Boundary conditions for two-sided fractional diffusion.

•Two-sided fractional diffusion equations are written in conservation form.•Mass-preserving, reflecting boundary conditions for these diffusion equations are a combination of fractional derivatives.•Stable explicit and implicit Euler schemes for two-sided fractional diffusion equations with any combination of absorbing and reflecting boundary conditions are presented.•Closed-form, steady-state solutions are derived.•Numerical experiments verify that the explicit and implicit Euler schemes converge to the analytical steady-state solution for large time.