Approximate Numerical solutions for the nonlinear dispersive shallow water waves as the Fornberg–Whitham model equations

Abstract The nonlinear partial differential equations having travelling or solitary wave solutions is numerically challenging, in which one of the important type is the Fornberg–Whitham model equation. This article aims to solve the Fornberg–Whitham type equations numerically via the variational iteration algorithm-I (MVIA-I). The MVIA-I gives approximate and exact solutions with easily computable terms to linear and nonlinear PDEs without the linearization or discretization, small perturbation and Adomian polynomials. To assess the precision, reliability and compactness of the recommended algorithm, we have compared the obtained results with the traditional variational iteration method (VIM), homotopy analysis method, reproducing kernel Hilbert space method and Adomian’s decomposition method which reveals that the MVIA-I is computationally attractive, exceptionally productive and is more reliable than the others techniques used in the literature.