The mBBM equation: a mathematical key to unlocking wave behavior in fluids

This research endeavors to analytically and numerically solve the nonlinear modified Benjamin–Bona–Mahony (mBBM) equation, a model of paramount importance in fluid dynamics, particularly for its application in describing unidirectional water waves with small amplitude that are influenced by dispersion and nonlinear effects. The objective of this study is to enhance the understanding of wave propagation in fluids and to establish a clear connection between the mBBM equation and other nonlinear evolution equations. Utilizing the extended auxiliary equation (EAE) and improved Kudryashov (IKud) methods, the research provides analytical solutions, while the extended cubic–B–spline (ECBS) method validates these solutions numerically. The results showcase the accuracy of the EAE and IKud methods in depicting the wave structures governed by the mBBM equation. The significance of this study lies in its potential to advance the application of the mBBM model in real–world scenarios, such as oceanography and coastal engineering. Conclusively, the research affirms the efficacy of the combined analytical and numerical approach in solving the mBBM equation, contributing novel insights into the field of applied mathematics and nonlinear partial differential equations.