Topological analysis with some techniques for solving a fractional tsunami shallow water mathematical model-based on Hausdorff–locally compact structures and their analytical implications

This paper provides a comprehensive analytical study of the fractional Whitham–Broer–Kaup equations (WBKEs), formulated using theAtangana–Baleanu–Caputo (ABC) fractional derivative to model nonlinearoscillatory behavior and the dynamics of tsunami shallow water waves. Within the framework of Banach spaces endowed with the compact–open topology, we rigorously establish the existence, uniqueness, and Hyers–Ulam stability of solutions by applying fixed–point theorems. To construct approximate analytical solutions, we develop a hybrid approach that integrates fractional power series expansions (FPSEs) with the new iterative method (NIM), referred to as the expansion new iterative method (ENIM). This methodology efficiently handles nonlinearities and fractional–order effects, yielding rapidly convergent series solutions that remain consistent with exact analytical results. Overall, the study demonstrates the robustness of the fractional WBKE model under small perturbations and confirms its effectiveness in capturing the complex dynamics of tsunami shallow water wave propagation.