Comparative Analysis of Time-Fractional Coupled System of Shallow-Water Equations Including Caputo’s Fractional Derivative

Mathematical replication of nonlinear physical and abstract systems is a very dynamic progression for calculating the solution nature of fractional partial differential equations (FPDEs) corresponding to numerous applications in science and engineering. The goal of this work is to solve the time-fractional coupled system of shallow-water equations (SWEs). The SWEs are helpful in clarifying the mechanics of water movement in marine or oceanic applications. Furthermore, the previously indicated system is characterized by a thin fluid layer with a uniform density that preserves the hydrostatic equilibrium. The vertical dimension of shallow-water flows is much smaller than the normal horizontal dimension. In the present study, we derive the solutions for these fractional systems by using three novel and efficient techniques: the variational iteration method, the homotopy perturbation method, and the Adomian decomposition method with the Yang transformation. Fractional derivative is taken in the Caputo sense. The comparison of these solutions with the actual solutions and the outcomes then proved the ease of use, effectiveness, and high degree of precision of the used techniques. Tabular and graphical simulations are used to demonstrate the outcomes of the suggested approaches. The existing framework describes the behavior of various fractional orders quite well. The results demonstrate the effectiveness of the suggested approach. At the end, the applied procedures are the most widely used and convergent techniques for resolving nonlinear FPDEs.