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Novel Analytical Methods for and Qualitative Analysis of the Generalized Water Wave Equation

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  • Haitham Qawaqneh

    (Department of Mathematics, Faculty of Science and Information Technology, Al-Zaytoonah University of Jordan, Amman 11733, Jordan)

  • Abdulaziz S. Al Naim

    (Accounting Department, Business School, King Faisal University, Al-Ahsa 31982, Saudi Arabia)

  • Abdulrahman Alomair

    (Accounting Department, Business School, King Faisal University, Al-Ahsa 31982, Saudi Arabia)

Abstract

For a significant fluid model and the truncated M-fractional (1 + 1)-dimensional nonlinear generalized water wave equation, distinct types of truncated M-fractional wave solitons are obtained. Ocean waves, tidal waves, weather simulations, river and irrigation flows, tsunami predictions, and more are all explained by this model. We use the improved ( G ′ / G ) expansion technique and a modified extended direct algebraic technique to obtain these solutions. Results for trigonometry, hyperbolic, and rational functions are obtained. The impact of the fractional-order derivative is also covered. We use Mathematica software to verify our findings. Furthermore, we use contour graphs in two and three dimensions to illustrate some wave solitons that are obtained. The results obtained have applications in ocean engineering, fluid dynamics, and other fields. The stability analysis of the considered equation is also performed. Moreover, the stationary solutions of the concerning equation are studied through modulation instability. Furthermore, the used methods are useful for other nonlinear fractional partial differential equations in different areas of applied science and engineering.

Suggested Citation

  • Haitham Qawaqneh & Abdulaziz S. Al Naim & Abdulrahman Alomair, 2025. "Novel Analytical Methods for and Qualitative Analysis of the Generalized Water Wave Equation," Mathematics, MDPI, vol. 13(14), pages 1-26, July.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:14:p:2280-:d:1702288
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    References listed on IDEAS

    as
    1. Abdulrahman Alomair & Abdulaziz S. Al Naim & Ahmet Bekir, 2024. "Exploration of Soliton Solutions to the Special Korteweg–De Vries Equation with a Stability Analysis and Modulation Instability," Mathematics, MDPI, vol. 13(1), pages 1-17, December.
    2. Tariq, Kalim U. & Wazwaz, Abdul-Majid & Javed, Rizwan, 2023. "Construction of different wave structures, stability analysis and modulation instability of the coupled nonlinear Drinfel’d–Sokolov–Wilson model," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    3. Sahoo, S. & Saha Ray, S., 2016. "Solitary wave solutions for time fractional third order modified KdV equation using two reliable techniques (G′/G)-expansion method and improved (G′/G)-expansion method," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 448(C), pages 265-282.
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