Hybrid expansion methods for fractional non-linear mathematical systems with Erdelyi-Kober derivative operators in theory of tsunami wave modeling

This work develops two hybrid semi-analytical methods, namely the expansion new iterative method (ENIM) and the expansion homotopy perturbation method (EHPM), for solving the fractional Whitham-Broer-Kaup equations (WBKEs) involving the Erdélyi-Kober (EK) fractional derivative, since EK derivative is adopted due to its ability to incorporate scaling properties and nonlocal memory effects in a more flexible framework. The study focuses on the mathematical behavior of a fractional extension of the WBK system related to shallow water wave modeling. Several analytical properties of EK operators and their action on fractional power series expansions are established to support the proposed frameworks. By combining EK fractional integration with nonlinear operator decomposition and power series representations, ENIM and EHPM provide approximate solutions to nonlinear fractional partial differential equations. The methods are applied to the fractional WBK system, and numerical results demonstrate good agreement with the classical solution in the benchmark case ([Formula: see text]). For fractional cases ([Formula: see text]), the results are approximate and model-dependent. The comparison indicates that EHPM yields smaller absolute errors than ENIM within the tested parameter ranges. The influence of the fractional order α on the solution behavior is also illustrated, showing a transition between diffusive and classical wave patterns. These findings highlight the effectiveness of the proposed methods in terms of numerical accuracy within the considered framework.