I am trying to solve the following system of coupled integral equations in Mathematica, where $J_1$ is the Bessel function of first kind:
\begin{align}f(y) &= -y\int_0^{\infty}\mbox{d}rJ_1(yr)\mbox{e}^{cf(r)}\cos(cg(r)-xr^2/2)\\ g(y) &= -y\int_0^{\infty}\mbox{d}rJ_1(yr)\mbox{e}^{cf(r)}\sin(cg(r)-xr^2/2)\end{align}
My strategy is to start by inserting an initial guess for $f$ and $g$ on the RHS. I tried with $f_0 = 0$ and $g_0 = 0$. Then I fix a value for $c$, a value for $x$, and a range of values for $y$, so I also get a set of discrete values for the first iteration of my solutions $f_1$ and $g_1$, using the NIntegrate for numerical integration. Next, I interpolate the range of values for $f_1$ and $g_1$ and insert again the resulting functions on the RHS of the equations. My hope is that after some iterations of this process, $f$ and $g$ converge to an approximate solution for the system of equations.
Although, the first step of my iterative method works, I have some problems in the subsequent steps:
Because I set a range for the values of $y$, which needs to be necessarily bounded, I don't know how to deal with the next iteration. As you can see, the integrals in the system run from $0$ to $\infty$, therefore I need all the values from $0$ to $\infty$ on the LHS at each of the iterations. Any idea on how to fix this?
Mathematica does not seem happy for certain ranges of $y$, i.e., for certain upper and lower bounds for the values of $y$ and certain separations between the values of $y$, the numeric integration does not work and it produces errors of the form:
NIntegrate::ncvb: NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in $r$ near $\{r\} = \{101.4\}$. NIntegrate obtained $1.169743334653884`$ and $0.0028671699249602856`$ for the integral and error estimates.
Both 1) and 2) do not allow me to proceed to the next iteration, and therefore, I cannot obtain a solution to my problem.
I would very much appreciate your help.