With the help of Maple, the precise traveling wave solutions of three fractal-order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended ()-expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional-order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.

中文翻译：

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机械求解几个分数阶偏微分方程，并讨论分数阶的影响

借助Maple，利用函数展开法获得了与水波有关的三个分形阶模型方程的精确行波解，包括双曲解，三角解和有理解。从每个非线性色散波模型方程的解中选择一个孤立波解，并讨论分数阶变化对这些孤立波解的影响。结果表明，分数导数可以调制三个模型方程的孤立解的波形，局部周期性和结构。我们还指出了扩展（）-展开方法。在“说明和讨论”部分中，使用了更通用的辅助方程式来进一步强调规则，这些规则对于构造新的辅助方程式具有一定的参考价值。通过选择其他可解方程作为辅助方程，可以丰富分数阶非线性偏微分方程的解。