Application of the Atangana–Baleanu operator in Caputo sense for numerical solutions of the time-fractional Burgers–Fisher equation using finite difference approaches

Abstract

This research investigates the numerical solution of the time-fractional Burgers–Fisher equation, utilizing the Atangana–Baleanu differential operator in the Caputo sense. This study addresses the need to comprehend the dynamics of nonlinear phenomena encountered in various scientific and engineering contexts, specifically within the Burgers–Fisher equation, which intertwines diffusion and reaction processes. Our findings reveal that the application of the Atangana–Baleanu operator significantly alters the behavior of the system, exhibiting distinct characteristics compared to traditional methods. Notably, we identify unique patterns of propagation, such as enhanced wave speed and altered front dynamics, that emerge due to the fractional dynamics. The simulations demonstrate improved stability and convergence properties when utilizing the Atangana–Baleanu operator, allowing for more accurate representations of physical processes. Additionally, we observe the emergence of non-local effects and the potential for multiple equilibrium states, enriching our understanding of the complex interactions within the system. Through the finite difference method, we efficiently discretize the continuous problem, facilitating simulations that illustrate the intricate temporal behavior of the time-fractional system. This methodology not only enhances the understanding of the physical processes involved but also contributes a novel framework for studying time-fractional equations, emphasizing the rich dynamics introduced by the Atangana–Baleanu operator in conjunction with the Caputo fractional derivative.