# 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.

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Hi, welcome to Mathematica.SE, please consider taking the tour so you learn the basics of the site. I'm sorry to see that your question didn't receive attention. Can you share the code you have been working on? – rhermans Dec 1 '14 at 0:02