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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.

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2 Answers 2

I'm not an expert here but it seems like since the integral is a convolution a Fourier or Laplace-based method could work. Might get you into the Wiener-Hopf technique.

Search for transform-based methods for integral equations in Google..

Good luck, Tom

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up vote 1 down vote accepted

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 and code here.

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