A New Approach for an Analytical Solution for a System of Multi-term Linear Fractional Differential Equations

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.