A normalized variable-order time-fractional diffusion equation

In this article, we propose a normalized variable-order time-fractional diffusion equation that effectively models anomalous diffusion phenomena characterized by evolving memory effects. The normalization ensures consistent scaling of the fractional operator across varying orders and thus allows more accurate and stable numerical simulations. Unlike conventional fixed-order models, the proposed formulation permits temporal variation of the fractional order, which results in a more accurate and flexible description of nonlocal and heterogeneous diffusion phenomena observed in various physical and biological systems. A normalization factor is introduced to ensure that the associated weight function maintains a unit sum, which preserves the consistency of the memory representation. To solve the resulting equation, a finite difference discretization method is developed that accounts for the time-varying nature of the fractional derivative. Numerical simulations are carried out to investigate the influence of different variable-order functions on the solution dynamics. The computational results demonstrate that even when the time-averaged fractional order is the same, the temporal variation in the order significantly affects the evolution of the diffusion profile. In particular, the proposed model captures transitions between fast and slow diffusion regimes that are not attainable using fixed-order approaches. Additionally, it is shown that when the fractional order reaches zero at the final time, the numerical solution coincides with the result obtained from the fully implicit Euler method using a single large time step. This observation provides further insight into the interplay between memory decay and solution behavior. The proposed method offers a computationally feasible and physically consistent framework for modeling time-dependent anomalous diffusion.