Advances in Mathematical Physics
Volume 2015 (2015), Article ID 567842, 10 pages
http://dx.doi.org/10.1155/2015/567842

Research Article

Exact Solutions for Some Fractional Differential Equations

Abdullah Sonmezoglu

Department of Mathematics, Faculty of Science and Arts, Bozok University, 66100 Yozgat, Turkey

Received 10 February 2015; Accepted 29 April 2015

Academic Editor: Andrei D. Mironov

Copyright © 2015 Abdullah Sonmezoglu. 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 extended Jacobi elliptic function expansion method is used for solving fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative. By means of this approach, a few fractional differential equations are successfully solved. As a result, some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions are established. The proposed method can also be applied to other fractional differential equations.

1. Introduction

Fractional differential equations attracted attention in physics, biology, engineering, signal processing, systems identification, control theory, finance, and fractional dynamics [1–3]. Also, they are employed in social sciences such as food supplement, climate, finance, and economics.

Finding approximate and exact solutions to fractional differential equations is an important task. Various analytical and numerical methods have been introduced to obtain solutions of fractional differential equations, such as the Adomian decomposition method [4, 5], the variational iteration method [6–8], the homotopy analysis method [9–12], the homotopy perturbation method [13–15], the Lagrange characteristic method [16], the finite difference method [17], the finite element method [18], the differential transformation method [19], the fractional subequation method [20–24], the first integral method [25], the -expansion method [26–29], the fractional complex transform method [30], and the modified simple equation method [31–33].

In [34], Jumarie proposed a modified Riemann-Liouville derivative. With this kind of fractional derivative and some useful formulas, we can convert fractional differential equations into integer-order differential equations by variable transformation.

In this paper, we used extended Jacobi elliptic function expansion method [35–37] to establish exact solutions for three nonlinear space-time fractional differential equations in the sense of Jumarie’s modified Riemann-Liouville derivative, namely, the space-time fractional generalized reaction duffing equation, the space-time fractional bidirectional wave equations, and the space-time fractional symmetric regularized long wave (SRLW) equation. Also, we included figures to show the properties of some Jacobi elliptic function solutions of these fractional differential equations.

2. Jumarie’s Modified Riemann-Liouville Derivative and the Extended Jacobi Elliptic Function Expansion Method

In this section, we first give the definition and some properties of the modified Riemann-Liouville derivative which are used further in this paper.

The Jumarie modified Riemann-Liouville derivative of order is defined by the expression [34]

where , denote a continuous (but not necessarily differentiable) function.

Some properties of the fractional modified Riemann-Liouville derivative were summarized and three useful formulas of them are [34]

Next, let us consider nonlinear partial fractional differential equation

where is an unknown function and is a polynomial of . In this equation, the partial fractional derivatives involving the highest order derivatives and the nonlinear terms are included.

Li and He [38] presented a fractional complex transform to convert fractional differential equations into ordinary differential equations (ODEs), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. By using the traveling wave variable

where , are nonzero arbitrary constants and is the wave speed, we can rewrite (3) as the following nonlinear ODE:

where the prime denotes the derivation with respect to . If possible, we should integrate (5) term by term one or more times.

Our main goal is to derive exact or at least approximate solutions, if possible, for this ODE. For this purpose, using the extended Jacobi elliptic function expansion method, can be expressed as a finite series of Jacobi elliptic functions, , that is, the ansatz:

The parameter is determined by balancing the linear term(s) of highest order with the nonlinear one(s). And

where and are the Jacobi elliptic cosine function and the Jacobi elliptic function of the third kind, respectively, with the modulus . Therefore, the highest degree of is taken as

Substituting (6)–(8) into (5) and comparing the coefficients of each power of in both sides, we get an overdetermined system of nonlinear algebraic equations with respect to , , and . Solving this system, with the aid of Mathematica, then , , and can be determined. Substituting these results into (6), then some new Jacobi elliptic function solutions of (3) can be obtained. We can get other kinds of Jacobi doubly periodic wave solutions.

Since

degenerates, respectively, as the following form.

(1)Solitary wave solutions:

(2)Triangular function formal solution:

3. Applications of the Method

In this section, we present three examples to demonstrate the effectiveness of our approach to solve nonlinear fractional partial differential equations.

3.1. Space-Time Fractional Generalized Reaction Duffing Equation

We have applied the extended Jacobi elliptic function expansion method to construct the exact solutions of space-time fractional generalized reaction duffing equation [39, 40] in the form

where , , , and are all constants. Equation (12) reduces many well-known nonlinear fractional wave equations such as the following.

(i)Fractional Klein-Gordon equation:

(ii)Fractional Landau-Ginzburg-Higgs equation:

(iii)Fractional equation:

(iv)Fractional duffing equation:

(v)Fractional Sine-Gordon equation:

For our purpose, we introduce the following transformations:

where is a wave variable and and are constants; all of them are to be determined. Substituting (18) into (12), (12) is reduced into an ODE:

where . Suppose that the solution of (19) can be expressed by

Considering the homogeneous balance between the highest order derivative and the highest order nonlinear term in (19), we obtain . So

Substituting (21) into (19) and comparing the coefficients of each power of in both sides, we get an overdetermined system of nonlinear algebraic equations with respect to , , , and . Solving this system with Mathematica, we get the following results.

Case 1. Consider

Case 2. Consider

Case 3. Consider

Thus, we obtain the following solutions of (12).

Solution 1. See Figure 1:

(a)

(b)

(c)

Figure 1: Profiles of in (25) corresponding to the values , , ; , , ; and , , from (a) to (c).

Solution 2. See Figure 2:

(a)

(b)

(c)

Figure 2: Profiles of in (26) corresponding to the values , , ; , ; and , , , from (a) to (c).

Solution 3. Consider

3.1.1. Soliton Solutions

When the modulus approaches to 1 in (25), (26), and (27), we can obtain solitary wave solutions of space-time fractional generalized reaction duffing equation, respectively:

3.1.2. Triangular Periodic Solutions

When the modulus approaches to zero in (26), (27), we can obtain trigonometric function solutions of space-time fractional generalized reaction duffing equation, respectively:

3.2. Space-Time Fractional Bidirectional Wave Equations

Let us apply our method to the space-time fractional bidirectional wave equations in the form [41, 42]

where represents the distance along the channel, is the elapsed time, the variable is the dimensionless horizontal velocity, is the dimensionless deviation of the water surface from its undisturbed position, and , , , and are real constants. When , (30) is the generalization of bidirectional wave equations, which can be used as a model equation for the propagation of long waves on the surface of water with a small amplitude by Bona and Chen [43].

For our purpose, we use the following transformation:

where and are nonzero constants. Substituting (31) into (30), we obtain

where . Suppose that the solutions of (32) can be expressed by

Balancing the highest order derivative terms and nonlinear terms in (32), we can obtain . So we have

Proceeding as in the previous case, we get the following results.

Case 1. Consider

Case 2. Consider

Thus, we obtain the following solutions of (30).

Solution 1. Consider

Solution 2. Consider

where

3.2.1. Soliton Solutions

When the modulus approaches to 1 in (37), (38), we can obtain solitary wave solutions of the space-time fractional bidirectional wave equations, respectively:

where

3.3. The Space-Time Nonlinear Fractional SRLW Equation

We consider the space-time nonlinear fractional SRLW equation [44, 45]

which arises in several physical applications including ion sound waves in plasma. For our purpose, we use the following transformation:

where , , and are constants with , . Substituting (43) into (42), we obtain

Suppose that the solutions of (44) can be expressed by

Considering the homogeneous balance between the highest order derivative and the highest order nonlinear term in (44), we obtain . So we have

Proceeding as in the previous cases, we get the following results.

Case 1. Consider

where

Case 2. Consider

where

Case 3. Consider

where

Thus, we obtain the following solutions of (42).

Solution 1. Consider

Solution 2. Consider

Solution 3. Consider

3.3.1. Soliton Solutions

When the modulus approaches to 1 in (53), (54), and (55), we can obtain solitary wave solutions of the space-time nonlinear fractional SRLW equation, respectively:

3.3.2. Triangular Periodic Solutions

We can obtain trigonometric function solutions of the space-time nonlinear fractional SRLW equation, when the modulus approaches to zero; for example, (54), (55) give the same solution:

4. Conclusion

In this paper, we used the extended Jacobi elliptic function expansion method for solving fractional differential equations and applied it to find exact solutions of the space-time fractional generalized reaction duffing equation, the space-time fractional bidirectional wave equations, and the space-time fractional symmetric regularized long wave (SRLW) equation. With the aid of Mathematica, we successfully obtained some new Jacobi elliptic function solutions including solitary wave solutions and trigonometric function solutions for these equations. This method is effective and can also be applied to other fractional differential equations.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author wishes to thank the referees for their valuable suggestions. He thanks Mr. Mehmet Ekici from the Department of Mathematics, Bozok University, Yozgat, Turkey. This paper is supported by the Scientific and Technological Research Council of Turkey (TUBITAK).

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