The Lipschitzian Fractional Derivative: Using the Lipschitz Exponent to Unify Spatial Irregularity and Temporal Memory

The study proposes a new fractional operator termed the Lipschitzian Fractional Derivative (LFD) that directly links the differentiation order to the Lipschitz Exponent (LE) of functions. This means that instead of enforcing a fixed fractional order, the operator adjusts to the function’s actual smoothness or irregularity. The derivative naturally connects fractional calculus (order-based) and fractal calculus (dimension-based), since the LE can be understood as a measure of local fractal dimension. It is therefore a bridge operator. It unifies spatial irregularity (fractal dimension) and temporal smoothness (fractional order). This implies that memory effects and fractal geometry can be described simultaneously by a single operator. The Atangana operators provide a natural generalization of the Marchaud derivative in fractal contexts. They allow modeling of systems where both temporal memory and spatial fractality overlap, for example, diffusion in porous media, or anomalous transport in astrophysical systems. A natural way to combine smoothness-based fractional operators with dimension-based fractal operators is to tie the differentiation order directly to the LE. In this manner, the operator uses only one parameter rather than the two of Atangana’s operators to encode both the regularity of the function and the fractality of the geometry. This could lead to a common framework for researching singularities, anomalous diffusion, and fractal dynamics in applied mathematics, physics, and engineering.