In this paper, we propose a novel reduced differential transform method (RDTM) to compute analytical and semianalytical approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions. The performance of the proposed method was analyzed and compared with a convergent series solution form with easily computable coefficients. The behavior of approximated series solutions at different values of fractional order and its modeling in 2-dimensional and 3-dimensional spaces are compared with exact solutions using MATLAB graphical method analysis. Moreover, the physical and geometrical interpretations of the computed graphs are given in detail within 2- and 3-dimensional spaces. Accordingly, the obtained approximate solutions of fractional order Airy’s ordinary differential equations and fractional order Airy’s and Airy’s type partial differential equations subjected to certain initial conditions exactly fit with exact solutions. Hence, the proposed method reveals reliability, effectiveness, efficiency, and strengthening of computed mathematical results in order to easily solve fractional order Airy’s type differential equations.

中文翻译：

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使用 RDTM 的模型的分数阶 Airy 型微分方程

在本文中，我们提出了一种新的简化微分变换方法（RDTM）来计算分数阶艾里常微分方程和分数阶艾里和艾里型偏微分方程在某些初始条件下的解析和半解析近似解。对所提出方法的性能进行了分析，并与具有易于计算系数的收敛级数解形式进行了比较。不同分数阶值的近似级数解的行为并使用MATLAB图形方法分析将其在2维和3维空间中的建模与精确解进行比较。此外，在 2 维和 3 维空间内详细给出了计算图的物理和几何解释。因此，得到的分数阶艾里常微分方程和分数阶艾里和艾里型偏微分方程在一定初始条件下的近似解与精确解完全吻合。因此，所提出的方法揭示了计算数学结果的可靠性、有效性、效率和加强，以便轻松求解分数阶艾里型微分方程。