Analysis of fractional viewpoints on the Jaulent–Miodek and Whitham–Broer–Kaup coupled equations

In this work, we use the Caputo fractional calculus to methodically examine the Coupled Jaulent–Miodek (CJM) fractional equation and the fractional Whitham–Broer–Kaup (WBK) system. The Sumudu residual power series approach and the Sumudu iteration transform method are used to analyze the nonlinear fractional differential equation systems, providing a comprehensive analytical analysis. The Sumudu iteration transform approach is used to achieve the fractional WBK system’s dynamics, as well as the Sumudu power series residual approach is utilized to investigate the CJM equation’s behavior for fractions. We thoroughly examine their interactions using known solutions, using both symbolic calculations and numerical simulations. This leads to the identification of new solutions and the clarification of the way in which certain systems of fractions behave in terms of the operator of Caputo. The outcomes demonstrate the efficacy of the strategies used to decipher the intricate dynamics of fractional nonlinear systems by demonstrating a strong convergence agreement between analytical and numerical solutions.