Analytical investigation of the fractional nonlinear shallow-water model

In this research article, we introduce an investigation for analytical approximate solutions of the time-fractional nonlinear shallow water model. This model is described as a system of coupled partial differential equations that characterize the dynamics of water motion below a pressure surface in oceanic or sea environments, which is distinguished by the fact that the vertical dimension is smaller in magnitude than the typical horizontal dimension. The modified generalized Mittag-Leffler function method is an effective and innovative analytical technique to acquire convenient approximate solutions for this fractional order model. The methodology of proposed method to solve general fractional nonlinear partial differential equations is presented. Also, the convergence of this method and estimated error analysis for the projected solutions are proved. The approximate solutions gained by our method when \(\alpha =1\) are compared with the recognized exact solutions and outcomes of other techniques in the same conditions published in the literature, including the residual power series method, natural transform decomposition method, modified homotopy analysis transform method and new iterative method. Moreover, some two and three-dimensional graphs and tabulated data display a simulation of acquired results. Also, the influence of \(\alpha \) on the behavior of solutions is exhibited. The findings demonstrate the effectiveness and advantages of the suggested method, including not requiring any linearization or perturbation and transformations, easily computable components, implemented directly to the problems, satisfactory approximate solutions and a small absolute error.