Convergence Analysis and Numerical Solution of the BBM Equation using the Kamal-Adomian Decomposition Method

Abstract

The Benjamin-Bona-Mahony (BBM) equation is a nonlinear dispersive partial differential equation widely used to describe the propagation of long waves in fluid media and other physical systems. Due to the nonlinear nature of the equation, obtaining exact analytical solutions can be challenging. Consequently, semi-analytical techniques are often employed to obtain accurate approximations with reduced computational complexity. In this study, series solutions of the BBM equation are obtained using the Kamal-Adomian Decomposition Method (KADM). This approach combines the Kamal transform with the Adomian Decomposition Method, enabling the equation to be handled systematically by separating its linear and nonlinear components. The resulting solution is constructed iteratively as a convergent series without requiring linearization or discretization. To assess the efficiency of the method, several initial value problems are considered and the resulting solutions are compared with previously reported results in the literature. The method demonstrates rapid convergence, yielding highly accurate approximations with only a few terms of the series. Overall, the results demonstrate that KADM effectively captures the essential wave characteristics of the BBM equation, particularly the interaction between nonlinear steepening and dispersive effects. These findings indicate that the method is a reliable and efficient tool for solving nonlinear partial differential equations arising in applied mathematics and fluid dynamics.