Delta u_0$ and $u(l,t) = u_0 + \Delta u_0$ with frequency components at multiples of $\omega$.
Thanks in advance! Any advice would be greatly appreciated!