Efficient Numerical Strategies for 2D Nonlinear Variable-Order Fractal Fractional Reaction-Diffusion Models

Abstract

This study addresses a nonlinear 2D reaction-diffusion system governed by variable-order fractal fractional derivatives in both the Caputo and Atangana-Baleanu Caputo (ABC) senses. These derivatives are characterized by non-singular kernels involving Mittag-Leffler functions, capturing memory and hereditary effects in complex media. To obtain numerical solutions, a novel Non-standard Weighted average Finite Difference Method (NWFDM) is developed and implemented. This approach allows flexibility in temporal and spatial discretization while accommodating the variable fractional and fractal orders. A comprehensive analysis of stability using a von Neumann-type method and detailed error estimates is provided for both fractional formulations. Numerical simulations confirm the stability and accuracy of the proposed schemes, revealing the distinct impact of fractional parameters on system dynamics. The proposed methods exhibit superior performance compared to traditional schemes, particularly in capturing anomalous diffusion behavior in nonlinear environments. This study investigates a nonlinear 2D reaction-diffusion system incorporating variable-order fractal.