A three-level average implicit finite difference scheme to solve equation obtained by coupling the Rosenau–KdV equation and the Rosenau–RLW equation

In the present work, a mathematical model to obtain the solution of the nonlinear wave by coupling the Rosenau–KdV equation and the Rosenau–RLW equation is proposed. The solution properties are also derived. A numerical tool is applied to the model by using a three-level average implicit finite difference technique. The fundamental conservative properties of the equation are preserved by the presented numerical scheme, and the existence and uniqueness of the numerical solution are proved. Moreover, the convergence and stability of the numerical solution are also shown. The new method give second-order accurate in time and space. Thus, the presented results can be constructed to demonstrate the viability of the new model.