Mathematical Problems in Engineering

Volume 2011, Article ID 587068, 14 pages

http://dx.doi.org/10.1155/2011/587068

## Solution to the Linear Fractional Differential Equation Using Adomian Decomposition Method

^{1}Department of Mathematics, Xiamen University, Xiamen 361005, China^{2}Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China

Received 25 March 2010; Accepted 1 June 2010

Academic Editor: Geraldo Silva

Copyright © 2011 Jin-Fa Cheng and Yu-Ming Chu. 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

We obtain the analytical general solution of the linear fractional differential equations with constant coefficients by Adomian decomposition method under nonhomogeneous initial value condition, which is in the sense of the Caputo fractional derivative.

#### 1. Introduction

Fractional differential equations are hot topics both in mathematics and physics. Recently, the fractional differential equations have been the subject of intensive research. There are several methods to obtain the solution, such as the Laplace transform method, power series method, and Green function method. Many remarkable results for the fractional differential equations can be found in the literature [1–11]. In particular, the Adomian decomposition method has attracted the attention of many mathematicians [12–15].

For a better understanding of the fractional derivatives and for a physical understanding of the fractional equations, the readers can refer to the recent publications in [16, 17]. Ebaid [18] suggested a modification of the Adomian method, and a few iterations lead to exact solution. Das [19] compared the variational iteration method with the Adomian method for fractional equations and found that the variational iteration method is much more effective. For other methods of the fractional differential equations, especially the homotopy perturbation method, variational iteration method and differential transform method were presented in [20, 21].

Consider the following -term fractional differential equation with constant coefficients: where and is a real constant. In [12], the authors obtain the particular solution of (1.1) of the homogeneous initial value problem of the form

However, it seems also more meaningful and more complicated for solving general solution of (1.1) under nonhomogeneous initial value condition. Therefore, in this paper, we will remove the restriction of the homogeneous initial value, consider the nonhomogeneous initial value problems of the form and obtain the analytical general solution of (1.1), which generalizes the result in [12].

We organize the paper as follows. In Section 2, we give some basic definitions and properties. In Section 3, we obtain the analytical general solution of the linear fractional differential equations by Adomian decomposition method. Some explicit examples are given in Section 4.

#### 2. Basic Definitions and Notations

*Definition 2.1 (see [1]). *The Riemann-Liouville integral of order is defined by
From Definition 2.1, we clearly see that
where and is a real number.

*Definition 2.2 (see [1]). *For , the Caputo fractional derivative of is defined by
Therefore,

Lemma 2.3 (see [1]). *If is continuous, then
*

Lemma 2.4 (see [1]). *If is continuous, then
*

Lemma 2.5. *If is continuous, then
**
where and .*

* Proof. * From Definition 2.2 and Lemmas 2.3–2.4, we get
It is easy to see that

Proposition 2.6 (see [12]). * One has
*

Proposition 2.7 (see [12]). * More over, one has
*

#### 3. The Analytical Solution of the Linear Constant Coefficient Fractional Differential Equation

For simplicity, if , then we denote or by or .

In this section, we use Adomian decomposition method to discuss the general form of the linear fractional differential equations with constant coefficients, and apply and some basic transformation and integration to obtain the solution of the equations.

Let us consider the following -term linear fractional differential equations with constant coefficients: where are real constants, denotes Caputo fractional derivative of order .

Applying to both sides of (1.1) and utilizing Lemma 2.5, we get

By the Adomian decomposition method, we obtain the recursive relationship as follows:

By Adomian decomposition method, adding all terms of the recursion, we obtain the solution of (1.1) as Let Then,

Next, we estimate and , respectively.

For , by [12] we obtain

For , by the initial conditions (2.10) we get

Using formulas (2.2) and (2.3), the above expression can be written as where denotes .

Using Propositions 2.6 and 2.7, the above solution is equivalent to the following form:

Therefore, where and is the Mittag-Leffler function Substituting the Green function into the above expression, we know that is the analytical general solution of (1.1).

#### 4. Illustrative Examples

In order to verify our conclusions, we give some examples.

Consider an initial value problem for the relaxation-oscillation equation (see [1]) where are real constants.

Utilizing Lemma 2.5 and applying to both sides of (4.1), we obtain

According to the above procedure of solving the linear fractional differential equations with constant coefficients and using the Adomian decomposition method, let

Adding all of the above terms, we obtain the solution of the equation by Adomian decomposition method as follows: where .

It is easy to see that

Consider an initial value problem for the nonhomogeneous Bagley-Torvik equation (see [5]) where are real constants.

Utilizing Lemma 2.5 and applying to both sides of (4.6), we obtain

According to the above procedure of solving the linear fractional differential equation with constant coefficients and using the Adomian decomposition method, let

Adding all of the above terms, we obtain the solution of the equation by Adomian decomposition method as follows: where

#### Acknowledgment

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 60850005), the Natural Science Foundation of Zhejiang Province (Nos. D7080080 and Y607128) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (No. T200924).

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