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International Journal of Differential Equations
Volume 2011 (2011), Article ID 814132, 8 pages
http://dx.doi.org/10.1155/2011/814132
Research Article

New Method for Solving Linear Fractional Differential Equations

Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt

Received 4 May 2011; Revised 21 July 2011; Accepted 25 July 2011

Academic Editor: Shaher M. Momani

Copyright © 2011 S. Z. Rida and A. A. M. Arafa. 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 develop a new application of the Mittag-Leffler Function method that will extend the application of the method to linear differential equations with fractional order. A new solution is constructed in power series. The fractional derivatives are described in the Caputo sense. To illustrate the reliability of the method, some examples are provided. The results reveal that the technique introduced here is very effective and convenient for solving linear differential equations of fractional order.

1. Introduction

Fractional differential equations have excited, in recent years, a considerable interest both in mathematics and in applications. They were used in modeling of many physical and chemical processes and engineering (see, e.g., [16]). In its turn, mathematical aspects of fractional differential equations and methods of their solutions were discussed by many authors: the iteration method in [7], the series method in [8], the Fourier transform technique in [9, 10], special methods for fractional differential equations of rational order or for equations of special type in [1116], the Laplace transform technique in [36, 16, 17], and the operational calculus method in [1823]. Recently, several mathematical methods including the Adomian decomposition method [1825], variational iteration method [2326] and homotopy perturbation method [27, 28] have been developed to obtain the exact and approximate analytic solutions. Some of these methods use transformation in order to reduce equations into simpler equations or systems of equations, and some other methods give the solution in a series form which converges to the exact solution.

The reason of using fractional order differential (FOD) equations is that FOD equations are naturally related to systems with memory which exists in most biological systems. Also they are closely related to fractals which are abundant in biological systems. The results derived from the fractional system are of a more general nature. Respectively, solutions to the fractional diffusion equation spread at a faster rate than the classical diffusion equation and may exhibit asymmetry. However, the fundamental solutions of these equations still exhibit useful scaling properties that make them attractive for applications.

The concept of fractional or noninteger order derivation and integration can be traced back to the genesis of integer order calculus itself [29]. Almost all of the mathematical theory applicable to the study of noninteger order calculus was developed through the end of the 19th century. However, it is in the past hundred years that the most intriguing leaps in engineering and scientific application have been found. The calculation techniques in some cases meet the requirement of physical reality. The use of fractional differentiation for the mathematical modeling of real-world physical problems has been widespread in recent years, for example, the modeling of earthquake, the fluid dynamic traffic model with fractional derivatives, and measurement of viscoelastic material properties. Applications of fractional derivatives in other fields and related mathematical tools and techniques could be found in [3041]. In fact, real-world processes generally or most likely are fractional order systems.

The derivatives are understood in the Caputo sense. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses.

2. Fractional Calculus

There are several approaches to the generalization of the notion of differentiation to fractional orders, for example, the Riemann-Liouville, Grünwald-Letnikov, Caputo, and generalized functions approach [42]. The sRiemann-Liouville fractional derivative is mostly used by mathematicians but this approach is not suitable for real-world physical problems since it requires the definition of fractional order initial conditions, which have no physically meaningful explanation yet. Caputo introduced an alternative definition, which has the advantage of defining integer order initial conditions for fractional order differential equations [42]. Unlike the Riemann-Liouville approach, which derives its definition from repeated integration, the Grünwald-Letnikov formulation approaches the problem from the derivative side. This approach is mostly used in numerical algorithms.

Here, we mention the basic definitions of the Caputo fractional-order integration and differentiation, which are used in the upcoming paper and play the most important role in the theory of differential and integral equation of fractional order.

The main advantages of Caputo approach are the initial conditions for fractional differential equations with the Caputo derivatives taking on the same form as for integer order differential equations.

Definition 2.1. The fractional derivative of 𝑓(𝑥) in the Caputo sense is defined as 𝐷𝛼𝑓(𝑥)=𝐼𝑚𝛼𝐷𝑚=1𝑓(𝑥)Γ(𝑚𝛼)𝑥0(𝑥𝑡)𝑚𝛼+1𝑓(𝑚)(𝑡)𝑑𝑡(2.1) for 𝑚1<𝛼𝑚, 𝑚𝑁, 𝑥>0.
For the Caputo derivative we have 𝐷𝛼𝐶=0, 𝐶 is constant, 𝐷𝛼𝑡𝑛=0,(𝑛𝛼1),Γ(𝑛+1)𝑡Γ(𝑛𝛼+1)𝑛𝛼,(𝑛>𝛼1)..(2.2)

Definition 2.2. For 𝑚 to be the smallest integer that exceeds 𝛼, the Caputo fractional derivative of order 𝛼>0 is defined as 𝐷𝛼𝜕𝑢(𝑥,𝑡)=𝛼𝑢(𝑥,𝑡)𝜕𝑡𝛼=1Γ(𝑚𝛼)𝑡0(𝑡𝜏)𝑚𝛼+1𝜕𝑚𝑢(𝑥,𝜏)𝜕𝜏𝑚𝜕𝑑𝜏,for𝑚1<𝛼<𝑚𝑚𝑢(𝑥,𝑡)𝜕𝑡𝑚.,for𝛼=𝑚𝑁(2.3)

3. Analysis of the Method

The Mittag-Leffler (1902–1905) functions𝐸𝛼 and 𝐸𝛼,𝛽 [42], defined by the power series𝐸𝛼(𝑧)=𝑘=0𝑧𝑘Γ(𝛼𝑘+1),𝐸𝛼,𝛽(𝑧)=𝑘=0𝑧𝑘Γ(𝛼𝑘+𝛽),𝛼>0,𝛽>0,(3.1) have already proved their efficiency as solutions of fractional order differential and integral equations and thus have become important elements of the fractional calculus theory and applications.

In this paper, we will explain how to solve some of differential equations with fractional level through the imposition of the generalized Mittag-Leffler function𝐸𝛼(𝑧). The generalized Mittag-Leffler method suggests that the linear term 𝑦(𝑥) is decomposed by an infinite series of components:𝑦=𝐸𝛼(𝑎𝑥𝛼)=𝑛=0𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1).(3.2) We will use the following definitions of fractional calculus:𝐷𝛼𝑦=𝑛=1𝑎𝑛𝑥(𝑛1)𝛼,𝐷Γ((𝑛1)𝛼+1)(3.3)2𝛼𝑦=𝑛=2𝑎𝑛𝑥(𝑛2)𝛼Γ.((𝑛2)𝛼+1)(3.4) This is based on the Caputo fractional is derivatives. The convergence of the Mittag Leffler function discussed in [42].

4. Applications and Results

In this section, we consider a few examples that demonstrate the performance and efficiency of the generalized Mittag-Leffler function method for solving linear differential equations with fractional derivatives.

Example 4.1. Consider the following fractional differential equation [43]: 𝑑𝛼𝑦𝑑𝑥𝛼=𝐴𝑦.(4.1) By using (3.3) into (4.1) we find 𝑛=1𝑎𝑛𝑥(𝑛1)𝛼Γ((𝑛1)𝛼+1)𝐴𝑛=0𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1)=0.(4.2)Combining the alike terms and replacing (n) by (𝑛+1) in the first sum, we assume the form𝑛=0𝑎𝑛+1𝑥𝑛𝛼Γ(𝑛𝛼+1)𝐴𝑛=0𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1)=0,𝑛=0𝑎𝑛+1𝐴𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1)=0.(4.3)With the coefficient of 𝑥𝑛𝛼 equal to zero and identifying the coefficients, we obtain recursive 𝑎𝑛+1𝐴𝑎𝑛=0𝑎𝑛+1=𝐴𝑎𝑛,at𝑛=0,𝑎1=𝐴𝑎0=𝐴,at𝑛=1,𝑎2=𝐴𝑎1𝑎2=𝐴2,at𝑛=2,𝑎3=𝐴𝑎2𝑎3=𝐴3.(4.4) Substituting into (3.2) 𝑦(𝑥)=𝑎0+𝑎1𝑥𝛼Γ(𝛼+1)+𝑎2𝑥2𝛼Γ(2𝛼+1)+𝑎3𝑥3𝛼𝑥Γ(3𝛼+1)+,𝑦(𝑥)=1+𝐴𝛼Γ(𝛼+1)+𝐴2𝑥2𝛼Γ(2𝛼+1)+𝐴3𝑥3𝛼Γ(3𝛼+1)+.(4.5)The general solution is 𝑦(𝑥)=𝑛=0𝐴𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1).(4.6) We can write the general solution in the Mittag-Leffler function form as 𝑦(𝑥)=𝐸𝛼(𝐴𝑛𝑥𝛼).(4.7) As 𝛼=1, we have the exact solution: 𝑦(𝑥)=𝑛=0(𝐴𝑥)𝑛Γ(𝑛+1)=𝑒𝐴𝑥,(4.8) which is the exact solution of the standard form.

Example 4.2. Consider the fractional differential equation [44] 𝑑2𝛼𝑦𝑑𝑥2𝛼𝑦=0.(4.9) By using (3.2) and (3.4) into (4.9) we find 𝑛=2𝑎𝑛𝑥(𝑛2)𝛼Γ((𝑛2)𝛼+1)𝑛=0𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1)=0.(4.10)Combining the alike terms and replacing (𝑛) by (𝑛+2) in the first sum, we assume the form𝑛=0𝑎𝑛+2𝑥𝑛𝛼Γ(𝑛𝛼+1)𝑛=0𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1)=0,𝑛=0𝑎𝑛+2𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1)=0.(4.11)With the Coefficient of 𝑥𝑛𝛼 equal to zero and identifying the coefficients, we obtain recursive 𝑎𝑛+2=𝑎𝑛.(4.12) Substituting into (3.2), we find that: 𝑥𝑦(𝑥)=1+𝑎𝛼+𝑥Γ(𝛼+1)2𝛼Γ(2𝛼+1)+𝑎2𝑥3𝛼Γ(3𝛼+1)+.(4.13) If 𝑎=1, we can write the general solution in the Mittag-Leffler function form as 𝑦(𝑥)=𝑛=0𝑥𝛼Γ(𝑛𝛼+1)=𝐸𝛼(𝑥𝛼)(4.14) which is the exact solution of the linear fractional differential equation (4.9).

Example 4.3. Consider the fractional differential equation [43] 𝑑2𝛼𝑦𝑑𝑥2𝛼+𝑑𝛼𝑦𝑑𝑥𝛼2𝑦=0.(4.15) By using (3.2) and (3.4) into (4.15) we find 𝑛=2𝑎𝑛𝑥(𝑛2)𝛼+Γ((𝑛2)𝛼+1)𝑛=1𝑎𝑛𝑥(𝑛1)𝛼Γ((𝑛1)𝛼+1)2𝑛=0𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1)=0.(4.16) Combining the alike terms and replacing (𝑛) by (𝑛+2) in the first sum, we assume the form 𝑛=0𝑎𝑛+2𝑥𝑛𝛼+Γ(𝑛𝛼+1)𝑛=0𝑎𝑛+1𝑥𝑛𝛼Γ(𝑛𝛼+1)2𝑛=0𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1)=0,𝑛=0𝑎𝑛+2+𝑎𝑛+12𝑎𝑛𝑥𝑛𝛼Γ(𝑛𝛼+1)=0(4.17) With the coefficient of 𝑥𝑛𝛼 equal to zero and identifying the coefficients, we obtain recursive 𝑎𝑛+2=2𝑎𝑛𝑎𝑛+1.(4.18) Substituting into (3.2), we find that: 𝑥𝑦(𝑥)=1+𝑎𝛼𝑥Γ(𝛼+1)+(2𝑎)2𝛼𝑥Γ(2𝛼+1)+(𝑎2)3𝛼Γ(3𝛼+1)+.(4.19) If 𝑎=1, we can write the general solution in the Mittag-Leffler function form as 𝑦(𝑥)=𝑛=0𝑥𝛼Γ(𝑛𝛼+1)=𝐸𝛼(𝑥𝛼)(4.20) which is the solution of the linear fractional differential equation (4.15).

5. Conclusions

A new generalization of the Mittag-Leffler function method has been developed for linear differential equations with fractional derivatives. The new generalization is based on the Caputo fractional derivative. It may be concluded that this technique is very powerful and efficient in finding the analytical solutions for a large class of linear differential equations of fractional order.

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