In this study, we derive a new fractional Gardner equation with the conformable fractional derivative for the first time. Using the extended F-expansion method, different traveling wave solutions expressed in the form of the hyperbolic solutions such as sech, tanh, csch, coth, sinh and their combination, and the triangular solutions like csc, sec, tan, cot and their combination are obtained. Finally, the solutions are illustrated through the 3D plots. The results strongly prove that the proposed approach is effective and can help us understand the nonlinear problems arising in physics better.

中文翻译：

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扩展 F 展开法在求解具有一致分数导数的分数阶 Gardner 方程中的应用

在这项研究中，我们首次推导出了一个具有一致分数导数的新分数加德纳方程。使用扩展的F-展开法，不同的行波解以sech、tanh、csch、coth、sinh及其组合等双曲解形式表示，以及csc、sec、tan、cot及其组合等三角解形式表示获得。最后，通过 3D 图说明了解决方案。结果有力地证明了所提出的方法是有效的，可以帮助我们更好地理解物理学中出现的非线性问题。