Working on thermodynamics, I arrived to an equation like this in spatial and frequency domain:
$$ \partial_z G(z,\omega) + \left( G(z,\omega) F(\omega) \right) \ast G(z,\omega) = 0 $$
with $0 < l_1 \leq z \leq l_2$ the position (in a tube filled with gas), $\omega \geq 0$ the frequency and the convolution product defined as:
$$ F_1(\omega) \ast F_2(\omega) = F_3(\omega) $$
I have the form of $F(w)$, it is a function like:
$$ F(\omega) \propto \frac{1}{i \omega + \nu_0} $$
with $\nu_0 > 0 $ a real number. I want a general solution, since I need to test different boundary conditions on $G$. Do you have an idea ?
HINT
Last time I had a similar equation but linear, Abezhiko confirmed me that the convolution exponential form was right. This time it is non linear in $G$, so more difficult. We can also applied the inverse Fourier transform to have:
$$ \partial_z g(z,t) + \left( g(z,t) \ast f(t) \right) g(z,t) = 0 $$
which seems to be difficult also due to the convolution to the function $f$. Note that this convolution is just "a retardation effect" due to the form of the function $f$:
$$ f \propto \theta(t) e^{- \nu_0 t} $$
with $\theta(t)$ Heaviside step function. So maybe we can approximate something... ?