The Laplace Residual Power Series Method for Solving Fractional Fifth-Order Korteweg–de Vries Equations

This study explores a modern analytical approach for solving the fractional fifth-order Korteweg–de Vries (KdV) equations, which describe intricate wave phenomena influenced by nonlinearity, dispersion, and memory effects. Specifically, the Laplace residual power series method (LRPSM) is utilized to obtain accurate approximate analytical solutions for three fundamental fractional equations: the fractional Sawada–Kotera (SK) equation, the fractional Caudrey–Dodd–Gibbon (CDG) equation, and the fractional Kaup–Kuperschmidt (KK) equation. These equations represent special cases of the broader fractional fifth-order KdV equation. The novelty of this study lies in the application of LRPSM, which addresses the limitations of traditional methods by combining analytical precision with computational efficiency. The method successfully captures fractional dynamics, including soliton-like behaviors and memory effects, demonstrating its capability to model wave attenuation and smoothness influenced by fractional orders. The numerical results demonstrate that this method achieves minimal error margins, validating its robustness and precision in solving nonlinear fractional systems. Numerical examples validate the efficiency and robustness of this method, achieving high accuracy in solving nonlinear fractional systems. The results establish LRPSM as a versatile and reliable tool for solving fractional differential equations, paving the way for advancements in modern wave theory and applications across disciplines such as plasma physics, fluid mechanics, and nonlinear optics.OPEN ACCESS Received: 12/10/2024 Accepted: 21/01/2025 Published: 14/07/2025