Mass-conserving tempered fractional diffusion in a bounded interval

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. A reflecting boundary condition enforces mass conservation on a bounded interval. In this work, explicit and implicit Euler schemes for tempered fractional diffusion with discrete reflecting or absorbing boundary conditions are constructed. Discrete reflecting boundaries are formulated such that the Euler schemes conserve mass. Conditional stability of the explicit Euler methods and unconditional stability of the implicit Euler methods are established. Analytical steady-state solutions involving the Mittag-Leffler function are derived and shown to be consistent with late-time numerical solutions. Several numerical examples are presented to demonstrate the accuracy and usefulness of the proposed numerical schemes.