# Non-linear integral equation

I'm trying to solve with Mathematica an integral equation. I found this excellent answer (How to solve a non-linear integral equation?) solving with a collocation method a problem which can be restated as:

$$\int_a^b f \left( \phi \left(x \right) \right) \mathrm{d} x= 1$$

for some function $f$, $\phi(x)$ being the unknown. My problem is slightly more complicated:

$$\phi(x) = \int_a^b K \left( x, y \right) f \left( \phi \left( y \right) \right) \mathrm{d} y$$

and the kernel $K(x,y)$ is singular for $x=y$, it behaves as $\sim \frac{1}{\left| x - y \right|}$. I would like to know if it is possible to extend the collocation method to this case, or, alternatively, which other methods can be used to numerically solve my integral equation.

$$f(\phi(x)) = \frac{\phi(x)}{\sqrt{\phi^2(x) + C^2}}$$ $$K(x,y) = y \frac{\mathrm{e}^{-\left| x - y \right|}}{\left| x - y \right|}$$
@george2079: it should converge, at least it is eq. 5 in this paper: journals.aps.org/prl/pdf/10.1103/PhysRevLett.74.1633 The authors refer to an unpublished paper for the numerical part, commenting as follows: letting $x=tan(\beta)$, we set up a Gaussian-quadrature grid for $\beta$ and convert the above equations into a matrix form which can be solved iteratively. The logarithmic singularities are treated separately. – zakk Mar 1 '14 at 1:54