Is there any intelligent guy who could solve this equation for $p(k,t)$? $p(k,t)$ is the Fourier transport of $p(x,t)$ which is the probability of the particle to be at time $t$ at position $x$. $p(x,t)$ must be normalized in the interval $(0,l)$. $a$ is a constant and $i$ is used as the imaginary unit.
$p(k,t)+ \partial_t p(k,t)= -\frac{iak}{l}$
$$p(k,t)+ \partial_t p(k,t)= -\frac{iak}{l}$$ Since there is only one derivative with respect to only one variable $ t$, the equation is a simple ODE. Solving it is straightforward : $$p(k,t)=\frac{iak}{l}+C(k)e^{-t}$$ where $C(k)$ is any differentiable function of $k$.