In this brief review, we present the results of the fractional differential approach in cosmology in the context of the exact models of cosmological accelerated expansion obtained by several authors to date. Most of these studies are devoted to the problem of introducing fractional derivatives or fractional integrals into the classical General Relativity (GR). There are several observational and theoretical motivations to investigate the modified or alternative theories of GR. Among other things, we cover General Relativity modified by a phenomenological approach dealing with fractional calculus. At the same time, a sufficiently large number of exact solutions of the cosmological equations modified by this approach were obtained. Some of these models may be especially relevant in the light of solving the problem of late accelerated expansion of the Universe. These studies are largely motivated by rapid progress in the field of observational cosmology that now allows, for the first time, precision tests of fundamental physics on the scale of the observable Universe. The purpose of this review is to provide a reference tool for researchers and students in cosmology and gravitational physics, as well as a self-contained, comprehensive and up-to-date introduction to the subject as a whole.

中文翻译：

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加速膨胀宇宙模型中的分数阶导数

在这篇简短的回顾中，我们在迄今为止几位作者获得的宇宙学加速膨胀的精确模型的背景下，展示了宇宙学中分数微分方法的结果。这些研究中的大多数致力于将分数导数或分数积分引入经典广义相对论 (GR) 的问题。有几个观察和理论动机来研究 GR 的修改或替代理论。除其他外，我们涵盖了通过处理分数微积分的现象学方法修改的广义相对论。同时，用这种方法修正的宇宙学方程得到了足够多的精确解。鉴于解决宇宙后期加速膨胀的问题，其中一些模型可能特别相关。这些研究的主要动机是观测宇宙学领域的快速进展，现在首次允许在可观测宇宙的规模上对基础物理学进行精确测试。这篇综述的目的是为宇宙学和引力物理学的研究人员和学生提供一个参考工具，以及对整个学科的自成一体、全面和最新的介绍。