There is an equation
$$
w(x) = g(x)+\int\limits\_0^M w(y)f(x-y)\,dy
$$
where $f\geq 0$, $f\in C^\infty(\mathbb R\setminus\{c\})$ for some point $c$ and $\int\limits\_{-\infty}^\infty f(t)\,dt\leq 1$. With regards to $g$ we know that $0\leq g(t)\leq 1$. This equation should be solved for $w(x)$ on $[0,M]$. Functions $g,f$ are given. I also know a priori that $w\in C([0,M])$ and bounded by $0$ and $1$.


I guess that it is impossible to solve it analytically. For the numerical methods
I know just one method - Neumann series, moreover $$\sup\limits\_{x\in[0,M]}\int\limits\_0^M f(x-y)dy = \alpha<1$$
but $1-\alpha\approx 0.001$ so the convergence of these series is very slow. Could you advise me any other method for the solution of this problem - or maybe you can refer me to the appropriate literature?


I am looking for the procedure which can solve this equation with any precision in a sense that
$$
 \sup\limits\_{[0,M]}|w(x) - w^\*(x)|\leq\varepsilon
$$ 
and faster (or less computationally demanding) than von Neumann series, since $\alpha$ is very close to $1$.


I also asked it on [MSE](https://math.stackexchange.com/questions/52735/integral-equation-solution-fredholm-second-type).



Integral Fredholm equation of the second type
Just in the case someone will be interested in a problem of such a kind. Very nice methods are developed by Prof. Kendall E. Atkinson. I read some of his papers and also used his toolbox for MATLAB which solves these problems very precise. One can find the description of a toolbox [here](http://homepage.math.uiowa.edu/~atkinson/ftp/Fie_paper.pdf) and code [here](http://homepage.cs.uiowa.edu/~atkinson/fred.html).


