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

Time-fractional initial-boundary value problems of the form D(t)(alpha)u - p Delta u + cu = f are considered, where D(t)(alpha)u is a Caputo fractional derivative of order alpha is an element of (0, 1) and the spatial domain lies in R-d for some d is an element of 11, 2, 31. As alpha -> 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 D(t)(alpha)u is replaced by partial derivative u/partial derivative t. Nevertheless, most of the rigorous analyses of numerical methods for this time-fractional problem have error bounds that blow up as alpha -> 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 alpha -> 1(-).