Journal of Applied Mathematics

Volume 2013, Article ID 482419, 12 pages

http://dx.doi.org/10.1155/2013/482419

## Numerical Solutions for the Time and Space Fractional Nonlinear Partial Differential Equations

^{1}Mathematics Department, Faculty of Science, Taif University, Taif, Saudi Arabia^{2}Mathematics Department, Faculty of Science, Zagazig University, Zagazig, Egypt^{3}Mathematics Department, Faculty of Science, El-Minia University, El-Minia, Egypt

Received 27 June 2013; Accepted 24 September 2013

Academic Editor: Mehmet Sezer

Copyright © 2013 Khaled A. Gepreel et al. 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 implement relatively analytical techniques, the homotopy perturbation method, and variational iteration method to find the approximate solutions for time and space fractional Benjamin-Bona Mahony equation. The fractional derivatives are described in the Caputo sense. These methods are used in applied mathematics to obtain the analytic approximate solutions for the nonlinear Bejamin-Bona Mahoney (BBM) partial fractional differential equation. We compare between the approximate solutions obtained by these methods. Also, we present the figures to compare between the approximate solutions. Also, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved -expansion function method to find exact solutions of nonlinear fractional BBM equation.

#### 1. Introduction

In recent years, there has been a great deal of interest in fractional differential equations. First there were almost no practical applications of fractional calculus, and it was considered by many as an abstract area containing only mathematical manipulations of little or no use. Nearly 30 years ago, the paradigm began to shift from pure mathematical formulations to applications in various fields. During the last decade, fractional calculus has been applied to almost every field of science, engineering, and mathematics. Several fields of application of fractional differentiation and fractional integration are already well established, and some others have just started. Many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, and astrophysics [1–11]. There has been some attempt to solve linear problems with multiple fractional derivatives (the so-called multiterm equations) [1, 12]. Not much work has been done for nonlinear problems, and only a few numerical schemes have been proposed to solve nonlinear fractional differential equations. More recently, applications have included classes of nonlinear equation with multiorder fractional derivative, and this motivates us to develop a numerical scheme for their solutions [13]. Numerical and analytical methods have included Adomian decomposition method (ADM) [14–17], variational iteration method (VIM) [18–20], homotopy perturbation method [21–24], homotopy analysis method [25–27], and the fractional complex transformation [28, 29] to get some special exact solutions for nonlinear partial fractional differential equation.

Recently, Zhao [30] investigated the nature of protein dynamics and thermodynamics. Olszewski [31] studied the uncertainty relation between intervals of energy and time derived for the electromagnetic radiation of a harmonic oscillator. El-Naggar et al. [32] investigated the initial stress, magnetic field, voids, and rotation effects on plane waves in generalized thermoelasticity. Khrennikov [33] discussed “Einstein’s Dream” Quantum Mechanics as Theory of Classical Random Fields.

The main objective of this paper is to use two different methods such as homotopy perturbation method and variational iteration method for calculating the analytic approximate solutions of the Bejamin-Bona Mahoney equation. We compare between the approximate solutions which obtained from these methods. We make the figures to compare the approximate solutions. Also, we find some exact solutions to time and space fractional Bejamin-Bona Mahoney equation by using the improved -expansion function method.

#### 2. Preliminaries and Notations

We give some basic definitions and properties of the fractional calculus theory which are used further in this paper [1, 2]. For the finite derivative [a,b], we define the following fractional integral and derivatives.

*Definition 1. *If , the set of all integrable functions, and then, the Riemann-Liouville fractional integral of order , denoted by , is defined by

*Definition 2. *For , the Caputo fractional derivative of order , denoted by , is defined by
where is such that and .

If is an integer, then this derivative takes the ordinary derivative

Finally the Caputo fractional derivative on the whole space is defined by the following.

*Definition 3. *For , the Caputo fractional derivative of order on the whole space, denoted by , is defined by

In recent years, in order to investigate the local behavior of fractional models, several local versions of fractional derivatives have been proposed, that is, the Kolwankar-Gangal local fractional derivative [35], Chen’s fractal derivative [36], Cresson’s derivative [37], and Jumarie’s modified Riemann-Liouville derivative [38, 39]. Jumarie’s derivative is defined as where , denotes a continuous (but not necessarily first-order-differentiable) function. We can obtain the following properties.

*Property 1. *Let satisfy the definition of the modified Riemann-Liouville derivative, and let be a -order differentiable function. The generalized Taylor series is given by [38, 40]

*Property 2. *Assume that denotes a continuous function. We use the following equality for the integral with respect to [39, 40]:

*Property 3. *Some useful formulas include

Function should be differentiable with respect to , and is fractional differentiable in (8). The above results are employed in the following sections. The modified Riemann-Liouville derivative has been successfully applied in the probability calculus [41], the fractional Laplace problems [42], the fractional variational approach with several variables [43], the fractional variational iteration method [44], the fractional variational approach with natural boundary conditions [45], and the fractional Lie group method [46].

#### 3. Basic Idea of the Homotopy Perturbation Method (HPM)

In this section, we illustrate some basic concepts of the homotopy perturbation method for the following nonlinear partial fractional differential equation [21–24]: subject to the initial and boundary conditions where is a linear operator, while is a nonlinear operator, is a known analytical function, and denotes the fractional derivative in the Caputo sense. The solution is assumed to be a causal function of time, that is, vanishing for . Also, is the th derivative of , are the specified initial conditions, and is a boundary operator. Using He’s homotopy perturbation technique [47], we can construct the following simple homotopy: or

The homotopy parameter always changes from zero to unity. In the case of , (13) or (14) becomes and when , (13) or (14) turns out to be the original fractional differential equation.

Applying the homotopy perturbation method, we use the homotopy parameter to expand the solution into the following form:

For nonlinear problems, let us set . Substituting (16) into (13) or (14) and equating the terms with identical powers of , we can obtain a series of equations of the form and so on, where the functions satisfy the following equations:

Applying the operator on both sides of (17) and considering the initial and boundary conditions, the terms of the series solution can be given by

On setting , we get an accurate approximate solution in the following form:

#### 4. Basic Idea of Variational Iteration Method (VIM)

The basic concepts of the variational iteration method can be expressed as follows. Consider the differential equation of the form where is a linear operator, is a nonlinear operator, and is the inhomogeneous term. According to VIM, a correction functional for (23) can be constructed as follows: where is a general Lagrange multiplier, which can be identified optimally via the variational theory, the subscript denotes the th approximations, and is considered as restricted variation; that is, . The successive approximations , , of the solution can be obtained after finding the Lagrange multiplier and by using the selective function which is usually selected from initial conditions. Consequently, the solution is obtained as

#### 5. Approximate Solutions for the Time and Space Fractional BBM Equation

In this section, we use the homotopy perturbation method and variational iteration method to calculate the approximate solutions for the time and space fractional Bejamin-Bona Mahoney equation. Also, we compare the approximate solutions obtained from these methods.

##### 5.1. The HPM for the Time and Space Fractional Nonlinear BBM Equation

In this subsection, we use the HPM to construct the approximate solution of the time and space fractional nonlinear BBM equation in the form where , , and are constants.

With the initial condition,

By the homotopy perturbation technique, we construct a homotopy which satisfies

According to the homotopy perturbation method, we can first use the embedding parameter as a small parameter and assume that the solution of (28) can be written as a power series in as follows:

Substituting (29) into (28) and arranging the coefficients of the powers of , after some calculation we obtain and so on. We substitute the initial condition (27) into the system (30), then the approximate solutions of (26) take the following form: and so on. Thus the approximate solution of (26) is given by where

##### 5.2. The VIM for the Time-Space Fractional Nonlinear BBM Equation

In this subsection, we use the VIM to construct the approximate solution of the time and space fractional nonlinear BBM equation (26).

The correction functional for (26) can be approximately expressed as follows: where is a general Lagrange multiplier, is considered as restricted variations, and is considered as a restricted variation. Making the above correction functional stationary and noticing that , we obtain Therefore, it can be identified as , and the following variational iteration formula can be obtained:

By the iteration formula (36), we can obtain the other components as and so on, and in the same manner, the rest of components of the iteration formula (36) can be obtained using Mathematica package where

The comparison between the approximate solutions which obtained, by using HPM (32) and VIM (38) is shown in Table 1.

Now we make the comparison between the approximate solutions obtained by the variation iteration method and the approximate solution obtained by the homotopy perturbation method as shown in Figures 1, 2, 3, 4, and 5.

*Remark 4. *From Figures 2–5, the approximate solution obtained by the homotopy perturbation method is different from the approximate solutions obtained by variational iteration when and are not positive integers.

#### 6. The Improved -Expansion Function Method

Consider the following nonlinear partial fractional differential equation: where is an unknown function and is a polynomial of and its partial fractional derivatives, in which the highest order derivatives and the nonlinear terms are involved.

In the following, we give the main steps of the improved -expansion method.

*Step 1. *Li and He [48] proposed a fractional complex transformation to convert fractional differential equations into ordinary differential equations (ODE), so all analytical methods devoted to the advanced calculus can be easily applied to the fractional calculus. The traveling wave variable
where , , , and are nonzero arbitrary constants, permits us to reduce (40) to an ODE of in the form

If possible, we should integrate (43) term by term one or more times.

*Step 2. *Suppose that the solution of (43) can be expressed as a polynomial of in the form [49–51]
where are constants, while satisfies the following second-order linear ODE:
with and being constants.

*Step 3. *The positive integer can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in (43). More precisely, we define the degree of as , which gives rise to the degrees of the other expressions as follows:

Therefore, we can obtain the value of in (44).

*Step 4. *Substituting (44) into (43), using (45), collecting all terms with the same order of together and equating each coefficient of the resulting polynomial to zero, we obtain a set of algebraic equations for , , , , , , and .

*Step 5. *Since the general solutions to (45) are well known, then substituting , , , , , , , and the general solutions of (45) into (44), we get more traveling wave solutions of the nonlinear partial fractional derivatives (40).

#### 7. Applications

In this section, we use the improved -expansion method to construct the exact solutions for some nonlinear partial fractional differential equations, via the time-space fractional nonlinear BBM equation.

We can see that the fractional complex transformation where and are constants, permits us to reduce (26) to an ordinary differential equation. After that, we integrate to obtain where is the integration constant.

Considering the homogeneous balance between the highest order derivative and the nonlinear term in (48), we deduce that . Thus, we get where , , , , and are arbitrary constants to be determined later. Substituting (49) into (48), collecting all the terms of powers of , and setting each coefficient to zero, we get a system of algebraic equations. With the aid of Maple or Mathematica, we can solve this system of algebraic equations to obtain the following sets of solutions.

*Case** 1*
where , , , , , and are arbitrary constants. Substituting (50) into (49) yields

From the solutions of (45) and (51), we deduce the following types of traveling wave solutions for (26).

*Family 1. *If , then we have the hyperbolic solution

*Family 2. *If , then we have the trigonometric solution

*Family 3. *If , then we have
where

*Case** 2*
where , , , , and are arbitrary constants. Substituting (56) into (49) yields

From the solutions of (45) and (57), we deduce the following types of traveling wave solutions for (26).

*Family 1. *If , then we have the hyperbolic solution

*Family 2. *If , then we have the trigonometric solution

*Family 3. *If , then we have
where

#### 8. Conclusion

In this paper, we use the homotopy perturbation method and variational iteration method to obtain the analytic approximate solutions for linear and nonlinear partial fractional differential equations. Section 5 shows the comparative study between the factional variational itearation methods and the homotopy perturbation method. The two methods are powerful and effective tools for the solutions of fractional partial differential equations, and both give two different approximate solutions of higher accuracy and closed form solutions if existing. Also, we use the improved -expansion function method to calculate the exact solutions for the time and space fractional nonlinear BBM equation. This method is reliable and simple and gives many new exact solutions for the nonlinear fractional differential equations. When the parameters take certain special values, the solitary waves are derived from the traveling waves. This method is very efficient and powerful in finding the exact solutions for the nonlinear fractional differential equations.

#### References

- I. Podlubny,
*Fractional Differential Equation*, Academic Press, London, UK, 1999. - S. G. Samko, A. A. Kilbas, and O. I. Marichev,
*Fractional Integrals and Derivatives: Theory and Applications*, Gordon and Breach, Langhorne, Pennsylvania, 1993. View at MathSciNet - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204 of*North-Holland Mathematical Studies*, Elsevier, Amsterdam, The Netherlands, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - A. M. A. El-Sayed, “Fractional-order diffusion-wave equation,”
*International Journal of Theoretical Physics*, vol. 35, no. 2, pp. 311–322, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. E. Herzallah, A. M. A. El-Sayed, and D. Baleanu, “On the fractional-order diffusion-wave process,”
*Romanian Journal of Physics*, vol. 55, no. 3-4, pp. 274–284, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. E. Herzallah, S. I. Muslih, D. Baleanu, and E. M. Rabei, “Hamilton-Jacobi and fractional like action with time scaling,”
*Nonlinear Dynamics*, vol. 66, no. 4, pp. 549–555, 2011. View at Publisher · View at Google Scholar · View at Scopus - R. L. Magin,
*Fractional Calculus in Bioengineering*, Begell House Publisher, West Redding, Conn, USA, 2006. - B. J. West, M. Bologna, and P. Grigolini,
*Physics of Fractal Operators*, Springer, New York, NY, USA, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - I. S. Jesus and J. A. Tenreiro MacHado, “Fractional control of heat diffusion systems,”
*Nonlinear Dynamics*, vol. 54, no. 3, pp. 263–282, 2008. View at Publisher · View at Google Scholar · View at Scopus - O. P. Agrawal and D. Baleanu, “A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems,”
*Journal of Vibration and Control*, vol. 13, no. 9-10, pp. 1269–1281, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. E. Tarasov, “Fractional vector calculus and fractional Maxwell's equations,”
*Annals of Physics*, vol. 323, no. 11, pp. 2756–2778, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,”
*Bulletin of Science, Technology and Society*, vol. 15, no. 2, pp. 86–90, 1999. View at Google Scholar - V. S. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 13, no. 8, pp. 1642–1654, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Daftardar-Gejji and S. Bhalekar, “Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 202, no. 1, pp. 113–120, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Daftardar-Gejji and H. Jafari, “Solving a multi-order fractional differential equation using Adomian decomposition,”
*Applied Mathematics and Computation*, vol. 189, no. 1, pp. 541–548, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. E. Zayed, T. A. Nofal, and K. A. Gepreel, “Homotopy perturbation and Adomain decomposition methods for solving nonlinear Boussinesq equations,”
*Communications on Applied Nonlinear Analysis*, vol. 15, no. 3, pp. 57–70, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. E. Herzallah and K. A. Gepreel, “Approximate solution to the time-space fractional cubic nonlinear Schrodinger equation,”
*Applied Mathematical Modelling*, vol. 36, no. 11, pp. 5678–5685, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,”
*Physics Letters A*, vol. 371, no. 1-2, pp. 26–33, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. He, “Preliminary report on the energy balance for nonlinear oscillations,”
*International Journal of Non-Linear Mechanics*, vol. 34, no. 4, pp. 699–708, 1999. View at Google Scholar - G.-C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,”
*Physics Letters A*, vol. 374, no. 25, pp. 2506–2509, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “Homotopy perturbation method for solving boundary value problems,”
*Physics Letters A*, vol. 350, no. 1-2, pp. 87–88, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Golbabai and K. Sayevand, “Fractional calculus: a new approach to the analysis of generalized fourth-order diffusion-wave equations,”
*Computers & Mathematics with Applications*, vol. 61, no. 8, pp. 2227–2231, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Golbabai and K. Sayevand, “The homotopy perturbation method for multi-order time fractional differential equations,”
*Nonlinear Science Letters A*, vol. 1, pp. 147–154, 2010. View at Google Scholar - K. A. Gepreel, “The homotopy perturbation method applied to the nonlinear fractional Kolmogorov-Petrovskii-Piskunov equations,”
*Applied Mathematics Letters*, vol. 24, no. 8, pp. 1428–1434, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. A. Gepreel and M. S. Mohamed, “Analytical approximate solution for nonlinear space-time fractional Klein-Gordon equation,”
*Chinese Physics B*, vol. 22, no. 1, Article ID 010201, 2013. View at Google Scholar - S. J. Liao,
*The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problem [Ph.D. thesis]*, Shanghai Jiao Tong University, 1992. - S. J. Liao, “An approximate solution technique not depending on small parameters: a special example,”
*International Journal of Non-Linear Mechanics*, vol. 30, no. 3, pp. 371–380, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. He, “Asymptotic methods for solitary solutions and compactons,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 916793, 130 pages, 2012. View at Publisher · View at Google Scholar - K. A. Gepreel and S. Omran, “Exact solutions for nonlinear partial fractional differential equations,”
*Chinese Physics B*, vol. 21, Article ID 110204, 2012. View at Google Scholar - Q. Zhao, “Nature of protein dynamics and thermodynamics,”
*Reviews in Theoretical Science*, vol. 1, pp. 83–101, 2013. View at Google Scholar - S. Olszewski, “A look on the scale of time useful in non-relativistic quantum mechanics,”
*Quantum Matter*, vol. 2, pp. 481–483, 2013. View at Google Scholar - A. M. El-Naggar, Z. Kishka, A. M. Abd-Alla, I. A. Abbas, S. M. Abo-Dahab, and M. Elsagheer, “On the initial stress, magnetic field, voids and rotation effects on plane waves in generalized thermoelasticity,”
*Journal of Computational and Theoretical Nanoscience*, vol. 10, pp. 1408–1417, 2013. View at Google Scholar - A. Khrennikov, “‘Einstein's Dream’-quantum mechanics as theory of classical random fields,”
*Reviews in Theoretical Science*, vol. 1, pp. 34–57, 2013. View at Google Scholar - R. Asghari, “Application of the homotopy perturbation method to the modified BBM equation,”
*Middle East Journal*, vol. 10, no. 2, pp. 274–276, 2011. View at Google Scholar - K. M. Kolwankar and A. D. Gangal, “Local fractional Fokker-Planck equation,”
*Physical Review Letters*, vol. 80, no. 2, pp. 214–217, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Chen and H. G. Sun, “Multiscale statistical model of fully-developed turbulence particle accelerations,”
*Modern Physics Letters B*, vol. 23, no. 3, article 449, 2009. View at Google Scholar - J. Cresson, “Non-differentiable variational principles,”
*Journal of Mathematical Analysis and Applications*, vol. 307, no. 1, pp. 48–64, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “Lagrange characteristic method for solving a class of nonlinear partial differential equations of fractional order,”
*Applied Mathematics Letters*, vol. 19, no. 9, pp. 873–880, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G.-C. Wu, “A fractional characteristic method for solving fractional partial differential equations,”
*Applied Mathematics Letters*, vol. 24, no. 7, pp. 1046–1050, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations,”
*Mathematical and Computer Modelling*, vol. 44, no. 3-4, pp. 231–254, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Almeida, A. B. Malinowska, and D. F. M. Torres, “A fractional calculus of variations for multiple integrals with application to vibrating string,”
*Journal of Mathematical Physics*, vol. 51, no. 3, Article ID 033503, p. 12, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - K. A. Gepreel and A. A. Al-thobaiti, “Exact solutions of nonlinear partial fractional sub-equation method,”
*Indian Journal of Physics*, 2014. View at Publisher · View at Google Scholar - A. B. Malinowska, M. R. Sidi Ammi, and D. F. M. Torres, “Composition functionals in fractional calculus of variations,”
*Communications in Fractional Calculus*, vol. 1, no. 1, pp. 32–40, 2010. View at Google Scholar - G.-C. Wu, “A fractional lie group method for anomalous diffusion equations,”
*Communications in Fractional Calculus*, vol. 1, pp. 27–31, 2010. View at Google Scholar - J. H. He, “Some asymptotic methods for strongly nonlinear equations,”
*International Journal of Modern Physics B*, vol. 20, no. 10, pp. 1141–1199, 2006. View at Google Scholar - Z.-B. Li and J.-H. He, “Fractional complex transform for fractional differential equations,”
*Mathematical & Computational Applications*, vol. 15, no. 5, pp. 970–973, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Wang, X. Li, and J. Zhang, “The $({G}^{\text{'}}/G)$-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 · View at Google Scholar · View at MathSciNet - E. M. E. Zayed and K. A. Gepreel, “The $({G}^{\text{'}}/G)$-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics,”
*Journal of Mathematical Physics*, vol. 50, no. 1, p. 013502, 12, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - H. Zhang, “New exact complex travelling wave solutions to nonlinear Schrödinger (NLS) equation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 3, pp. 668–673, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet