A time-stepping finite element method for a time-fractional partial differential equation of hidden-memory space-time variable order

. We analyze a time-stepping finite element method for a time-fractional partial differential equation with hidden-memory space-time variable order. Due to the coupling of the space-dependent variable order with the finite element formulation and the hidden memory, the variable fractional order cannot be split from the space and destroys the monotonicity in the time-stepping discretization. Because of these difficulties, the numerical analysis of a fully-discrete finite element method of the proposed model remained untreated in the literature. We develop an alternative analysis to address these issues and to prove an optimal-order error estimate of the fully-discrete finite element scheme without any assumption on the regularity of the true solution and perform numerical experiments to substantiate the theoretical findings.