Let $s\in (0,1)$, how i can solve the equation:
$$ (-\Delta)^su=1,\quad\text{in}\quad(-1,1)?$$
I have no idea, any help would be appreciated.



Solving an equation with fractional laplacian
The solution to
$$(-\Delta)^s u(x)=1,\;\;-1<x<1,
$$
with $u(x)=0$ for $|x|\geq 1$ is
$$
u(x)=\frac{\sqrt{\pi}(1-x^2)^s }{4^{s}\Gamma(\tfrac{1}{2}+s)\Gamma(1+s)},$$
as follows from the [integral definition](https://en.wikipedia.org/wiki/Fractional_Laplacian) of the fractional Laplacian.  

$$\text{The $n$-dimensional generalization is} \qquad u(x)=\frac{\Gamma(n/2)(1-x^2)^s }{4^{s}\Gamma(\tfrac{n}{2}+s)\Gamma(1+s)}.$$


An early reference for this result is R. Getoor, [First passage times for symmetric stable processes in space](https://www.jstor.org/stable/1993412%0A) (1961) and a comment by Mateusz Kwaśnicki points to an earlier [paper](http://acta.bibl.u-szeged.hu/13487/1/math_009_001-042.pdf) by Marcel Riesz.


The OP asks for a simple derivation. A calculation using only elementary calculus is in [Some observations on the Green function for the ball in the fractional Laplace framework.](https://arxiv.org/abs/1502.06468)


