Approximation of generalized time fractional derivatives: Error analysis via scale and weight functions

In this manuscript, we investigate various properties of the Generalized time fractional derivative (GTFD). We establish several regularity results and derive bounds in specific spaces like C1,W1,1 that characterize the behavior of the GTFD operator. Furthermore, we propose an efficient hybrid computational approximation to approximate the GTFD of order α∈(0,1), applicable to smooth and non-smooth solutions. This approximation is based on a Newton interpolation polynomial of arbitrary finite degree. The role of scale and weight functions in influencing the local truncation error and convergence order is thoroughly analyzed, with numerical experiments providing validation of these theoretical insights. The proposed approximation is further utilized to construct a computational scheme for solving the general time fractional diffusion equation (GTFDE), for which we rigorously establish uniqueness and convergence, while stability is proven specifically for linear interpolation. Numerical examples are utilized to verify that our scheme is more efficient compared to the existing schemes (Stynes et al., 2017, Z. wang, 2025 and Xu et al., 2013). Without loss of generality, numerical results are presented for linear and quadratic interpolation, confirming the approximation’s accuracy and consistency with theoretical predictions.