I'm attempting to solve a simple Dirichlet problem on the fractional Laplacian with boundary conditions:

$r^{+}(\nabla^s) v = f$

where $0 \leq s \leq 1/2$, $v$ is zero outside of $[0,1]$, $r^{+}$ restricts a function to $[0,1]$, and $f:[0,1] \rightarrow \mathbb{R}$ with $f(t) = t^{-2s}$ or more generally, $f(t) = t^{k}$ for some fixed value $k$.

Most references I can find concern themselves with proving the regularity of solutions to the fractional Laplacian equation. Is there a simple way to solve this equation? I've looked in references such as: https://arxiv.org/pdf/1712.01196.pdf which purport to solve these equations, but I could not find a place in the text where a method for solving such equations is given.

I'm attempting to solve a simple Dirichlet problem on the fractional Laplacian with boundary conditions:

$r^{+}(\nabla^s) v = f$

where $0 \leq s \leq 1/2$, $v$ is zero outside of $[0,1]$, $r^{+}$ restricts a function to $[0,1]$, and $f:[0,1] \rightarrow \mathbb{R}$ with $f(t) = t^{-s}$ or more generally, $f(t) = t^{k}$ for some fixed value $k$.

Most references I can find concern themselves with proving the regularity of solutions to the fractional Laplacian equation. Is there a simple way to solve this equation? I've looked in references such as: https://arxiv.org/pdf/1712.01196.pdf which purport to solve these equations, but I could not find a place in the text where a method for solving such equations is given.