Given the following PDE:
$$\partial\_t\Psi(x,t)=\partial\_{xx}\Psi(x,t)+k\partial\_x\Psi(x,t)+g(x,t)-\beta\Psi(x,t)=0$$
where:
$$g(x,t)=g\_0\left(\frac{a\_0}{2}+\sum\_{k=1}^\infty\left[a\_k\sin(kt)+b\_kcos(kt)\right]\right)$$ with:
$\Psi(0,t)=0,\Psi(L,t)=0$
and $\Psi(0,x)=\Psi\_0$, $\beta$ constant $(\beta\gt\gt1)$ and $k$ constant,
is it possible to find analytically a solution of this equation?
Thanks in advance.



Solution of a partial differential equation containing a Fourier series
Let us write the operator, with the notation $D\_x=-i\partial\_x$,
$$
L=D\_x^2-ik D\_x+\beta,\quad\text{the equation is}\quad\partial\_t\Psi+L\Psi=g.
$$
We start with noticing
$
\partial\_t-ik D\_x+\beta=e^{iktD\_x}e^{-t\beta}\partial \_t e^{t\beta}e^{-iktD\_x}
$
so that with $\Psi=e^{iktD\_x}e^{-t\beta} u$, we get the new equation, using the commutation properties with $L$,
$$
\partial\_t e^{iktD\_x}e^{-t\beta} u+Le^{iktD\_x}e^{-t\beta} u=g\Longleftrightarrow
e^{-iktD\_x}e^{t\beta}\partial\_t e^{iktD\_x}e^{-t\beta} u+Lu=e^{-iktD\_x}e^{t\beta}g
$$
i.e.
$$
\partial\_t u-\beta u+ikD\_x u+D\_x^2 u-ikD\_x u+\beta u=\partial\_t u+D\_x^2 u=e^{-iktD\_x}e^{t\beta}g
$$
which is the heat equation:
we get for positive times $t$
$$
u=e^{-tD\_x^2}u(0)+\int\_0^t e^{-(t-s)D\_x^2}e^{-istD\_x}e^{s\beta}g(s) ds.
$$


