In this paper, we consider a version of the fractional Clifford–Fourier transform (FrCFT) and study its several properties and applications to partial differential equations in Clifford analysis. First, we give the definition of the FrCFT and its inverse transform in the form of integral. Then, we discuss the relationship between the FrCFT and the Clifford–Fourier transform (CFT) and give some properties of the FrCFT, including Plancherel identity, differential properties, etc. Especially we give a new form of differential formula. Finally, we give an application of these results to a partial differential equation.

中文翻译：

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分数克利福德-傅立叶变换及其应用

在本文中，我们考虑分数Clifford-Fourier变换（FrCFT）的一个版本，并研究其几个性质以及在Clifford分析中将其应用于偏微分方程。首先，我们以积分形式给出FrCFT的定义及其逆变换。然后，我们讨论了FrCFT与Clifford-Fourier变换（CFT）之间的关系，并给出了FrCFT的一些属性，包括Plancherel身份，微分性质等。特别是，我们给出了一种新的微分公式形式。最后，我们将这些结果应用于偏微分方程。