  We propose a new fractional Laplacian for bounded domains, expressed as a
conservation law and thus particularly suited to finite-volume schemes. Our
approach permits the direct prescription of no-flux boundary conditions. We
first show the well-posedness theory for the fractional heat equation. We also
develop a numerical scheme, which correctly captures the action of the
fractional Laplacian and its anomalous diffusion effect. We benchmark numerical
solutions for the L\'evy-Fokker-Planck equation against known analytical
solutions. We conclude by numerically exploring properties of these equations
with respect to their stationary states and long-time asymptotics.

Numerical Analysis;Numerical Analysis