In this paper, we introduce the two-sided fractional quaternion Fourier transform (FrQFT) and give some properties of it. The main results of this paper are divided into three parts. Firstly we give a definition of the FrQFT. Secondly based on properties of the two-sided QFT, we study the relationship between the two-sided QFT and the two-sided FrQFT, and give some differential properties of the two-sided FrQFT and the Parseval identity. Finally, we give an example to illustrate the application of the two-sided FrQFT and its inverse transform in solving partial differential equations.

中文翻译：

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双边分数四元数傅里叶变换及其应用

在本文中，我们介绍了双边分数四元数傅里叶变换（FrQFT）并给出了它的一些性质。本文的主要结果分为三个部分。首先我们给出FrQFT的定义。其次，基于双面QFT的性质，研究了双面QFT与双面FrQFT的关系，并给出了双面FrQFT和Parseval恒等式的一些微分性质。最后，我们举例说明了双边FrQFT及其逆变换在求解偏微分方程中的应用。