# Integral Fredholm equation of the second type

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.