Analysis of Nonlinear Coupled Jaulent-Miodek and Whitham-Broer-Kaup Equations Within Fractional Derivative

This paper investigates the nonlinear coupled Jaulent-Miodek (JM) and Whitham-Broer-Kaup (WBK) equations through the lens of fractional calculus, employing the Mohand Variational Iteration Method (MVIM) and q-Homotopy Mohand Transform Method (q-HMTM). These equations, pivotal in describing nonlinear wave phenomena and fluid dynamics, are studied in their fractional-order forms using the Caputo operator to extend traditional models. The proposed methods efficiently yield analytical and approximate solutions, showcasing their reliability and accuracy. The solutions derived are presented through numerical simulations and graphical depictions, revealing the intricate dependence of system behavior on fractional-order parameters. This sensitivity provides a mechanism for tuning physical phenomena modeled by JM and WBK equations, offering valuable insights into wave propagation, fluid dynamics, and other nonlinear coupled systems. The study establishes the efficacy of q-HMTM and MVIM in handling complex fractional-order systems, underlining their potential for broader applications in science and engineering. By bridging classical and fractional models, this work contributes to the ongoing development of advanced mathematical tools for analyzing nonlinear phenomena.