Approximately Solutions of the Generalized Regular Long Wave Equation Utilizing Four Different Operator Splitting Algorithms Combined with the Finite Element Method

The one-dimensional nonlinear generalized regular wave GRLW equation with physical boundary conditions will be taken into consideration in this article. This study employs operator splitting algorithms in together with the finite element based Galerkin method, which is a practical and highly adaptable method for the approximately solution of the equation. The study's goal is to generate more precise and appealing results. B-splines serve as the foundation for the application of Galerkin methods in this study. Four distinct numerical schemes are utilized for the numerical results obtained throughout the article: Lie-Trotter splitting techniques $L_{\Delta t}=P-M,$ $L_{\Delta t}=M-P$ and Strang splitting techniques $S_{\Delta t}=P-M-P,$ $S_{\Delta t}=M-P-M$ and it is evaluated which of these methods yields the most accurate results.  The work take into consideration the algorithm's output that yields the best results.  Using the suggested techniques, the study has been put into practice to solve the single solitary wave and the interaction of two solitary waves problems. Error norms have been calculated in this article, and the newly acquired results are compared with those discovered by some researchers in the literature to show the precision and appeal of the numerical techniques employed.  Furthermore, The current method is indicated to be unconditionally stable through the use of the Von-Neumann analysis. This study, which will be carried out after the literature review, will create a new and more effective solution method for the (GRLW) equation.