Application of optimal homotopy asymptotic method with use of Daftardar Jeffery Polynomials to Benjamin-Bona-Mahony equation

The Benjamin-Bhona-Mahony equation is a non-linear partial differential equation arising in the study of waves, oceanography, Plasma physics, and shallow water theory. In the present work, we looked at the approximate solution of non-linear Benjamin-Bona-Mahony (BBM) problem by the Optimal Homotopy Asymptotic Method with Daftardar Jeffery Polynomials (OHAM-DJ). The BBM result is compared to analytic evaluation, the Homotopy Perturbation Technique (HPM), the Adomian Decomposition Method (ADM), and the Optimal Homotopy Asymptotic Method (OHAM-DJ). Figures of precise versus approximate solutions are also created, and it is established that OHAM-DJ's solution is substantially closer to the approximative than the precise. Additionally, the outcome demonstrates the effectiveness, simplicity, ease of use, and explicitness of OAM-DJ and provides a good means of controlling the convergence of approximations.