Dynamical and computational analysis of fractional Whitham-Broer-Kaup equations arising in shallow-water waves

In this work, we extend the modified homotopy analysis transform method (MHATM) for studying the three different coupled time-fractional physical problems. The first one is coupled time-fractional Whitham-Broer-Kaup (W-B-K) equations, and others are coupled modified Boussinesq equations and coupled approximate long wave equations as the special cases of W-B-K equations. The fractional W-B-K model is a coupled structure that describes the nonlinear evolution of shallow-water waves. The novelty of the proposed algorithm, the fractional derivative, is taken in the Caputo-Fabrizio (CF) sense, which consists of an exponential form of the non-singular kernel. With the aid of Banach's fixed point theory, the uniqueness and convergence analysis for the coupled W-B-K equations is presented through the theorems. With the help of Picard's stable approach, the stability analysis of the proposed technique is shown. We study the comparison for the solutions of MHATM for CF time-fractional derivative with the solutions derived with the aid of other techniques. The main advantage of the MHATM with the assistance of CF derivative is that, it offers solutions to problems in a rapidly convergent series leading to ideal solutions. The accuracy and efficiency of the present method have been shown through different graphical as well as tabulated analyses. However, the results indicate that MHATM with CF fractional derivative is a good organization and applicable to solve highly nonlinear various fractional physical problems like W-B-K equations.