# 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!

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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.