# Code for solving numerically an integro-differential equation

First of all, I want to greet the community. This is my first question, but I hope I will be able to help answering others members questions, although I am quite new working with Mathematica.

I would like to work out the numerical solution of this integro-differential equation:

$$\partial_t P(t)=-\frac{1}{\hbar^2}\int_0^tg(t,\tau)P(\tau) e^{\frac{i}{\hbar}k(t-\tau)}d\tau$$

where $k$ is a constant. The function is defined as

$$g(t,\tau)=\frac{\hbar^4}{4}\sum_{n=1}^N|\alpha_n|^2\exp[-i \omega_n\cdot(t-\tau)]=F_1(t-\tau)+iF_2(t-\tau)$$

Before deciding to solve numerically, I applied the Laplace transform to my equation (having in mind that I have a convolution product) and got $P(s)$ but I can't go any further because to restore $P(t)=\mathcal{L}^{-1}[P(s)]$ I need to solve the Bromwich integral and finding the poles of my function has no analytical solution. Again, at certain point I should use numerical methods so I prefer to start solving the equation numerically from the beginning and learn how to do that.

To be honest, this is my first try in Mathematica and I would appreciate any help, comments focused on code writing. Thank you very much.