Numerical Simulation of Fractal-Fractional Logistic Equation

This study presents a numerical analysis of the logistic growth equation incorporating a Caputo-type fractal-fractional derivative. The proposed operator blends a power-law memory kernel (with order a = (۰, ۱]) and a local fractal derivative of Hausdorff type (with order ẞ= (۰, ۱]). We develop a grid-explicit, dimensionally consistent scheme for computing the history convolution of fractional and fractal-fractional derivatives. Numerical simulations reveal that reducing a and/or ẞ delays saturation and smooths transient dynamics. The proposed approach can act as a prototype for simulating real-world processes that involve memory and fractal-like temporal patterns.

کلیدواژه ها:

fractal-fractional calculus ، Caputo-type derivative ، logistic growth ، numerical methods ، memory effect