Fractional-order wave dynamics: analytical solutions for coupled Burgers' and KdV systems using advanced transform techniques

This study explores advanced analytical methods for solving nonlinear fractional differential equations, focusing on the fractional coupled Burgers' equations and the fractional system of nonlinear Korteweg-de Vries (KdV) equations. The Residual Power Series Transform Method (RPSTM) and the New Iteration Transform Method (NITM) are employed to obtain approximate analytical solutions. These methods combine the advantages of power series expansion and integral transforms, ensuring efficient convergence without requiring restrictive assumptions or discretization. The fractional derivatives are considered in the Caputo sense to accurately describe the memory and hereditary properties inherent in physical processes. Comparative analysis between RPSTM and NITM demonstrates their effectiveness, with numerical simulations and graphical representations validating the accuracy and efficiency of the obtained solutions. The study highlights the significance of fractional-order parameters in nonlinear wave propagation and fluid dynamics, offering insights into real-world applications in physics and engineering.