Exact and approximate solutions of the fractional dynamical system via transform techniques

This study investigates the Jaulent–Miodek system of partial differential equations (PDEs) using two powerful analytical techniques: the iteration transform method (ITM) and the residual power series transform method (RPSTM). These methods are employed to derive approximate and exact solutions for the system under the framework of the fractional Caputo operator. The fractional formulation provides a more generalized and flexible approach to modeling complex physical phenomena, particularly in nonlinear wave propagation and fluid dynamics. The effectiveness of the proposed methods is demonstrated through numerical examples, where solutions are analyzed for different fractional orders. Comparative analysis highlights the accuracy and efficiency of ITM and RPSTM in handling fractional PDEs. The results confirm that these techniques offer a robust and reliable means for solving nonlinear fractional models, paving the way for further research into their applications in mathematical physics and engineering.