This is probably a quick one for someone familiar with integral equations:

I have an equation of the form $\int_{a}^{b}du \, g(u,v) \, f(u)=c$ with $a$, $b$, and $c$ constant; and $f(u)$ a known function (with $g(u,v)$ unknown).

Since this is a simple form it seems that someone at some point probably studied it extensively, but I can't find any info on it by searching because I don't know what it would be called. It looks like the "reverse" case, with the unknown function being of a single variable and the known being of two, is fairly well known (Fredholm equation).

Many thanks!

  • $\begingroup$ en.wikipedia.org/wiki/Volterra_integral_equation ? $\endgroup$ – Bennett Gardiner Nov 3 '14 at 10:34
  • $\begingroup$ @BennettGardiner Hey, thanks for the comment, but the wiki says that for the Volterra equation it is the unknown function which has one variable and the known one which has two. Do you know what the opposite is called? $\endgroup$ – quantum_loser Nov 3 '14 at 10:38
  • $\begingroup$ Oh, I see. That's difficult... surely $c$ would be at least a function of $v$? $\endgroup$ – Bennett Gardiner Nov 3 '14 at 11:42
  • $\begingroup$ @BennettGardiner Actually no, not in my case. In fact it is this property of "marginal flatness" that I am hoping places some constraints on a particular physical system I am studying. To be exact the function $g$ is actually some unknown probability measure I want to try to find some constraints for. $\endgroup$ – quantum_loser Nov 3 '14 at 11:47
  • $\begingroup$ Have you considered that $\displaystyle g(u,v) = \frac{c}{f(u)(b-a)}$ is a solution? $\endgroup$ – Bennett Gardiner Nov 3 '14 at 12:10

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