Blow-up of error estimates in time-fractional initial-boundary value problems

Time-fractional initial-boundary value problems of the form Dₜ^α u-p u +cu=f are considered, where Dₜ^α u is a Caputo fractional derivative of order α ∈ (0,1) and the spatial domain lies in Rᵈ for some d∈ \1,2,3\. As α → 1⁻ we prove that the solution u converges, uniformly on the space-time domain, to the solution of the classical parabolic initial-boundary value problem where Dₜ^α u is replaced by ∂ u/∂ t. Nevertheless, most of the rigorous analyses of numerical methods for this time-fractional problem have error bounds that blow up as α → 1⁻, as we demonstrate. We show that in some cases these analyses can be modified to obtain robust error bounds that do not blow up as α → 1⁻.