Fractional modeling of nonlinear dispersive systems: on the comparative study of Whitham–Broer–Kaup equations using various derivatives

This paper introduces an analytical framework for solving the coupled fractional Whitham–Broer–Kaup (WBK) equations. This framework integrates the Sumudu Transform (ST) with the Decomposition Method (DM), employing fractional derivatives defined in the Caputo (C), Atangana–Baleanu Caputo (ABC), and Caputo–Fabrizio (CF) senses. The Sumudu Transform serves as a potent analytical instrument, streamlining the solution process and upholding both initial and boundary conditions. Our methodology’s precision and convergence are validated through the derivation of exact and approximate solutions for WBK equations. A thorough comparison with existing methods confirms the simplicity and physical validity of the SDM framework. This research advances fractional modeling and bridges a gap in Caputo (C), Atangana–Baleanu Caputo (ABC), and Caputo–Fabrizio (CF) fractional calculus by incorporating integral transform methods. It offers a new perspective for tackling complex nonlinear fractional partial differential equations (PDEs). The evidence presented in the tables and figures collectively substantiates that the proposed method approaches are highly precise, robust, and adaptable across diverse fractional orders, rendering them suitable for investigating intricate waves and nonlinear dispersive systems.