Farooq Ayesha, Shafique Tooba, Abbas Muhammad, Birhanu Asnake, Hamed Y S
Department of Mathematics, University of Sargodha, Sargodha, 40100, Pakistan.
Department of Mathematics, College of Science, Hawassa University, Hawassa, Ethiopia.
Sci Rep. 2025 Jan 11;15(1):1683. doi: 10.1038/s41598-025-86177-7.
In applied research, fractional calculus plays an important role for comprehending a wide range of intricate physical phenomena. One of the Klein-Gordon model's peculiar case yields the Phi-four equation. Additionally, throughout the past few decades it has been utilized to explain the kink and anti-kink solitary waveform contacts that occur in biological systems and in the field of nuclear mechanics. In this current work, the key objective is to analyze the consequences of fractional variables on the soliton wave dynamic behavior in a nonlinear time-fractional Phi-four equation. Using the formulation of the conformable fractional derivative it illustrates some of the recovered solutions and analyze their dynamic behavior. The analytical solutions are drawn by using the extended direct algebraic and the Bernoulli Sub-ODE scheme. Various types of soliton solutions are proficiently expressed. Adjusting the specific values of fractional parameters allows to produce the periodic, kink, bell shape, anti-bell shape and W-shaped solitons. The impact of the conformable derivative on the precise solutions of the fractional Phi-four equation is demonstrated with a series of 2D, 3D and contour graphical representations.
在应用研究中，分数阶微积分对于理解各种复杂的物理现象起着重要作用。克莱因 - 戈登模型的一个特殊情况产生了φ⁴方程。此外，在过去几十年中，它已被用于解释生物系统和核力学领域中出现的扭结和反扭结孤立波形接触。在当前这项工作中，关键目标是分析分数变量对非线性时间分数阶φ⁴方程中孤子波动力学行为的影响。使用一致分数阶导数的公式，它展示了一些得到的解并分析了它们的动力学行为。通过使用扩展的直接代数方法和伯努利子常微分方程方案得出解析解。各种类型的孤子解都能被有效地表示出来。调整分数参数的特定值可以产生周期、扭结、钟形、反钟形和W形孤子。通过一系列二维、三维和等高线图形表示展示了一致导数对分数阶φ⁴方程精确解的影响。
