My current research has lead to me solving a fractional Laplacian equation on $\mathbb{R}^d$. For Laplace's equation $-\Delta u = f$, I know we can solve it by an integral,
$$ u(x) = \int_{\mathbb{R}^d} \Phi(x-y) f(y) dy $$
where $\Phi$ is the fundamental solution to Laplace's equation. I desire a similar solution for the fractional Laplacian $(-\Delta)^{\alpha/2} u = f$, for $\alpha \in (0,2)$. Is there an integral representation of the solution similar to Laplace's equation? If it is helpful, I am specifically wanting to solve this equation for $\alpha = 1$.
Thank you for taking the time to read this post. Any insight into this is greatly appreciated!
The fundamental solution of $(-\Delta )^{\frac \alpha 2} u = f$, $\alpha \in (0,2)$ is given by (up to a scaling constant) the Reisz potential of order $\alpha$: $$ \Phi_\alpha(x) = \frac 1 {\vert x \vert^{n-\alpha} } .$$
If you would like to read further, I would recommend looking at Fractional Thoughts by Nicola Garofalo and references therein. In particular, there is a section on the fundamental solution.