Fractional analysis of nonlinear dynamics in Korteweg-de Vries-Burgers’ and modified Korteweg-de Vries equations

This paper investigates the fractional-order KdV-Burgers’ and modified KdV (mKdV) equations using two advanced mathematical techniques: the iterative transform method (ITM) and the residual power series transform method (RPSTM). These methods are employed to solve nonlinear fractional-order differential equations, with the Caputo operator providing the framework for fractional differentiation. The study highlights the effectiveness of ITM and RPSTM in obtaining approximate analytical solutions for these complex equations, which play a significant role in modeling wave propagation, fluid dynamics, and nonlinear systems. The proposed methods demonstrate robust convergence, accuracy, and computational efficiency in addressing the complexities of fractional-order systems. Through numerical examples and graphical representations, the paper provides insight into the behavior of solutions for different fractional orders, offering valuable contributions to the fields of mathematical physics and applied mathematics.