Exact Solutions of the Space Time Fractional Symmetric Regularized Long Wave Equation Using Different Methods

Güner, Özkan; Eser, Dursun

doi:https://doi.org/10.1155/2014/456804

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2014 | Article ID 456804 | https://doi.org/10.1155/2014/456804

Exact Solutions of the Space Time Fractional Symmetric Regularized Long Wave Equation Using Different Methods

Özkan Güner¹and Dursun Eser²

Academic Editor: Hossein Jafari

Received04 Apr 2014

Accepted22 Jun 2014

Published22 Jul 2014

Abstract

We apply the functional variable method, exp-function method, and -expansion method to establish the exact solutions of the nonlinear fractional partial differential equation (NLFPDE) in the sense of the modified Riemann-Liouville derivative. As a result, some new exact solutions for them are obtained. The results show that these methods are very effective and powerful mathematical tools for solving nonlinear fractional equations arising in mathematical physics. As a result, these methods can also be applied to other nonlinear fractional differential equations.

1. Introduction

Fractional calculus is a field of mathematics that grows out of the traditional definitions of calculus. Fractional calculus has gained importance during the last decades mainly due to its applications in various areas of physics, biology, mathematics, and engineering. Some of the current application fields of fractional calculus include fluid flow, dynamical process in self-similar and porous structures, electrical networks, probability and statistics, control theory of dynamical systems, systems identification, acoustics, viscoelasticity, control theory, electrochemistry of corrosion, chemical physics, finance, optics, and signal processing [1–3].

There are several definitions of the fractional derivative which are generally not equivalent to each other. Some of these definitions are Sun and Chen’s fractal derivative [4, 5], Cresson’s derivative [6, 7], Grünwald-Letnikov’s fractional derivative [8], Riemann-Liouville’s derivative [8], and Caputo’s fractional derivative [9]. But the Riemann-Liouville derivative and the Caputo derivative are the most used ones.

Lately, both mathematicians and physicists have devoted considerable effort to the study of explicit solutions to nonlinear fractional differential equations. Many powerful methods have been presented. Among them are the fractional -expansion method [10–13], the fractional exp-function method [14–16], the fractional first integral method [17, 18], the fractional subequation method [19–22], the fractional functional variable method [23], the fractional modified trial equation method [24, 25],andthe fractional simplest equation method [26].

The paper suggests the functional variable method, the exp-function method, the -expansion method, and fractional complex transform to find the exact solutions of nonlinear fractional partial differential equation with the modified Riemann-Liouville derivative.

This paper is organized as follows. In Section 2, basic definitions of Jumarie’s Riemann-Liouville derivative are given; in Section 3, description of the methods for FDEs is given. Then, in Section 4, these methods have been applied to establish exact solutions for the space-time fractional symmetric regularized long wave (SRLW) equation. Conclusion is given in Section 5.

2. Jumarie’s Modified Riemann-Liouville Derivative

Recently, a new modified Riemann-Liouville derivative is proposed by Jumarie [27, 28]. This new definition of fractional derivative has two main advantages: firstly, comparing with the Caputo derivative, the function to be differentiated is not necessarily differentiable; secondly, different from the Riemann-Liouville derivative, Jumarie’s modified Riemann-Liouville derivative of a constant is defined to be zero. Jumarie’s modified Riemann-Liouville derivative of order is defined by where , denotes a continuous (but not necessarily first-order-differentiable) function. Some useful formulas and results of Jumarie’s modified Riemann-Liouville derivative can be found in [28, 29] which are direct consequences of the equality In the above formulas (3)–(5), is nondifferentiable function in (3) and (4) and differentiable in (5). The function is nondifferentiable, and is differentiable in (4) and nondifferentiable in (5). Because of these, the formulas (3)–(5) should be used carefully. The above equations play an important role in fractional calculus in Sections 3 and 4.

3. Description of the Methods for FDEs

We consider the following general nonlinear FDEs of the type where is an unknown function. is a polynomial of and its partial fractional derivatives, in which the highest order derivatives and the nonlinear terms are involved.

The fractional complex transform [30–32] is the simplest approach to convert the fractional differential equations into ordinary differential equations. This makes the solution procedure extremely simple. The traveling wave variable is where where , , and are nonzero arbitrary constants. We can rewrite (7) in the following nonlinear ODE: where the prime denotes the derivation with respect to . Now we consider three different methods.

3.1. Basic Idea of Functional Variable Method

The features of this method are presented in [33]. We describe functional variable method to find exact solutions of nonlinear space-time fractional differential equations as follows.

Let us make a transformation in which the unknown function is considered as a functional variable in the form and some successive derivatives of are where “” stands for . The ODE (10) can be reduced in terms of , , and its derivatives by using the expressions of (12) into (10) as

The key idea of this particular form (13) is of special interest since it admits analytical solutions for a large class of nonlinear wave type equations. Integrating (13) gives the expression of . This and (11) give the appropriate solutions to the original problem.

3.2. Basic Idea of Exp-Function Method

According to exp-function method, developed by He and Abdou [34], we assume that the wave solution can be expressed in the following form: where , , , and are positive integers which are known to be further determined and and are unknown constants. We can rewrite (14) in the following equivalent form:

This equivalent formulation plays an important and fundamental part in finding the analytic solution of problems. To determine the value of and , we balance the linear term of highest order of (10) with the highest order nonlinear term. Similarly, to determine the value of and , we balance the linear term of lowest order of (10) with lowest order nonlinear term [35–40].

3.3. Basic Idea of -Expansion Method

According to -expansion method, developed by Wang et al. [41], the solution of (10) can be expressed by a polynomial in as where are constants, while satisfies the following second-order linear ordinary differential equation: where and are constants. The positive integer can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in (10). By substituting (16) into (10) and using (17) we collect all terms with the same order of . Then by equating each coefficient of the resulting polynomial to zero, we obtain a set of algebraic equations for , , , , and . Finally solving the system of equations and substituting , , , , , , and the general solutions of (17) into (16), we can get a variety of exact solutions of (7) [42, 43].

4. Exact Solutions of Space-Time Fractional Symmetric Regularized Long Wave (SRLW) Equation

We consider the space-time fractional symmetric regularized long wave (SRLW) equation [44] which arises in several physical applications including ion sound waves in plasma. For , it is shown that this equation describes weakly nonlinear ion acoustic and space-charge waves, and the real-valued corresponds to the dimensionless fluid velocity with a decay condition [45].

We use the following transformations: where and are nonzero constants.

Substituting (20) with (1) into (18), equation (18) can be reduced into an ODE: where .

4.1. Exact Solutions by Functional Variable Method

Integrating (21) twice and setting the constants of integration to be zero, we obtain or

Then we use the transformation (11) and (12) to convert (22) to

The solution of (21) is obtained as So we have which is the exact solution of space-time fractional symmetric regularized long wave (SRLW) equation. One can see that the result is different than results of Alzaidy [44].

4.2. Exact Solutions by Exp-Function Method

Balancing the order of and in (22), we obtain where are determined coefficients only for simplicity. Balancing highest order of exp-function in (27) we have which leads to the result: In the same way, we balance the linear term of the lowest order in (22), to determine the values of and where are determined coefficients only for simplicity. From (30), we have and this gives

For simplicity, we set and , so (15) reduces to

Substituting (33) into (22) and using Maple, we obtain where

Solving this system of algebraic equations by using Maple, we get the following results: where and are arbitrary parameters. Substituting these results into (33), we get the following exact solution: where and are arbitrary parameters and .

Finally, if we take and , (37) becomes and we obtain the hyperbolic function solution of the space-time fractional symmetric regularized long wave (SRLW) equation. Comparing our result to the results in [46], it can be seen that our solution has never been obtained.

4.3. Exact Solutions by -Expansion Method

Recently, Zayed et al. [47] obtained solitary wave solutions to SRLW equation by means of improved -expansion method. But they applied this method to (22). Namely, they took the constants of integration as zero.

In our study, we integrate (21) twice with respect to and we get where and are constants of integration.

Use ansatz (39), for the linear term of highest order with the highest order nonlinear term . By simple calculation, balancing with in (39) gives so that

Suppose that the solutions of (41) can be expressed by a polynomial in as follows:

By using (17) and (42) we have

Substituting (42) and (43) into (39), collecting the coefficients of , and setting it to zero, we obtain the following system:

Solving this system by using Maple gives where , , , and are arbitrary constants.

By using (42), expression (45) can be written as

Substituting general solutions of (17) into (46) we have three types of travelling wave solutions of space-time fractional symmetric regularized long wave (SRLW) equation. These are the following.

When , where .

When ,

In particular, if , , , , then and become

Comparing our results to Zayed’s results [47], it can be seen that these results are new.

5. Conclusion

In this paper, the functional variable method, the exp-function method, and -expansion method have been successfully employed to obtain solution of the space-time fractional symmetric regularized long wave (SRLW) equation. These solutions include the generalized hyperbolic function solutions, generalized trigonometric function solutions, and rational function solutions, which may be very useful to understand the nonlinear FDEs and our result can turn into hyperbolic solution when suitable parameters are chosen. To the best of our knowledge, the solutions obtained in this paper have not been reported in literature. Maple has been used for programming and computations in this work.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY. USA, 1993.
View at: MathSciNet
I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
View at: MathSciNet
A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, Netherlands, 2006.
View at: MathSciNet
H. Sun and W. Chen, “Fractal derivative multi-scale model of fluid particle transverse accelerations in fully developed turbulence,” Science in China, Series E: Technological Sciences, vol. 52, no. 3, pp. 680–683, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
W. Chen and H. Sun, “Multiscale statistical model of fully-developed turbulence particle accelerations,” Modern Physics Letters B, vol. 23, no. 3, pp. 449–452, 2009.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
J. Cresson, “Scale calculus and the Schrödinger equation,” Journal of Mathematical Physics, vol. 44, no. 11, pp. 4907–4938, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
J. Cresson, “Non-differentiable variational principles,” Journal of Mathematical Analysis and Applications, vol. 307, no. 1, pp. 48–64, 2005.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993.
View at: MathSciNet
M. Caputo, “Linear models of dissipation whose Q is almost frequency independent II,” Geophysical Journal International, vol. 13, no. 5, pp. 529–539, 1967.
View at: Publisher Site | Google Scholar
B. Zheng, “ ${(G}^{'} / G)$ -expansion method for solving fractional partial differential equations in the theory of mathematical physics,” Communications in Theoretical Physics, vol. 58, no. 5, pp. 623–630, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
K. A. Gepreel and S. Omran, “Exact solutions for nonlinear partial fractional differential equations,” Chinese Physics B, vol. 21, no. 11, Article ID 110204, 2012.
View at: Publisher Site | Google Scholar
A. Bekir and Ö. Güner, “Exact solutions of nonlinear fractional differential equations by $(G^{'} / G)$ -expansion method,” Chinese Physics B, vol. 22, no. 11, Article ID 110202, 2013.
View at: Google Scholar
N. Shang and B. Zheng, “Exact solutions for three fractional partial differential equations by the $(G^{'} / G)$ method,” IAENG International Journal of Applied Mathematics, vol. 43, no. 3, pp. 114–119, 2013.
View at: Google Scholar | MathSciNet
S. Zhang, Q.-A. Zong, D. Liu, and Q. Gao, “A generalized exp-function method for fractional riccati differential equations,” Communications in Fractional Calculus, vol. 1, no. 1, pp. 48–51, 2010.
View at: Google Scholar
A. Bekir, Ö. Güner, and A. C. Cevikel, “Fractional complex transform and exp-function methods for fractional differential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 426462, 8 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
B. Zheng, “Exp-function method for solving fractional partial differential equations,” The Scientific World Journal, vol. 2013, Article ID 465723, 8 pages, 2013.
View at: Publisher Site | Google Scholar
B. Lu, “The first integral method for some time fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 395, no. 2, pp. 684–693, 2012.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
M. Eslami, B. F. Vajargah, M. Mirzazadeh, and A. Biswas, “Application of first integral method to fractional partial differential equations,” Indian Journal of Physics, vol. 88, no. 2, pp. 177–184, 2014.
View at: Google Scholar
S. Zhang and H.-Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
B. Zheng and C. Wen, “Exact solutions for fractional partial differential equations by a new fractional sub-equation method,” Advances in Difference Equations, vol. 2013, article 199, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
J. F. Alzaidy, “Fractional sub-equation method and its applications to the space—time fractional differential equations in mathematical physics,” British Journal of Mathematics & Computer Science, vol. 3, no. 2, pp. 153–163, 2013.
View at: Publisher Site | Google Scholar
H. Jafari, H. Tajadodi, N. Kadkhoda, and D. Baleanu, “Fractional subequation method for Cahn-Hilliard and Klein-Gordon equations,” Abstract and Applied Analysis, vol. 2013, Article ID 587179, 5 pages, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
A. Nazarzadeh, M. Eslami, and M. Mirzazadeh, “Exact solutions of some nonlinear partial differential equations using functional variable method,” Pramana—Journal of Physics, vol. 81, no. 2, pp. 225–236, 2013.
View at: Publisher Site | Google Scholar
H. Bulut, M. Baskonus H, and Y. Pandir, “The modified trial equation method for fractional wave equation and time fractional generalized burgers equation,” Abstract and Applied Analysis, vol. 2013, Article ID 636802, 8 pages, 2013.
View at: Publisher Site | Google Scholar
Y. Pandir, Y. Gurefe, and E. Misirli, “New exact solutions of the time-fractional nonlinear dispersive KdV equation,” International Journal of Modeling and Optimization, vol. 3, no. 4, pp. 349–352, 2013.
View at: Google Scholar
N. Taghizadeh, M. Mirzazadeh, M. Rahimian, and M. Akbari, “Application of the simplest equation method to some time-fractional partial differential equations,” Ain Shams Engineering Journal, vol. 4, no. 4, pp. 897–902, 2013.
View at: Publisher Site | Google Scholar
G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1367–1376, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,” Applied Mathematics Letters, vol. 22, no. 3, pp. 378–385, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
G. Jumarie, “Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann—Liouville derivative,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1659–1664, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Li and J. He, “Fractional complex transform for fractional differential equations,” Mathematical & Computational Applications, vol. 15, no. 5, pp. 970–973, 2010.
View at: Google Scholar | MathSciNet
Z. B. Li and J. H. He, “Application of the fractional complex transform to fractional differential equations,” Nonlinear Science Letters A: Mathematics, Physics and Mechanics, vol. 2, pp. 121–126, 2011.
View at: Google Scholar
J. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,” Physics Letters A, vol. 376, no. 4, pp. 257–259, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
A. Zerarka, S. Ouamane, and A. Attaf, “On the functional variable method for finding exact solutions to a class of wave equations,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 2897–2904, 2010.
View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
J. He and M. A. Abdou, “New periodic solutions for nonlinear evolution equations using Exp-function method,” Chaos, Solitons & Fractals, vol. 34, no. 5, pp. 1421–1429, 2007.
View at: Publisher Site | Google Scholar | MathSciNet
J. He and X. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006.
View at: Publisher Site | Google Scholar | Zentralblatt MATH
A. Bekir and A. Boz, “Application of He's exp-function method for nonlinear evolution equations,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2286–2293, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
A. Ebaid, “An improvement on the Exp-function method when balancing the highest order linear and nonlinear terms,” Journal of Mathematical Analysis and Applications, vol. 392, no. 1, pp. 1–5, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
I. Aslan, “On the application of the Exp-function method to the KP equation for $N$ -soliton solutions,” Applied Mathematics and Computation, vol. 219, no. 6, pp. 2825–2828, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
S. Yu, “ $N$ -soliton solutions of the KP equation by Exp-function method,” Applied Mathematics and Computation, vol. 219, no. 8, pp. 3420–3424, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
S. Zhang and H. Zhang, “An Exp-function method for a new {$N$}-soliton solutions with arbitrary functions of a {$(2+1)$}-dimensional vcBK system,” Computers & Mathematics with Applications, vol. 61, no. 8, pp. 1923–1930, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
M. Wang, X. Li, and J. Zhang, “The $(G^{ʹ} / G)$ -expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters. A, vol. 372, no. 4, pp. 417–423, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
A. Bekir, “Application of the $(G^{ʹ} / G)$ -expansion method for nonlinear evolution equations,” Physics Letters A, vol. 372, no. 19, pp. 3400–3406, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
H. Jafari, N. Kadkhoda, and A. Biswas, “The $G^{ʹ} / G$ -expansion method for solutions of evolution equations from isothermal magnetostatic atmospheres,” Journal of King Saud University-Science, vol. 25, no. 1, pp. 57–62, 2013.
View at: Publisher Site | Google Scholar
J. F. Alzaidy, “The fractional sub-equation method and exact analytical solutions for some nonlinear fractional PDEs,” American Journal of Mathematical Analysis, vol. 1, no. 1, pp. 14–19, 2013.
View at: Publisher Site | Google Scholar
H. Jafari, A. Borhanifar, and S. A. Karimi, “New solitary wave solutions for generalized regularized long-wave equation,” International Journal of Computer Mathematics, vol. 87, no. 1–3, pp. 509–514, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
F. Xu, “Application of Exp-function method to symmetric regularized long wave (SRLW) equation,” Physics Letters A, vol. 372, no. 3, pp. 252–257, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
E. M. E. Zayed, Y. A. Amer, and R. M. A. Shohib, “Exact traveling wave solutions for nonlinear fractional partial differential equations using the improved (G′/G)-expansion method,” International Journal of Engineering and Applied Science, vol. 7, pp. 18–31, 2014.
View at: Google Scholar

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Copyright © 2014 Özkan Güner and Dursun Eser. 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.

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