In this study, the fractional reduced differential transform method (FRDTM) is employed to solve three-dimensional fourth-order time-fractional parabolic partial differential equations with variable coefficients. The fractional derivative used in this study is in the Caputo sense. A few important lemmas which are essential to solve the problems using the proposed method are proved. The novelty of this method is that it uses appropriate initial conditions and finds the solution to the problems without any discretization, linearization, perturbation, or any restrictive assumptions. Two numerical examples are considered in order to validate the efficiency and reliability of the method. Furthermore, the FRDTM solution when α = 1 is compared with other analytical methods available in the existing literature. Computational results are shown in tables and graphs. The obtained results revealed that the method is capable and simple to solve fractional partial differential equations. The software used for the calculations in this study is Mathematica 7.

中文翻译：

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三维四阶时间分数阶抛物型偏微分方程及其解析解

在这项研究中，采用分数缩减微分变换方法（FRDTM）来求解具有可变系数的三维四阶时间分数阶抛物型偏微分方程。在这项研究中使用的分数导数是Caputo的。提出了一些重要的引理，这些引理对于使用该方法解决问题至关重要。这种方法的新颖之处在于它使用适当的初始条件，并且无需任何离散，线性化，摄动或任何限制性假设即可找到问题的解决方案。考虑两个数值示例，以验证该方法的效率和可靠性。此外，当α为FRDTM时 将= 1与现有文献中可用的其他分析方法进行比较。计算结果显示在表格和图表中。所得结果表明，该方法能够求解分数阶偏微分方程。本研究中用于计算的软件是Mathematica 7。