Nazari-Golshan A
Physics Department, Shahed University, Tehran, Iran.
Sci Rep. 2024 Aug 12;14(1):18642. doi: 10.1038/s41598-024-66298-1.
Complex and nonlinear fractal equations are ubiquitous in natural phenomena. This research employs the fractal Euler-Lagrange and semi-inverse methods to derive the nonlinear space-time fractal Fornberg-Whitham equation. This derivation provides an in-depth comprehension of traveling wave propagation. Consequently, the nonlinear space-time fractal Fornberg-Whitham equation is pivotal in elucidating fundamental phenomena across applied sciences. A novel analytical technique, the generalized Kudryashov method, is presented to address the space-time fractal Fornberg-Whitham equation. This method combines the fractional complex approach with the modified Kudryashov method to enhance its effectiveness. We derive an analytical solution for the space-time fractal Fornberg-Whitham equation to elucidate how various parameters influence the propagation of new traveling wave solutions. Furthermore, Figures 1 through 6 analyze the impact of parameters , , and on these new traveling wave solutions. Our results show that the solitary wave solutions remain intact for both case 1 and case 2, regardless of the time fractional orders . At the end, the manuscript discusses the implications of these findings for understanding complex wave phenomena, paving the way for further exploration and applications in wave propagation studies.
复杂的非线性分形方程在自然现象中无处不在。本研究采用分形欧拉 - 拉格朗日方法和半逆方法来推导非线性时空分形福恩伯格 - 惠特姆方程。这一推导为行波传播提供了深入理解。因此，非线性时空分形福恩伯格 - 惠特姆方程对于阐明应用科学中的基本现象至关重要。提出了一种新颖的分析技术——广义库德里亚绍夫方法，以求解时空分形福恩伯格 - 惠特姆方程。该方法将分数复变方法与改进的库德里亚绍夫方法相结合，以提高其有效性。我们推导了时空分形福恩伯格 - 惠特姆方程的解析解，以阐明各种参数如何影响新行波解的传播。此外，图1至图6分析了参数 、 和 对这些新行波解的影响。我们的结果表明，无论时间分数阶 如何，在情况1和情况2中孤立波解都保持不变。最后，本文讨论了这些发现对理解复杂波现象的意义，为波传播研究的进一步探索和应用铺平了道路。
