In this paper, we obtain an analytical solution for a system of multi-term linear fractional differential equations by using the analytic expression of inverse of multi-term fractional integral operator with matrix coefficients. The system of multi-term linear fractional differential equations is an efficient tool for solving the multi-term fractional partial differential equations. Firstly, several properties of multivariate Mittag–Leffler matrix function are given. Then, the reversibility of multi-term fractional integral operator with matrix coefficients and continuity of its inverse are proved, and the analytic expression of its inverse is obtained by the multivariate Mittag–Leffler matrix function. Finally, by using above results, the analytical solution for a system of multi-term linear fractional differential equations is presented, and some examples are shown to illustrate our results.

本文利用矩阵系数的多项式分数阶积分算子的逆解的解析表达式，得到了多元线性分数阶微分方程组的解析解。多项式线性分数阶微分方程组是求解多项式分数阶偏微分方程的有效工具。首先，给出了多元Mittag-Leffler矩阵函数的几个性质。然后，证明了具有矩阵系数的多项式分数阶积分算子的可逆性和其逆的连续性，并通过多元Mittag-Leffler矩阵函数获得了其逆的解析表达式。最后，利用以上结果，提出了多项线性分数阶微分方程组的解析解，