In this paper we study the exact solutions of a class of fractional delay differential equations. We consider the fractional derivative of the order between 1 and 2 in the sense of Caputo. In the first part, we introduce two novel matrix functions, namely, the generalized cosine-type and sine-type delay Mittag-Leffler matrix functions. Then we obtain the explicit solutions for the linear homogeneous equations subjecting to corresponding initial conditions, by means of undetermined coefficients. In the second part, we first obtain a particular solution by means of the Laplace transform for the inhomogeneous equations with null initial conditions. Then we give an analytical representation of the general solution of the inhomogeneous equations through the sum of its particular solution and the general solution of the corresponding homogeneous equation.

在本文中，我们研究了一类分数延迟微分方程的精确解。我们考虑 Caputo 意义上的 1 到 2 阶的分数导数。在第一部分，我们介绍了两个新的矩阵函数，即广义余弦型和正弦型延迟 Mittag-Leffler 矩阵函数。然后我们通过待定系数得到了相应初始条件下线性齐次方程的显式解。在第二部分中，我们首先通过拉普拉斯变换获得初始条件为零的非齐次方程的特解。