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

Faten H. Damag; Fozaiyah Alhubairah; Maryam F. Alshammari; Khaled M. Saad; Adem Kiliçman; Amin Saif; Najah Alshammari; Faten H. Damag; Fozaiyah Alhubairah; Maryam F. Alshammari; Khaled M. Saad; Adem Kiliçman; Amin Saif; Najah Alshammari

doi:10.3934/math.2026159

AIMS Mathematics

2026, Volume 11, Issue 2: 3957-3985. doi: 10.3934/math.2026159

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

1.
Department of Mathematics, Faculty of Sciences, Ha'il University, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, College of Sciences and Arts, Najran University, Najran, Saudi Arabia
3.
Department of Mathematics, Faculty of Applied Sciences, Taiz University, Taiz 6803, Yemen
4.
School of Mathematical Sciences, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA, Shah Alam 40450, Selangor, Malaysia

Received: 12 December 2025 Revised: 21 January 2026 Accepted: 03 February 2026 Published: 09 February 2026
MSC : 35R11, 35B35, 44A10, 76B15

This paper provides a comprehensive analytical study of the fractional Whitham–Broer–Kaup equations (WBKEs), formulated using the Atangana–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.
- fractional WBKEs,
- power series,
- ABC operator,
- stability
Citation: Faten H. Damag, Fozaiyah Alhubairah, Maryam F. Alshammari, Khaled M. Saad, Adem Kiliçman, Amin Saif, Najah Alshammari. Topological analysis with some techniques for solving a fractional tsunami shallow water mathematical model-based on Hausdorff–locally compact structures and their analytical implications[J]. AIMS Mathematics, 2026, 11(2): 3957-3985. doi: 10.3934/math.2026159

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Abstract

This paper provides a comprehensive analytical study of the fractional Whitham–Broer–Kaup equations (WBKEs), formulated using the Atangana–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.

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