Numerical and analytical results of the 1D BBM equation and 2D coupled BBM-system by finite element method

Abstract

In this paper, the numerical solutions are proposed for the 1D Benjamin–Bona–Mahony (BBM) equation and 2D coupled BBM system by using Galerkin finite element technique. In this regard, the cubic B-splines and linear triangular elements are used, respectively. In 1D space, a proposed numerical scheme is implemented to a test problem including the motion of a single solitary solution. To verify practicality and robustness of our new procedure, the error norms [Formula: see text], [Formula: see text] and three constants [Formula: see text] and [Formula: see text] are evaluated. Stability analysis of the linearized technique indicates that it is unconditionally stable. Moreover, a tsunami wave in 2D space is used to investigate the efficiency of the considered method. Also, the improved [Formula: see text]-[Formula: see text] function technique (IThChT) and the combined [Formula: see text]-[Formula: see text] function technique (ITCT) are obtained in the mentioned BBM equation. The presented methods are seen to be robust, impressive and economical to employment as compared to the existing finite difference techniques and other earlier papers for discovering the numerical solutions for numerous types of linear and nonlinear PDEs.