Analysis of the seventh-order Caputo fractional KdV equation: applications to the Sawada–Kotera–Ito and Lax equations

Abstract

In this study, we investigate the seventh-order nonlinear Caputo time-fractional KdV equation. The suggested model’s solutions, which have a series form, are obtained using the hybrid ZZ-transform under the aforementioned fractional operator. The proposed approach combines the homotopy perturbation method (HPM) and the ZZ-transform. We consider two specific examples with suitable initial conditions and find the series solution to test their applicability. To demonstrate the utility of the presented technique, we explore its applications to the fractional Sawada–Kotera–Ito problem and the Lax equation. We observe the impact of a few fractional orders on the wave solution evolution for the problems under consideration. We provide the efficiency and reliability of the ZZHPM by calculating the absolute error between the series solution and the exact solution of both the Sawada–Kotera–Ito and Lax equations. The convergence and uniqueness of the solution are portrayed via fixed-point theory.