I am looking at a paper "Exact Solutions of a Model For Crowding and Information Transmission in Financial Markets" by R.D'Hulst and G. J. Rodgers. They use a model from bond percolation theory based on the Smoluchowski coagulation equation. I have some background in ODEs and PDEs, but not really any in integro-differential equations such as this. It seems that this particular equation is solved using generating functions--a la recurrence relations, etc.
Here is the classical Smoluchowski coagulation equation:
$$ \frac{\partial n(x_i,t)}{\partial t}=\frac{1}{2}\sum^{i-1}_{j=1} K(x_i-x_j,x_j)n(x_i-x_j,t)n(x_j,t) - \sum^\infty_{j=1}K(x_i,x_j)n(x_i,t)n(x_j,t) $$
Here is the model presented in the paper. The particular constants or variablee are not so important, but the form of the equations looks very similar:
$$ \frac{\partial{n_s}}{\partial{t}} = -asn_s + \frac{(1-a)}{N_0}\sum_{r=1}^{s-1}rn_r(s-r)n_{s-r} - \frac{2(1-a)sn_s}{N_0}\sum_{r=1}^{\infty}rn_r $$
Does anyone have a good reference on how to solve this type of equation using generating functions? The paper provides the result of this method, but no explanation of the method or intuition behind it.
Also, is the usual method for solving integro-differential equations usually analytical? Or are these usually solved numerically like most other differential equations.
Thanks.