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I want to find a closed-form solution to the following Fredholm integral equation of the second kind: $$ \phi(t) = 1 + \int_0^1 \frac1{1+|t-s|} \phi(s) ds, \qquad t \in [0,1]. $$

A good first step might be to split the integral into $\int_0^t$ and $\int_t^1$ to make the absolute value disappear...

Apart from that, I have not been succesful. The resolvent formalism did not take me anywhere. Neither did differentiating on both sides.

Any hint is very much appreciated!

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  • $\begingroup$ any reason to believe that a closed form exists? $\endgroup$ – tired Jun 3 '17 at 9:59
  • $\begingroup$ Nothing but optimism. There is a closed form if you replace $\frac1{1+|t-s|}$ with $e^{-|t-s|}$... $\endgroup$ – Elias Strehle Jun 3 '17 at 13:39
  • $\begingroup$ you can refer to this book books.google.com/books/about/… at page 558, looks very similar $\endgroup$ – DuFong Jul 21 '17 at 4:25

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