On systems of linear fractional differential equations with constant coefficients

Abstract

This paper deals with the study of linear systems of fractional differential equations such as the following system: (1) Y ¯ ( α = A ( x ) Y ¯ + B ¯ ( x ) , where Y ¯ ( α is the Riemann–Liouville or the Caputo fractional derivative of order α (0 < α ≦ 1), and (2) A ( x ) = a 11 ( x ) · · · a 1 n ( x ) … · · · · … · · · · … · · · · a n 1 ( x ) · · · a nn ( x ) ; B ¯ ( x ) = b 1 ( x ) … … … b n ( x ) are matrices of known real functions. In a way analogous to the usual case, we show how a generalized matrix exponential function and certain fractional Green function, in connection with the Mittag–Leffler type functions, would allow us to obtain an explicit representation of the general solution to the system (1) when A is a constant matrix.