On the Solutions of the Fractional-Order Sawada–Kotera–Ito Equation and Modeling Nonlinear Structures in Fluid Mediums

This study investigates the wave solutions of the time-fractional Sawada–Kotera–Ito equation (SKIE) that arise in shallow water and many other fluid mediums by utilizing some of the most flexible and high-precision methods. The SKIE is a nonlinear integrable partial differential equation (PDE) with significant applications in shallow water dynamics and fluid mechanics. However, the traditional numerical methods used for analyzing this equation are often plagued by difficulties in handling the fractional derivatives (FDs), which lead to finding other techniques to overcome these difficulties. To address this challenge, the Adomian decomposition (AD) transform method (ADTM) and homotopy perturbation transform method (HPTM) are employed to obtain exact and numerical solutions for the time-fractional SKIE. The ADTM involves decomposing the fractional equation into a series of polynomials and solving each component iteratively. The HPTM is a modified perturbation method that uses a continuous deformation of a known solution to the desired solution. The results show that both methods can produce accurate and stable solutions for the time-fractional SKIE. In addition, we compare the numerical solutions obtained from both methods and demonstrate the superiority of the HPTM in terms of efficiency and accuracy. The study provides valuable insights into the wave solutions of shallow water dynamics and nonlinear waves in plasma, and has important implications for the study of fractional partial differential equations (FPDEs). In conclusion, the method offers effective and efficient solutions for the time-fractional SKIE and demonstrates their usefulness in solving nonlinear integrable PDEs.