System of linear fractional differential equations and the Mittag-Leffler functions with matrix variable

Abstract

The solution for system of linear fractional differential equations is derived in terms of the Mittag-Leffler functions with matrix variable. Three different methods for calculating the Mittag-Leffler functions with matrix variable are obtained with the help of inverse Laplace transform, Jordan canonical matrix and minimal polynomial, respectively. The solution for system of linear first-order differential equations is obtained as a special case. The results show that the Mittag-Leffler functions with matrix variable are powerful tools for solving system of linear fractional differential equations.