An Efficient Analysis of Fractional Derivatives for Solving the Nonlinear Time-Fractional Kuramoto–Sivashinsky Equation Using the Laplace Transform and Adomian Decomposition Method

This work presents a comparative analysis of the nonlinear time-fractional Kuramoto–Sivashinsky (KS) equation under three fractional derivative operators: Caputo, Caputo–Fabrizio, and Atangana–Baleanu in the Caputo sense. The objective is to examine how these operator definitions affect the qualitative behavior of the KS equation, particularly in terms of stability, damping, and wave propagation. For this purpose, the Laplace–Adomian decomposition method (L-ADM) is employed to construct semi-analytical series solutions expressed as rapidly convergent expansions. A unified framework is developed to apply L-ADM to all three fractional operators, supported by a rigorous convergence, existence, and uniqueness analysis, ensuring the stability and reliability of the obtained solutions. The comparison evaluates absolute error, convergence rate, and computation time to assess both accuracy and efficiency. Two numerical examples are presented for fractional orders α on the solution behavior. The outcomes are further compared with those of the new iterative transform method (NITM) and the q-homotopy analysis transform method (q-HATM). The results demonstrate distinct differences in solution behavior across the three fractional derivatives, emphasizing the importance of operator selection in fractional modeling.