Younas Tayyaba, Ahmad Jamshad
Department of Mathematics, Faculty of Science, University of Gujrat, Gujrat, Pakistan.
PLoS One. 2025 May 21;20(5):e0320190. doi: 10.1371/journal.pone.0320190. eCollection 2025.
The Boussinesq equation is essential for studying the behavior of shallow water waves, surface waves in oceans and rivers, and the propagation of long waves in nonlinear systems. Its fractional form allows for a more accurate representation of wave dynamics by incorporating the effects of nonlocal interactions and memory. In this paper, we focus on obtaining exact traveling wave solutions for the space-time fractional Boussinesq equation using two well-established methods: the modified Sardar sub-equation method and the new extended direct algebraic method, both implemented with Atangana's beta derivative. By applying these methods, we derive a variety of soliton solutions, including kink, anti-kink, periodic, dark, bright, and singular solitary waves. These solutions are presented in different mathematical forms, such as rational, hyperbolic, trigonometric, and exponential functions. This study not only provides new solutions but also enhances the understanding of wave propagation in fractional models, demonstrating the efficiency and applicability of the chosen methods. A comparative analysis of the methods and results is presented, along with an examination of the impact of fractional derivatives by adjusting their values. The study also includes 2D and 3D plots that illustrate the temporal behavior of the solutions. This study demonstrates that the methods employed are applicable to other nonlinear models in mathematical physics. A detailed analysis of the model's behavior is conducted, focusing on bifurcation, chaos, and stability. Phase portrait analysis at critical points reveals shifts in the system's dynamics, and introducing an external periodic force generates chaotic patterns. The solutions provided offer new insights into shallow water wave models, presenting effective tools for in-depth investigation of wave dynamics. All solutions are verified through MATHEMATICA and MATLAB simulations, ensuring their accuracy and reliability.
布辛涅斯克方程对于研究浅水波、海洋和河流中的表面波以及非线性系统中长波的传播行为至关重要。其分数阶形式通过纳入非局部相互作用和记忆效应，能够更准确地描述波动动力学。在本文中，我们专注于使用两种成熟的方法来获得时空分数阶布辛涅斯克方程的精确行波解：改进的萨达尔子方程方法和新的扩展直接代数方法，这两种方法均采用了阿坦加纳的贝塔导数。通过应用这些方法，我们推导出了多种孤子解，包括扭结、反扭结、周期、暗、亮和奇异孤波。这些解以不同的数学形式呈现，如实数、双曲、三角和指数函数。本研究不仅提供了新的解，还增进了对分数阶模型中波传播的理解，证明了所选方法的有效性和适用性。对方法和结果进行了比较分析，并通过调整分数阶导数的值来研究其影响。研究还包括二维和三维图，展示了解的时间行为。本研究表明所采用的方法适用于数学物理中的其他非线性模型。对模型行为进行了详细分析，重点关注分岔、混沌和稳定性。临界点处的相图分析揭示了系统动力学的变化，引入外部周期力会产生混沌模式。所提供的解为浅水波模型提供了新的见解，为深入研究波动力学提供了有效的工具。所有解均通过MATHEMATICA和MATLAB模拟进行了验证，确保了其准确性和可靠性。
