Advanced semi-analytical techniques for fractional shallow water wave models through the analogical structure of generalized $$\phi$$-Caputo derivative operators

Abstract

This paper presents advanced semi-analytical solution techniques for fractional shallow water wave systems governed by the Whitham–Broer–Kaup equations using the generalized $$\phi$$-Caputo fractional derivative. By exploiting the analogical structure of the $$\phi$$-Caputo operator, two extended solution procedures are developed, namely the $$\phi$$-Caputo residual power series method ($$\phi$$-CRPSM) and the $$\phi$$-Caputo new iterative method ($$\phi$$-CNIM). These formulations generalize classical RPSM and NIM approaches to a wider class of nonlocal fractional operators, providing enhanced flexibility for modeling memory-dependent nonlinear dispersive wave dynamics. The proposed methods generate rapidly convergent fractional power-series solutions and can be systematically implemented for nonlinear fractional systems without linearization or discretization. Applications to fractional Whitham–Broer–Kaup models demonstrate that the obtained solutions exhibit high accuracy, strong convergence behavior, and close agreement with exact solutions in the integer-order limit. Numerical illustrations confirm the effectiveness, stability, and computational efficiency of the proposed approaches in capturing the nonlinear, dispersive, and memory-influenced characteristics of shallow-water wave propagation. These results establish the proposed $$\phi$$-CRPSM and $$\phi$$-CNIM techniques as efficient and robust analytical–numerical tools for a broad class of nonlinear fractional partial differential equations.