2
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.