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The time-fractional diffusion partial differential equations (tFPDEs) (of order 0 &lt;  α  &lt; 1) properly model the anomalous diffusive transport or memory effects. Recent work [ 23 ] showed that the first-order time derivatives of their solutions have a singularity of  O ( t   α −1 ) near the initial time  t  = 0, which makes the error estimates of their numerical approximations in the literature that were proved under full regularity assumptions of the true solutions inappropriate. A sharp error estimate was proved for a finite difference method (FDM) with a graded partition for a one-dimensional tFPDE without artificial regularity assumptions on true solutions, [ 23 ].

A modified time-fractional diffusion equation and its finite difference method: Regularity and error analysis