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. This paper develops appropriate boundary conditions for the two-sided fractional diffusion equation, where the usual second derivative in space is replaced by a weighted average of positive (left) and negative (right) fractional derivatives. Mass preserving, reflecting boundary conditions for two-sided fractional diffusion involve a balance of left and right fractional derivatives at the boundary. Stable, consistent explicit and implicit Euler methods are detailed, and steady state solutions are derived. Steady state solutions for two-sided fractional diffusion equations using both Riemann–Liouville and Caputo flux are computed. For Riemann–Liouville flux and reflecting boundary conditions, the steady-state solution is singular at one or both of the end-points. For Caputo flux and reflecting boundary conditions, the steady-state solution is a constant function. Numerical experiments illustrate the convergence of these numerical methods. Finally, the influence of the reflecting boundary on the steady-state behavior subject to both the Riemann–Liouville and Caputo fluxes is discussed.