Numerical solution to integro-differential equation

The time dynamics of an atom interacting with a reservoir of spectral density $$J(\omega)$$ are obtained by solving the following integro-differential equation:

$$\frac{\mathrm{d}c(t)}{\mathrm{d}t} = - \int\limits_0^t \mathrm{d}\tau~ c(\tau)f(t-\tau),$$

where

$$f(t-\tau) = \int\limits_0^{\infty}\mathrm{d}\omega ~ J(\omega) \exp(i(\omega_E-\omega)(t-\tau))$$

I have an analytical solution when $$J(\omega)$$ is a Lorentzian distribution, however, is there a way to solve the above numerically for $$c(t)$$, given an arbitrary spectral density?

In particular I don't have an analytical expression for $$J$$, only the numerical data.

• Both LHS and RHS are linear operators applied to function c(t), so if c is discretized on a grid in time, then the equation boils down to a linear algebra problem. Jul 11 at 15:15