The fractional order generalization of Shannon entropy proposed by Ubriaco has been studied for discrete distributions. In the current paper, we conduct a detailed study of the continuous analogue of this entropy termed as fractional differential entropy and find some interesting properties which makes it stand out among the existing entropies in literature. The studied entropy measure is evaluated analytically and numerically for some well-known continuous distributions, which will be quite useful in reliability analysis works and other statistical studies of complex systems. Further, it has been used to model the one-dimensional vertical velocity profile of turbulent flows in wide open channels. A one-parametric spatial distribution function is utilized for better estimation of the velocity distribution. The validity of the model has been established using experimental and field data through regression analysis. A comparative study is also presented to show the superiority of the proposed model over the existing entropy-based models.

翻译：Ubriaco提出的香农熵分数阶推广已在离散分布中得到研究。本文对该熵的连续形式（称为分数阶微分熵）进行了详细研究，发现其具有若干独特性质，使其在现有熵度量中脱颖而出。我们通过解析和数值方法评估了该熵度量在若干经典连续分布中的表现，这对于可靠性分析及复杂系统的统计研究具有重要价值。进一步地，我们将其应用于宽明渠湍流一维垂向流速剖面的建模。采用单参数空间分布函数以更精确地估计流速分布。通过回归分析，利用实验和现场数据验证了模型的有效性。比较研究表明，所提模型优于现有的基于熵的模型。