The Exact Solutions of Such Coupled Linear Matrix Fractional Differential Equations of Diagonal Unknown Matrices by Using Hadamard Product

In this paper, we present the general exact solutions of such coupled system of matrix fractional differential equations for diagonal unknown matrices in Caputo sense by using vector extraction operators and Hadamard product. Some illustrated examples are also given to show our new approach.


Introduction
Fractional calculus attracted the attention of researchers because of its application in physics as the nonlinear oscillation of earthquake can be modeled with fractional derivatives [1], and the fluid-dynamic traffic model with fractional derivatives [2] can eliminate the deficiency arising from the assumption of continuum traffic flow.Based on experimental data fractional, partial differential equations for seepage flow in porous media are suggested in [3], and differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena [4].A review of some applications of fractional derivatives in continuum and statistical mechanics is given by Mainardi [5].The analytic results on the existence and uniqueness of solutions of the fractional differential equations have been investigated by many authors see as an example [6].During the last decades, several methods have been used to solve fractional differential equations, fractional partial differential equations, fractional integro-differential equations and dynamic systems containing fractional derivatives, such as Adomian's decomposition method [7], variational iteration method [8]- [11], homotopy perturbation method [12], homotopy analysis method [13], spectral methods [14], and other methods [15].
Recently, Wang [16] studied the synchronized motions in a star network of coupled fractional order systems in which the major element is coupled to each of the non-interacting individual elements and Kilicman and Al-Zhour [17] studied several operational matrices for fractional integration and differentiation and expanded the Kronecker convolution product to the Riemann-Liouville fractional integral of matrices.Al-Zhour [18] introduced the exact solution of coupled fractional order systems by using Kronecker structure.
In the present paper, the exact solutions of coupled and uncoupled systems of matrix fractional differential equations for diagonal unknown matrices are presented by using a new attractive method and some illustrated examples are also given to show our new approach.

Basic Results and Preliminaries
In this section, we recall some basic results and definitions associated to Hadamard product, Mittage-Leffler function and Caputo fractional derivative that will be used to get our results later.
.Then the Hadamard product of A and B is defined by [19]- [26].

( ) ( )
Note that the Mittage-Leffler matrix function of m A M ∈ can be represented by using spectral decomposition method by [18].

Main Results
In this section, we present the general exact solutions of the coupled and uncoupled system of fractional differential equations for diagonal unknown matrices by using the using vector extraction operators and Hadamard product.Lemma 3.1.Let n A M ∈ be a given scalar matrix, ,1 n c M ∈ be a given scalar vector, ( ) given vector function and ( ) ,1 n y x M ∈ be an unknown vector function to be solved.Then the exact solution of the following non-homogenous linear fractional system of order 0 1 p < < is given by [18]- [20].
is given by: ( is given by: ( Hence, the vector extraction solution of (3.3) is given by: ( A C M ∈ be given diagonal scalar matrices, and Y x M ∈ be an unknown diagonal matrix functions.Then the general vector extraction solution of the following general system of linear matrix fractional differential equations of order 0 1 p < < : is given by: ( ( ) Proof.By using (2-3), then (3.5) can be represented by: ( .

Illustrated Examples
In the section, we give some illustrated examples to show our new approach as discussed in above section.Example 4.1.Consider the following matrix linear fractional differential equation: where and ( ) Y x is diagonal matrix.Then the exact solution of (4-1) by apply- ing Theorem 3.2 is given by: Example 4.2.Consider the following system of order ,0 1 p p < < : Now the exact solution of (4-3) by applying Theorem 3.2 is given by: Example 4.3.Consider the following matrix fractional differential equation: where ( ) ( ) ( ) and ( ) Y x is diagonal matrix.Then the exact solu- tion of (4-4) by applying Theorem 3.2 is given by: ( ( ) Example 4.4.Consider the following matrix fractional differential equations of order 0 1 p < < : where ( ) ( ) Then the exact solution of (4-5) by applying Corollary 3.5 is given by:   Then the exact solution by applying Corollary 3.5 is given by:

Conclusion
The general exact solutions of coupled system of matrix fractional differential equations with diagonal matrices coefficients by using vector extraction operators and Hadamard product in Caputo sense are presented with some illustrated examples.How to find the complexity of this method requires further research.
diagonal matrix function.Then the general vector extraction solution of the following non-homogeneous matrix fractional differential equation Then the system (4-2) can be rewritten as: