# Solving systems of integral equations using Volterra series

I came across this problem when trying to solve the following integral equations arising in direct scattering: \begin{align} n_{11}(x,z)=1+\int_{-\infty}^xe^{-izy}u(y)n_{21}(y,z)dy, \quad n_{21}(x,z)=\int_{-\infty}^xe^{izy}\bar{u}(y)n_{11}(y,z)dy \end{align}

I was suggested to iterate thoses two equations to obtain Volterra series representation. However I am not familiar with Volterra series, so is there anyone who can kindly provide me with some kind of recipes on how to do it? Thank you very much!

In order to iterate, you have to substitute the second equation into the first one. So, $$n_{11}(x,z)=1+\int_{-\infty}^xdye^{-izy}u(y)\int_{-\infty}^ydy_1e^{izy_1}{\bar u}(y_1)n_{11}(y_1,z).$$ This equation is generally the starting pointing for an iterative procedure, the main tool of perturbation techniques. E.g., you can choose for the first iterate $n_{11}(x,z)=1$ and you will get $$n_{11}(x,z)=1+\int_{-\infty}^xdye^{-izy}u(y)\int_{-\infty}^ydy_1e^{izy_1}{\bar u}(y_1)$$ $$+\int_{-\infty}^xdye^{-izy}u(y)\int_{-\infty}^ydy_1e^{izy_1}{\bar u}(y_1)\int_{-\infty}^{y_1}dy_2e^{-izy_2}u(y_2)\int_{-\infty}^{y_2}dy_3e^{izy_3}{\bar u}(y_3)+\ldots.$$ You can stop the procedure at any desired order to get an approximation to the solution of the integral equations. Then, you put this approximation to $n_{11}$ back into the equation for $n_{12}$ and you will get an approximate solution for it at the given order.