Two highly efficient methods for soliton solutions of the modified Fornberg–Whitham equation

In this study, exact solutions of the modified Fornberg-Whitham equation, which is popular for modeling wave breaking, have been studied using the Kudryashov method, while its numerical solutions have been studied by finite element method. To achieve these aims, general form of the Kudryashov method and septic B-spline basis functions have been employed as a polynomial approximation to the solution respectively. The error norms \( L_{2}\) $L_2$ and \(L_{\infty }\) $L_{\infty }$ are illustrated for the adequacy and efficiency of the proposed method for the model problem. Von-Neumann stability analysis has been implemented which ensures that the schemes are unconditionally stable. The numerical results are illustrated, and comparisons are made in tabular form to demonstrate the effectiveness of the scheme studied in this article. Additionally, graphics are presented to compare the exact and numerical results to show the behavior of the solutions. The graphical comparisons guarantee the power of numerical scheme presented for the nonlinear problem, it also confirms that the exact solution process is efficient and handy to handle highly nonlinear problems.