On the analytical and numerical solutions of the Benjamin–Bona–Mahony equation

In this article, we employed the powerful sine-Gordon expansion method in obtaining analytical solutions of the Benjamin–Bona–Mahony equation. We obtain some new solutions with the hyperbolic function structures. Benjamin–Bona–Mahony equation has a wide range of applications in modelling long surface gravity waves of small amplitude. We also plot the 2- and 3-dimensional graphics of all analytical solutions obtained in this paper. On the other hand, we analyze the finite difference method and operators, we obtain discretize equation using the finite difference operators. We consider one of the analytical solutions to the Benjamin–Bona–Mahony equation with the new initial condition. We observe that finite difference method is stable when Fourier–Von Neumann technique is used. We also analyze the accuracy of the finite difference method with terms of the errors \(L_{2}\) and \(L_{\infty }\). We use the finite difference method in obtaining the numerical solutions of the Benjamin–Bona–Mahony equation. We compare the numerical results and the exact solution that are obtained in this paper, we support this comparison with the graphic plot. We perform all the computations and graphics plot in this study with the help of Wolfram Mathematica 9.