Exact solutions of fractional differential Stein matrix equations via diagonalization and Mittag–Leffler functions

In this work, an explicit formula in terms of the Hadamard product was derived for the solutions to fractional differential Stein matrix equations (FDSMEs), assuming that the coefficient matrices are separately diagonalizable. To begin with, we gave a general formula using the Kronecker product notation, which does not need any diagonalizability assumption to show the existence and uniqueness of the solution. In the case of diagonalizable coefficient matrices, the matrix equation became decoupled scalar Caputo fractional differential equations. Solutions to these differential equations were given explicitly in terms of two-parameter Mittag–Leffler functions and put together using the Hadamard product. The results for integer-order differential equations are obtained as special cases. Finally, we give three examples.