Abstract
This study examines approximate long wave and the modified Boussinesq equations, as well as their complexities with the Atangana-Baleanu fractional derivative operator in the Caputo sense. The analytical solution of the aforementioned model is discussed using the Elzaki transform and the Adomian decomposition method. These problems are indispensable for defining the characteristics of surface water waves by applying a particular relationship of dispersion. We used Elzaki transformation on time-fractional approximate long wave and modified Boussinesq equations, followed by inverse Elzaki transformation, to achieve the results of the equations. To validate the methodology, we concentrated on two systems and compared them to the actual solutions. The numerical and graphical results demonstrate that the proposed method is computationally precise and straightforward for investigating and resolving fractionally coupled nonlinear phenomena that occur in scientific and technological.