Traveling wave solutions of generalized seventh-order time-fractional KdV models through He-Laplace algorithm

Abstract

Non-linear evolution equations play a prominent role in describing a wide range of phenomena in optical fibers, fluid dynamics, electromagnetic radiation, plasma and solid state physics. An important category of non-linear evolution models that characterizes shallow wave phenomena are the Korteweg-de Vries (KdV) models. In this regard, time-fractional Korteweg-de Vries models of seventh order are the main focus of this research. A general KdV seventh-order equation is considered with different coefficients to form Lax, Kaup–Kuperschimdt and Sawada–Kotera–Ito KdV models. An efficient semi-analytical algorithm named as He-Laplace (HLM) is applied for the solution of these models. In this algorithm, Laplace transform is hybrid with homotopy perturbation method (HPM). This study provides important results as non-linear evolution seventh-order models in fractional sense have not been captured through HLM in current literature. Absolute errors are computed and compared with already existing results to confirm the superiority of proposed algorithm over other existing techniques. Numerical and graphical investigations are conducted to evaluate the approximate series form solutions. The dynamic behavior of fractional parameter is observed by calculating residual errors and plotting two dimensional diagrams throughout the fractional-domain. Analysis confirms that the proposed methodology provides an effective and convenient way for solving fractional KdV models.