When studying the case of a quantum system driven by two different frequencies, I have two solve differential equations of the form
\begin{equation} \label{eq1} \dot{a}(t) = \left(\int_0^t f(t-t') \, a(t') \, dt'\right) e^{i Bt} \end{equation}
with $f(t-t') = e^{i A (t-t')}$.
I. When $B = 0$ (case of only one driving) the Laplace transform of the equation yields
$$ s A(s) - a(0) = F(s) \,A(s) $$
thanks to the fact that we have to deal with a convolution product, where $F(s) = 1/(s-iA)$. We then have
$$ A(s) = \frac{a(0)}{s- F(s)}$$
and using the inverse Laplace transform a solution for $a(t)$ can be found.
II. When $B \neq 0$ (case of two drivings), the Laplace transform yields
$$ s A(s) - a(0) = F(s-i B) \,A(s-iB) $$
thanks to the fact that the imaginary exponential yields a shift in Laplace space. However, we obtained a non-local equation for $A(s)$.
Question 1 : Is there a method to solve such non-local equation ?
Question 2 : If not, is there then an alternative method to solve the first equation, that is the one in time domain ?
Thank you very much in advance. I think it is quiet a general case of equations we can often encounter.