Journal of Applied Mathematics

Volume 2012 (2012), Article ID 193061, 21 pages

doi:10.1155/2012/193061

Research Article

Analysis of a System for Linear Fractional Differential Equations

Fang Wang,1,2 Zhen-hai Liu,³ and Ping Wang⁴

¹School of Mathematical Science and Computing Technology, Central South University, Hunan, Changsha 410075, China

²School of Mathematical Science and Computing Technology, Changsha University of Science and Technology, Hunan, Changsha 410076, China

³School of Mathematical Science and Computing Technology, Guangxi University for Nationalities, Guangxi, Nanning 530006, China

⁴Xiamen Institute of Technology, Huaqiao University, Fujian, Xiamen 361021, China

Received 3 March 2012; Revised 21 July 2012; Accepted 23 July 2012

Academic Editor: Chein-Shan Liu

Copyright © 2012 Fang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The main purpose of this paper is to obtain the unique solution of the constant coefficient homogeneous linear fractional differential equations and the constant coefficient nonhomogeneous linear fractional differential equations if is a diagonal matrix and and prove the existence and uniqueness of these two kinds of equations for any and . Then we give two examples to demonstrate the main results.

1. Introduction

System of fractional differential equations has gained a lot of interest because of the challenges it offers compared to the study of system of ordinary differential equations. Numerous applications of this system in different areas of physics, engineering, and biological sciences have been presented in [1–3]. The differential equations involving the Riemman-Liouville differential operators of fractional order appear to be more important in modeling several physical phenomena and therefore seem to deserve an independent study of their theory parallel to the well-known theory of ordinary differential equations. The existence and uniqueness of solution for fractional differential equations with any have been studied in many papers, see [4–28]. In [4] Daftradar-Gejji and Babakhani have studied the existence and uniqueness of

where denotes the standard Riemman-Liouville fractional derivative, , , which is an m dimensional linear space. They have obtained that the system (1.1) has a unique solution defined on if and . In [17] Belmekki et al. have studied the existence of periodic solution for some linear fractional differential equation in . In [21] Ahmad and Nieto have studied the Riemann-Liouwille fractional differential equations with fractional boundary conditions. In comparison with the earlier results of this type we get more general assumptions. We assume instead of and consider the following system of fractional differential equations:

where denotes the standard Riemman-Liouville fractional derivative, , ,

and is a constant vector. We completely generalize the results in [4] and obtain the new results if . Furthermore, we also obtain some results of the unique solution of the homogeneous and nonhomogeneous initial value problems with the classical Mittag-Leffler special function [5] which is similar to the ordinary differential equations. Now we introduce the first Mittag-Leffler function defined by

The function belongs to . Indeed, taking the norm in , we have

The formula remains valid for . In this case, . Then we introduce the second Mittag-Leffler function defined by

The formula remains also valid for . In this case, .

The paper is organized as follows. In Section 2 we recall the definitions of fractional integral and derivative and related basic properties and preliminary results used in the text. In Section 3 we obtain the unique solution of the constant coefficient homogeneous and nonhomogeneous linear fractional differential equations for being the diagonal matrix. In Section 4 we prove the existence and uniqueness of these two kinds of equations for any . In Section 5 we give some specific examples to illustrate the results.

2. Definitions and Preliminary Results

Let us denote by the space of all continuous real functions defined on , which turns out to be a Banach space with the norm

We define similarly another Banach space , in which function is continuous on and is continuous on with the norm:

is the space of real functions defined on which are Lebesgue integrable on .

Obviously .

The definitions and results of the fractional calculus reported below are not exhaustive but rather oriented to the subject of this paper. For the proofs, which are omitted, we refer the reader to [6] or other texts on basic fractional calculus.

Definition 2.1 (see [6]). The fractional primitive of order of function is given by

From [17] we know exists for all , when ; consider also that when then and moreover

Definition 2.2 (see [6]). The fractional derivative of order of a function is given by

We have for all .

Lemma 2.3 (see [6]). Let . If one assumes , then the fractional differential equation

has , as solutions.

From this lemma we can obtain the following law of composition.

Lemma 2.4 (see [6]). Assume that with a fractional derivative of order that belongs to . Then

for some . When the function is in , then .

Lemma 2.5 (see [6]). Let be a nonempty closed subset of a Banach space , and let for every and such converges. Moreover, let the mapping satisfy the inequality

for every and any . Then, has a uniquely defined fixed point . Furthermore, for any , the sequence converges to this fixed point .

Lemma 2.6 (see [12]). Let and have real eigenvalues . Then there exists a basis of in which the matrix representation of assumes Jordan form, that is, the matrix of is made of diagonal blocks of the form , where each consists of diagonal blocks of the form

Lemma 2.7 (see [12]). Let and have complex eigenvalues , with multiplicity. Then there exists a basis of , where has matrix form , where each consists of diagonal blocks of the type

Lemma 2.8 (see [12]). Let . Then has a basis giving a matrix representation composed of diagonal blocks of type and/or matrices , where and are as defined in the preceding lemmas.

Now, we will introduce Lemma 2.9 to prove the following Theorem 4.4 in Section 4.

Lemma 2.9. Let . Assume that and belong to . Then For the initial value problem

has a unique solution provided , where is a suitable constant depending on , , and .

Proof. The initial value problem (2.11) will be solved in two steps.

  (1) Local existence. Our problem is equivalent to the problem of determination of fixed points of the following operator:

 with

 It is immediate to verify that is also well defined. Indeed,

 for and belong to . Then we can also prove is a contraction operator. Indeed,

 for all . Let us assume

 that is,

 Taking sufficiently small, we also have

and then

 with . Therefore is a contraction operator. This shows that initial problem (2.11) has a unique solution.  (2) Continuation of solution. Since we know the value of on , then we can compute

 We can solve the integral problem

 obtaining a unique solution for all . Now and agree on . Thus the solution admits as its continuation. Hence the proof of Lemma 2.9 is complete.

3. Initial Value Problem: Continuous Solutions on

We open this section with some basic examples, concerning the case when the solutions in are submitted to an initial condition.

Theorem 3.1. Let . For all the initial value problem

admits

as unique solution in .

Proof. According to Lemma 2.4, the initial value problem (3.1) is equivalent to the following equations:

Hence the proof of Theorem 3.1 is complete.

Theorem 3.2. Let . Assume

Then for all the initial value problem

has a unique solution in

given by

with

Proof. According to Lemma 2.4, the initial value problem (3.5) is equivalent to the following equations:

Hence the proof of Theorem 3.2 is complete.

The result remains true even if . In this case, (3.5) is reduced to the ordinary differential equations

which have a unique solution in

given by

with

Theorem 3.3. Let . For all the initial value problem

where

has a unique solution in

given by

with

Proof. We can write (3.27) in the following form:

According to Lemma 2.4,

is equivalent to the following equations:

for some . From (3.21) we obtain, by iteration,

Letting , if . Indeed,

On the other hand,

then we can obtain

Since ,

Hence the proof of Theorem 3.3 is complete.

The result remains valid even if . In this case,

has a unique solution in

given by

with

Theorem 3.4. Let . For all the initial value problem

where

and has a unique solution in

given by

with

Proof. We can write (3.44) in the following form:

According to Lemma 2.4, the equation

is equivalent to the following equations:

for some . From (3.38) we obtain, by iteration,

Letting , if . Indeed,

On the other hand,

Then we can obtain

We know that is satisfied for the fractional nonhomogeneous linear differential equation . So we can also deduce that the general solution of the fractional nonhomogeneous linear differential equation is equal to the general solution of the corresponding homogeneous linear differential equation plus the special solution of the nonhomogeneous linear differential equation. If , , then

Hence the proof of Theorem 3.4 is complete.

The result remains valid even if . In this case,

where

and has a unique solution in

given by

with

4. Existence and Uniqueness of the Solution

In Section 3 we have obtained the unique solution of the constant coefficient homogeneous and nonhomogeneous linear fractional differential equations for being the diagonal matrix. In the present section we will prove the existence and uniqueness of these two kinds of equations for any .

Theorem 4.1. Let and . If the matrix has distinct real eigenvalues, then for all the initial value problem

has the unique solution .

Proof. Since the matrix has distinct real eigenvalues, there exists an invertible matrix such that

where are the eigenvalues of the matrix . If we define ,

with . From the above Theorem 3.3 we know the initial value problem

has a unique solution defined on . Then uniquely solves the equations (4.1), where . Hence the proof of Theorem 4.1 is complete.

Theorem 4.2. Let . For all the initial value problem

where

has the unique solution defined on .

Proof. Let us define

We can find that (4.5) is equivalent to the following equation

Obviously, if and belong to . From the above Theorem 3.3 in Section 3, we know the complex Equation (4.7) has a unique solution defined on . Hence the proof of Theorem 4.2 is complete.

Theorem 4.3. Let and . If has eigenvalues , for all the initial value problem

has a unique solution .

Proof. Since has eigenvalues , there exists an invertible matrix such that where

Define

then

From the above Theorem 4.2, we know the initial value problem

has a unique solution defined on . Hence the proof of result is complete.

Theorem 4.4. Let and be an elementary Jordan matrix:

The initial value problem

has a unique solution provided , where is a suitable constant depending on , , and .

Proof. From the (4.15), we can write the equations in the following form:

Consider the first equation

We can obtain the solution of this equation

Consider the second equation

where now is a known function. Since , according to Lemma 2.9, (4.19) has a unique solution in . Now and are known functions which will be substituted in

and so on. Thus the system of equations given in (4.15) has unique solution in .

Theorem 4.5. Let and . The initial value problem

has the unique solution provided , where is a suitable constant depending on , , and .

Proof. In view of Lemma 2.8, there exists an invertible matrix such that is composed of diagonal blocks of the type and , as defined in the preceding Lemmas 2.7 and 2.8. Let and . Consider the initial value problem:

Then in view of Theorems 4.1–4.5, (4.22) has a unique solution: . Therefore (4.21) has a unique solution .

Remark 4.6. All the above results are valid for . Moreover, we can also discuss the case if , in this case, we cannot consider the usual initial condition , but . We can also obtain some similar results by the same method, So we did not give the detailed process and conclusion in this paper.

5. Illustrative Examples

In this section, we give some specific examples to illustrate the above results.

Example 5.1. Consider the following system, where , , ,,

Here

having the eigenvalues 2, 3, and 6. Choose the eigenvectors  ,, and . Then

where

Define the . Then the system of equation in is decoupled, namely,

In view of (3.30), we can obtain

Hence

Example 5.2. Consider the following system, where , , ,,

Here

having the eigenvalues . Choose the eigenvectors   , and , Then

where

Define the . Then the system of equation in is decoupled, namely,

In view of (3.30), we can obtain

Hence

Acknowledgments

The authors are highly grateful for the referee's careful reading and comments on this paper. The present paper was supported by the NNSF of China Grants no. 11271087 and no. 61263006.

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