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?

Edited: Thanks to Paul's comment I made my question more explicit. Unknown is function $w$ on $[0,M]$ rather than the point $x$. I would like to stress that the solution I need must be $\varepsilon$-precise, i.e. $\|w-w^*\|\leq \varepsilon$ for the numerical solution $w^*$ . Here $\|\cdot\|$ is a sup-norm.

I also posted this question on mathoverflow.

  • 2
    $\begingroup$ If it truly is $x$ to be found, it is not accurate to say that this is a Fredholm equation, ... as the latter would solve for an unknown function (e.g., for your $w$ above, given $g$ and $f$). Is the question really posed as intended? $\endgroup$ – paul garrett Jul 21 '11 at 0:52
  • $\begingroup$ @paul garrett: thank you, it is indeed an equation on $w$ - now I edited to make this fact more explicit. $\endgroup$ – Ilya Jul 21 '11 at 6:08
  • $\begingroup$ Would a direct discretization of the integral (leading to a system of linear equations on $w(x_i)$) work here? $\endgroup$ – Andrew Jul 21 '11 at 8:05
  • $\begingroup$ @Andrew: maybe it would. I only wonder what will be the error in this case. Perhaps, I should also mention in the question that the solution I need must be $\varepsilon$-precise. $\endgroup$ – Ilya Jul 21 '11 at 8:10
  • $\begingroup$ Probably an answer would be just too long. I recommend to start with the book "Theory of Tikhonov Regularization for Fredholm Equations of the First Kind" by Groetsch. This should help in understanding if (or if not) you have an ill-posed problem. There should also be pointers to numerical methods there. You could consider CG here... $\endgroup$ – Dirk Jul 21 '11 at 14:36

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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