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On systems of linear fractional differential equations with constant coefficients. (English) Zbl 1121.34006
Summary: This paper deals with the study of linear systems of fractional differential equations such as the following system: $$\overline Y^{(\alpha}={\bold A}(x) \overline Y+\overline{\bold B}(x),\tag1$$ where $\overline Y^{(\alpha}$ is the Riemann-Liouville or the Caputo fractional derivative of order $\alpha(0<\alpha \le 1)$, and $${\bold A}(x)=\left( \matrix a_{11}(x) & \dots & a_{1n}(x)\\ \dots & \dots &\cdot\\ \dots & \cdots & \cdot\\ \dots & \cdots & \cdot\\ a_{n1}(x) & \cdots & a_{nn}(x) \endmatrix\right);\quad \overline{\bold B}(x)=\left(\matrix b_1(x)\\ \cdots \\ \cdots \\ \cdots\\ b_n(x)\endmatrix\right)$$ 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 ${\bold A}$ is a constant matrix.

34A30Linear ODE and systems, general
Full Text: DOI
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