Hussain Akhtar, Ibrahim Tarek F, Birkea F M Osman, Alotaibi Abeer M, Al-Sinan Bushra R, Mukalazi Herbert
Abdus Salam School of Mathematical Sciences, Government College University, 68-B New Muslim Town, Lahore, 54600, Pakistan.
Department of Mathematics, Faculty of Sciences and Arts (Mahayel), King Khalid University, Abha, Saudi Arabia.
Sci Rep. 2024 Jun 7;14(1):13100. doi: 10.1038/s41598-024-62961-9.
Despite the historical position of the F-expansion method as a method for acquiring exact solutions to nonlinear partial differential equations (PDEs), this study highlights its superiority over alternative auxiliary equation methods. The efficacy of this method is demonstrated through its application to solve the convective-diffusive Cahn-Hilliard (cdCH) equation, describing the dynamic of the separation phase for ternary iron alloys (Fe-Cr-Mo) and (Fe-X-Cu). Significantly, this research introduces an extensive collection of exact solutions by the auxiliary equation, comprising fifty-two distinct types. Six of these are associated with Weierstrass-elliptic function solutions, while the remaining solutions are expressed in Jacobi-elliptic functions. I think it is important to emphasize that, exercising caution regarding the statement of the term 'new,' the solutions presented in this context are not entirely unprecedented. The paper examines numerous examples to substantiate this perspective. Furthermore, the study broadens its scope to include soliton-like and trigonometric-function solutions as special cases. This underscores that the antecedently obtained outcomes through the recently specific cases encompassed within the more comprehensive scope of the present findings.
尽管F展开方法在历史上作为一种获取非线性偏微分方程（PDEs）精确解的方法，但本研究突出了它相对于其他辅助方程方法的优越性。通过将该方法应用于求解对流扩散Cahn-Hilliard（cdCH）方程，证明了其有效性，该方程描述了三元铁合金（Fe-Cr-Mo）和（Fe-X-Cu）的分离相动力学。值得注意的是，本研究通过辅助方程引入了大量精确解，包括五十二种不同类型。其中六种与魏尔斯特拉斯椭圆函数解相关，其余解则用雅可比椭圆函数表示。我认为必须强调的是，在使用“新”这个术语时要谨慎，这里给出的解并非完全前所未有的。本文研究了大量例子来证实这一观点。此外，该研究将范围扩大到包括孤子状和三角函数解等特殊情况。这强调了先前通过最近的特定情况获得的结果包含在本研究更全面范围内。
