Novel analysis of fractional-order nonlinear generalized coupled Hirota-Satsuma and Korteweg-de Vries (KdV) systems

This study presents an analytical investigation of the fractional-order nonlinear third-order Korteweg-de Vries (KdV) equation and the generalized coupled Hirota-Satsuma KdV (HS-KdV) system using two powerful semi-analytical approaches: the Conformable Residual Power Series Method (CRPSM) and the Conformable New Iterative Method (CNIM). By employing the conformable fractional operator, the inherent complexities of fractional derivatives are simplified while maintaining consistency with classical calculus properties. Through suitable similarity transformations, the governing nonlinear partial differential equations are reduced to conformable fractional ordinary differential equations, allowing the derivation of analytical and approximate series solutions. The CRPSM constructs convergent residual-based power series expansions with controllable error terms, while the CNIM iteratively refines approximate solutions to achieve higher accuracy and faster convergence. A comparative analysis between the two conformable techniques and Variational Iteration Transform Method (VITM) demonstrates their strong agreement and computational efficiency. Graphical and numerical results highlight the significant influence of the fractional order and nonlinear parameters on the amplitude, phase, and propagation characteristics of solitonic and periodic wave structures. The outcomes reveal that the proposed conformable analytical framework is robust, accurate, and versatile for solving a wide class of nonlinear fractional evolution equations, particularly those modeling dispersive and interacting wave phenomena in mathematical physics, fluid mechanics, and plasma dynamics.