In this paper we studied the fractional Euler-Bernoulli beam equation including a composition of the left and right fractional Caputo derivatives. We analyzed the equation with two types of boundary conditions (for the fixed-supported and fixed-free ends). The differential equation is converted into an integral one, taking into account the assumed boundary conditions. The obtained exact solutions contain a composition of the left and right Riemann-Liouville integrals. Finally, we presented three particular solutions for a constant, power and trigonometric function.

中文翻译：

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具有固定支撑和自由端的梁的分数阶Euler-Bernoulli方程的精确解

在本文中，我们研究了分数欧拉-伯努利梁方程，其中包括左右分数阶Caputo导数的组成。我们用两种类型的边界条件（对于固定支撑端和固定自由端）分析了方程。考虑到假定的边界条件，该微分方程被转换为一个积分方程。所获得的精确解包含左右Riemann-Liouville积分的组成。最后，我们为常数，幂和三角函数提供了三种特殊的解决方案。