ChapterZero

An integral equation of the second kind

Fredholm integral equations of the second kind are of the form

$\sigma(x) + \int_\Omega k(x, y) \sigma(y) \; dy = \phi(x),$

where the kernel and data $\phi$ are given, and the problem is to find $\sigma$ . Sometimes solutions exist and are unique, but that’s not guaranteed.

But you don’t need to know anything about IEs to solve this problem:

$\sigma(x) + \int_0^1 (x-y)^2 \sigma(y)\; dy = \cos(\pi/2 x)$

Give it a try!

Bonus! solve this one:

$\sigma(x) + \int_{-1}^1 |x-y| \sigma(y) \; dy = \cos(\pi/2 x)$

I was surprised that it’s relatively easy to get this solution in a closed form. There’s a trick to it…

This entry was posted on Wednesday, May 23rd, 2007 at 3:57 pm and is filed under Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

An integral equation of the second kind

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