In this work, a novel time-stepping \(\overline{L1}\) formula is developed for a hidden-memory variable-order Caputo’s fractional derivative with an initial singularity. This formula can obtain second-order accuracy and an error estimate is analyzed strictly. As an application, a fully discrete difference scheme is established for the initial-boundary value problem of a hidden-memory variable-order time fractional diffusion model. Numerical experiments are provided to support our theoretical results.
