Iterative Investigation of the Nonlinear Fractional Cahn–Allen and Fractional Clannish Random Walker’s Parabolic Equations by Using the Hybrid Decomposition Method

In this work, we numerically investigate the fractional clannish random walker’s parabolic equations (FCRWPEs) and the nonlinear fractional Cahn–Allen (NFCA) equation using the Hybrid Decomposition Method (HDM). The analysis uses the Atangana–Baleanu fractional derivative (ABFD) in the Caputo sense, which has a nonsingular and nonlocal Mittag–Leffler kernel (MLk) and provides a more accurate depiction of memory and heredity effects, to examine the dynamic behavior of the models. Using nonlinear analysis, the uniqueness of the suggested models is investigated, and distinct wave profiles are created for various fractional orders. The accuracy and effectiveness of the suggested approach are validated by a number of example cases, which also support the approximate solutions of the nonlinear FCRWPEs. This work provides significant insights into the modeling of anomalous diffusion and complex dynamic processes in fields such as phase transitions, biological transport, and population dynamics. The inclusion of the ABFD enhances the model’s ability to capture nonlocal effects and long-range temporal correlations, making it a powerful tool for simulating real-world systems where classical derivatives may be inadequate.