^{1}

^{2}

^{1}

^{*}

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.

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 [

Recently, Wang [

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.

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.

Definition 2.1. Let

Definition 2.2. Let

Theorem 2.3. Let

Definition 2.4. The one parameter Mittage-Leffler functions and Mittage-Leffler matrix functions of matrix

Note that the Mittage-Leffler matrix function of

where ^{T}, respectively.

Theorem 2.5. Let

Definition 2.6. The Caputo fractional derivative of

Theorem 2.7. The relationship between the Mittage-Leffler function and Caputo derivative are given by:

a)

b)

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

is given by:

Theorem 3.2. Let

is given by:

Proof. By using (2-3), then (3.3) can be represented by:

Hence, the vector extraction solution of (3.3) is given by:

Theorem 3.3. Let

is given by:

Proof. By using (2-3), then (3.5) can be represented by:

Now by letting

Hence by using Lemma 3.1 and simple computations, then we get the solution as in (3-6).

Below we will discuss some important special cases of the general system as in Theorem 3.3.

Theorem 3.4. Let

are given by:

Proof. By multiplying the second equation in (3-9) by

Then (3-9) can be written as

Now, by using

Now by using (3-6), then the solution of (3.13) is given by:

Now we deal with

Since

Then

But

and

So,

(3.18)

Similarly,

(3.19)

Now from (3-13), (3-18) and (3-19), we get

Since,

Then, we get the vector extraction solution as in (3-11).

Corollary 3.5. Let

are given by:

Proof. The proof is straightforward by applying Theorem 3.4 by letting

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

Example 4.2. Consider the following system of order

where

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

Example 4.4. Consider the following matrix fractional differential equations of order

where

Then the exact solution of (4-5) by applying Corollary 3.5 is given by:

Example 4.5. Consider the following coupled matrix fractional differential equations:

Then the exact solution by applying Corollary 3.5 is given by:

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.

The authors express their sincere thanks to referees for very careful reading and helpful suggestion of this paper.

ZayedAl-Zuhiri,ZeyadAl-Zhour,KhaledJaber, (2016) The Exact Solutions of Such Coupled Linear Matrix Fractional Differential Equations of Diagonal Unknown Matrices by Using Hadamard Product. Journal of Applied Mathematics and Physics,04,432-442. doi: 10.4236/jamp.2016.42049