Application of the Banach Contraction Method to the Benjamin-Bona Mahony Equation: Comparative Analysis with the Laplace Adomian Decomposition Method

The Benjamin-Bona Mahony (BBM) equation, a nonlinear dispersive wave model, plays a crucial role in fields such as fluid dynamics and plasma physics. Solving the BBM equation analytically is challenging, necessitating numerical and semi-analytical approaches. This study investigates the application of the Banach Contraction Method (BCM) to the BBM equation, comparing its performance with the Laplace Adomian Decomposition Method (LADM). By employing iterative approximations, BCM demonstrates convergence to unique solutions under specific conditions, ensuring reliability. Two examples of the BBM equation are analyzed, and absolute differences between BCM and LADM results are evaluated for varying spatial and temporal resolutions. The results reveal that both methods exhibit high accuracy, with smaller discrepancies observed for shorter time intervals. However, differences increase with spatial position, suggesting potential sensitivity to spatial dynamics. BCM shows an advantage over LADM due to its strong theoretical framework for ensuring convergence and uniqueness. While LADM offers flexibility in nonlinear term handling, BCM's robustness makes it preferable in critical applications. Recommendations for further research include exploring computational efficiency, extending the comparison to other wave equations, and analyzing higher-order solutions to broaden the applicability of these method