Application of the Natural Transform Decomposition Method to Fractional-Order KdV Equations

Abstract

In this paper, the analytical solutions of fractional-order Korteweg–de Vries (KdV) equations are investigated by applying the Natural Transform Decomposition Method (NTDM). The proposed method combines the Natural Transform with the Adomian decomposition technique to obtain approximate series solutions of linear and nonlinear dispersive fractional partial differential equations. Several illustrative examples of fractional KdV equations are considered and solved systematically using NTDM. The Caputo fractional derivative is employed to describe the fractional-order behavior of the models. The obtained solutions are expressed in rapidly convergent series forms and compared with the corresponding exact solutions in the integer-order case. The results demonstrate that the proposed method is accurate, efficient, and computationally simple for solving fractional KdV equations without requiring discretization, perturbation, or linearization. Furthermore, it is observed that the fractional-order solutions converge to the classical exact solutions as the fractional parameter approaches unity. The study confirms that NTDM is a reliable analytical technique for investigating nonlinear fractional dispersive equations arising in mathematical physics and engineering sciences.