2012_Physics-Letters-A
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Physics Letters A 376 (2012) 2588–2590
Contents lists available at SciVerse ScienceDirect
Physics Letters A
/locate/pla
A generalized fractional sub-equation method for fractional differential equations with variable coefficients ✩
The Jumarie’s modified Riemann–Liouville derivative is defined
as
D
α
x
f
(x)
=
⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩
1d
Γ (−α) dx
x 0
(x
−
ξ
)−α−1(
f
(ξ
α < 0,
1d
Γ (1−α) dx
0x(x − ξ )−α( f (ξ )
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
In recent years, fractional differential equations (FDEs) have been caught much attention due to their numerous applications in the areas of physics, biology and engineering [1–3]. Many important phenomena in non-Brownian motion, signal processing, systems identification, control problem, viscoelastic materials and polymers are well described by fractional differential equation [4–7]. The main reason consists in the fact that in a real physical phenomenon, the next state might depend on not only its current state but also all of its historical states (non-local property). For better understanding the mechanisms of the complicated nonlinear physical phenomena as well as further applying them in practical life, the solution of fractional differential equation is much involved. In the past, many analytical and numerical methods have been proposed to obtain solutions of FDEs, such as finite difference method [8], finite element method [9], differential transform method [10,11], Adomian decomposition method [12–14], variational iteration method [15–18], homotopy perturbation method [19,20] and so on. However, to our knowledge, most of aforementioned methods are related to the constant-coefficient models. Recently, the study of variable-coefficient FDEs has attracted much attention because they can describe many nonlinear physical phe-
describes a variety of wave phenomena in plasma and solid state
[32,33].
The rest of this Letter is organized as follows: In Section 2, we
will describe the modified Riemann–Liouville derivative ics Letters A 376 (2012) 2588–2590
2589
2. Description of modified Riemann–Liouville derivative and generalized fractional sub-equation method
✩ This work is supported by the NSF of China (No. 10971166).
* Corresponding author. Tel.: +86 29 82665242; fax: +86 29 83237910.
E-mail addresses: tangbo08@ (B. Tang), leileiwei09@ (L. Wei).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
/10.1016/j.physleta.2012.07.018
nomena more realistically than the constant-coefficient ones. More recently, Zhang et al. [21] introduced a new method called fractional sub-equation method to look for traveling wave solutions of FDEs. The method is based on the homogeneous balance principle [22], Jumarie’s modified Riemann–Liouville derivative [23–25] and symbolic computation. By using fractional sub-equation method, Zhang et al. successfully obtained more traveling wave solutions of two FDEs.
The present Letter is motivated by the desire to propose a generalized fractional sub-equation method to improve the work made in [21] and [26]. As one application of the generalized fractional sub-equation method, we will consider the space–time fractional Gardner equation with variable coefficients:
Keywords: Generalized fractional sub-equation method Modified Riemann–Liouville derivative Fractional differential equation
abstract
In this Letter, a generalized fractional sub-equation method is proposed for solving fractional differential equations with variable coefficients. Being concise and straightforward, this method is applied to the space–time fractional Gardner equation with variable coefficients. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. It is shown that the considered method provides a very effective, convenient and powerful mathematical tool for solving many other fractional differential equations in mathematical physics.
article info
Article history: Received 14 May 2012 Received in revised form 14 July 2012 Accepted 18 July 2012 Available online 21 July 2012 Communicated by R. Wu
the main steps of the method here. In Section 3, we illustrate the
method in detail with the space–time fractional Gardner equation
Contents lists available at SciVerse ScienceDirect
Physics Letters A
/locate/pla
A generalized fractional sub-equation method for fractional differential equations with variable coefficients ✩
The Jumarie’s modified Riemann–Liouville derivative is defined
as
D
α
x
f
(x)
=
⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩
1d
Γ (−α) dx
x 0
(x
−
ξ
)−α−1(
f
(ξ
α < 0,
1d
Γ (1−α) dx
0x(x − ξ )−α( f (ξ )
© 2012 Elsevier B.V. All rights reserved.
1. Introduction
In recent years, fractional differential equations (FDEs) have been caught much attention due to their numerous applications in the areas of physics, biology and engineering [1–3]. Many important phenomena in non-Brownian motion, signal processing, systems identification, control problem, viscoelastic materials and polymers are well described by fractional differential equation [4–7]. The main reason consists in the fact that in a real physical phenomenon, the next state might depend on not only its current state but also all of its historical states (non-local property). For better understanding the mechanisms of the complicated nonlinear physical phenomena as well as further applying them in practical life, the solution of fractional differential equation is much involved. In the past, many analytical and numerical methods have been proposed to obtain solutions of FDEs, such as finite difference method [8], finite element method [9], differential transform method [10,11], Adomian decomposition method [12–14], variational iteration method [15–18], homotopy perturbation method [19,20] and so on. However, to our knowledge, most of aforementioned methods are related to the constant-coefficient models. Recently, the study of variable-coefficient FDEs has attracted much attention because they can describe many nonlinear physical phe-
describes a variety of wave phenomena in plasma and solid state
[32,33].
The rest of this Letter is organized as follows: In Section 2, we
will describe the modified Riemann–Liouville derivative ics Letters A 376 (2012) 2588–2590
2589
2. Description of modified Riemann–Liouville derivative and generalized fractional sub-equation method
✩ This work is supported by the NSF of China (No. 10971166).
* Corresponding author. Tel.: +86 29 82665242; fax: +86 29 83237910.
E-mail addresses: tangbo08@ (B. Tang), leileiwei09@ (L. Wei).
0375-9601/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
/10.1016/j.physleta.2012.07.018
nomena more realistically than the constant-coefficient ones. More recently, Zhang et al. [21] introduced a new method called fractional sub-equation method to look for traveling wave solutions of FDEs. The method is based on the homogeneous balance principle [22], Jumarie’s modified Riemann–Liouville derivative [23–25] and symbolic computation. By using fractional sub-equation method, Zhang et al. successfully obtained more traveling wave solutions of two FDEs.
The present Letter is motivated by the desire to propose a generalized fractional sub-equation method to improve the work made in [21] and [26]. As one application of the generalized fractional sub-equation method, we will consider the space–time fractional Gardner equation with variable coefficients:
Keywords: Generalized fractional sub-equation method Modified Riemann–Liouville derivative Fractional differential equation
abstract
In this Letter, a generalized fractional sub-equation method is proposed for solving fractional differential equations with variable coefficients. Being concise and straightforward, this method is applied to the space–time fractional Gardner equation with variable coefficients. As a result, many exact solutions are obtained including hyperbolic function solutions, trigonometric function solutions and rational solutions. It is shown that the considered method provides a very effective, convenient and powerful mathematical tool for solving many other fractional differential equations in mathematical physics.
article info
Article history: Received 14 May 2012 Received in revised form 14 July 2012 Accepted 18 July 2012 Available online 21 July 2012 Communicated by R. Wu
the main steps of the method here. In Section 3, we illustrate the
method in detail with the space–time fractional Gardner equation
