# Explicit solution of a Fredholm integral equation of the second kind

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!

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