This paper investigates the analytical and semi‐analytical solutions of the time‐fractional Cahn–Allen equation, which describes the structure of dynamic for phase separation in Fe‐Cr‐X (X = Mo, Cu) ternary alloys. We apply a modified auxiliary equation method and the Adomian decomposition method to get distinct solutions to our suggested model. These solutions describe the dynamic of the phase separation in iron alloys and use in solidification and nucleation problems. The applications of this method arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. We apply a conformable fractional derivative to this fractional model to convert it into a nonlinear partial differential equation with integer order. We obtain many analytical wave solutions and also apply a semi‐analytical scheme to calculate the absolute value of error. All solutions are verified by using Mathematica software.

中文翻译：

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时间分数Cahn–Allen方程的解析和半解析解

本文研究了时间分数Cahn-Allen方程的解析和半解析解，该方程描述了Fe-Cr-X（X = Mo，Cu）三元合金中相分离的动力学结构。我们应用了改进的辅助方程方法和Adomian分解方法，以得到建议模型的不同解。这些解决方案描述了铁合金中相分离的动力学过程，并用于固化和成核问题。这种方法的应用出现在许多领域，例如等离子体物理学，量子力学，数学生物学和流体动力学。我们对这个分数模型应用了一个适度的分数导数，将其转换为具有整数阶的非线性偏微分方程。我们获得了许多解析波解，并且还应用了半解析方案来计算误差的绝对值。所有解决方案均通过使用Mathematica软件进行了验证。