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I wonder if anyone could let me know if there is a general solution to the following problem.

Suppose the functions $g(u,w)$ and $h(w)$ are known. I want to find a function $f(u)$ such that

$$ \int_0^1 g(u,w) f(u) du = h(w). $$

In addition I know that

$$ \int_0^1 f(u) du = 2 $$


$$ \int_{0}^1 u f(u)du = 1 $$

Is it possible to solve the above for the unknown function $f(u)$? Of course there might be several solutions but I would be happy to find one.

Many thanks in advance for any suggestions.


share|improve this question
Welcome to MSE. Do you know $\LaTeX$ but unaware that site supports it, by any chance? –  user21436 Feb 6 '12 at 17:52
Yes, I was not aware of that. –  Mikael Anderson Feb 6 '12 at 17:53
Then, we'll love you (:-)) if you edit the math in your question and $\TeX$ify it for us! –  user21436 Feb 6 '12 at 17:54
@JavaMan That was Quick. OP says he knew $\LaTeX$, may be this would have been a chance for him to learn about the site. Never mind! –  user21436 Feb 6 '12 at 17:56
Such equations are called Fredholm equations of the first kind. –  Julián Aguirre Feb 6 '12 at 18:22

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