Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

The (G’/G)-Expansion Method and Traveling Wave Solutions for Some Nonlinear PDEs

J.F. Alzaidy

Mathematics Department, Faculty of Science, Taif University, Kingdom of Saudi Arabia

Abstract: In the present paper, we construct the traveling wave solutions involving parameters of some nonlinear PDEs in mathematical physics via the nonlinear SchrÖdinger (NLS−) equation and the regularized long-wave (RLW) equation by using a simple method which is called the (G’/G) -expansion method, where G=G(ζ) satisfies the second order linear ordinary differential equation. When the parameters are taken special values, the solitary waves are derived from the traveling waves. The traveling wave solutions are expressed by hyperbolic, trigonometric and rational functions. This method is more powerful and will be used in further works to establish more entirely new solutions for other kinds of nonlinear PDEs arising in mathematical physics.

Keywords: the (G’/G) -expansion method, traveling wave solutions, nonlinear SchrÖdinger (NLS−) equation, regularized long-wave (RLW) equation, solitary wave solutions

Received April 18, 2013; Revised May 18, 2013; Accepted May 19, 2013

Citation: J.F. Alzaidy, The (G’/G)-Expansion Method and Traveling Wave Solutions for Some Nonlinear PDEs. American Journal of Mathematical Analysis, 1 (3). 28-32. January 2013. doi:10.12691/ajma-1-3-1

1. Introduction

In recent years, the exact solutions of nonlinear PDEs have been investigated by many authors ( see for example [1-30] ) who are interested in nonlinear physical phenomena. Manäcklund transformation [2], Darboux transformation [3], Hirota bilinear method [4], variable separation approach [5], various tanh methods [6,7,8,9], homogeneous balance method [10] , similarity reductions method [11,12] , the reduction mKdV equation method [13], the tri-function method [14,15], the projective Riccati equation method [16], the Weierstrass elliptic function method [17], the Sine- Cosine method [18,19], the Jacobi elliptic function expansion [20,21], the complex hyperbolic function method [22], the truncated Painleve’ expansion [23], the F-expansion method [24], the rank analysis method [25] and so on.

In the present paper, we shall use a simple method which is called the -expansion method [26,27]. This method is firstly proposed by the Chinese Mathematicians Wang et al [28] for which the traveling wave solutions of nonlinear equations are obtained. The main idea of this method is that the traveling wave solutions of nonlinear equations can be expressed by a polynomial in , where satisfies the second order linear ordinary differential equation where ,where and c are constants. The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives and the nonlinear terms appearing in the given nonlinear equations .The coefficients of this polynomial can be obtained by solving a set of algebraic equations resulted from the process of using the proposed method. This new method will play an important role in expressing the traveling wave solutions for nonlinear evolution equations via the NLS equation and the RLW equation in terms of hyperbolic, trigonometric and rational functions.

2. Description of the (G’/G)-Expansion Method

Suppose we have the following nonlinear PDE:

[picture] | (1)

where is an unknown function, P is a polynomial in and its various partial derivatives in which the highest order derivatives and nonlinear terms are involved. In the following we give the main steps of a deformation method :

Setp1. The traveling wave variable

[picture] | (2)

where k and c are the wave number and the wave speed, respectively. Permits us reducing Equation.(1) to an ODE for in the form

[picture] | (3)

where ,

Setp2. Suppose that the solution Equation.(3) can be expressed by a polynomial in as follows:

[picture] | (4)

while satisfies the second order linear differential equation in the form:

[picture] | (5)

where and are constants to be determined later.

Setp3. The positive integer “n” can be determined by considering the homogeneous balance between the highest derivative term and the nonlinear terms appearing in Equation.(3). Therefore, we can get the value of n in Equation.( 4).

Setp4. Substituting Equation.( 4) into Equation.( 3) and using Equation.(5), collecting all terms with the same power of together and then equating each coefficient of the resulted polynomial to zero, yield a set of algebraic equations for and k.

Setp5. Solving the algebraic equations by use of Maple or Mathematica, we obtain values for and k.

Setp6. Since the general solutions of Equation. (5) have been well known for us, then substituting the obtained coefficients and the general solution of Equation.(5) into Equation. (4), we have the travelling wave solutions of the nonlinear PDE (1).

3. Applications of the Method

In this section, we apply the -expansion method to construct the traveling wave solutions for some nonlinear PDEs, namely the equation and the RLW equation which are very important nonlinear evolution equations in mathematical physics and have been paid attention by many researchers.

3.1. Example1. The NLS Equationuation

We start with the following equation [29] in the form:

[picture] | (6)

where p is a positive constant.

Now we convert Equation.(6) into the real form :

[picture] | (7)

The traveling wave variable (2) permits us converting Equation.(7) into the following ODEs.

[picture] | (8)

Suppose that the solution of ODEs(8) can be expressed by polynomial in terms of as follows:

[picture] | (9)

where and are arbitrary constants, while satisfies the second order linear ODE (5).Considering the homogeneous balance between the highest order derivatives and the nonlinear terms in Equation.(9), we get . Thus, we have

[picture] | (10)

where and are constants to be determined later. Substituting Equation.(10) with Equation. (5) into Equation.(8) and collecting all terms with the same power of . Setting each coefficients of this polynomial to be zero, we have the following system of algebraic equations:

[picture] | (11)

On solving the above algebraic Equations.(11) by using the Maple or Mathematica, we have:

[picture] | (12)

where and .

Substituting Equation.(12) into Equation.(10) yields

[picture] | (13)

where

[picture] | (14)

On solving Equation.( 5), we deduce that

[picture] | (15)

where A and B are arbitrary constants and .

On substituting Equation.(15) into Equation.(13), we deduce the following two types of traveling wave solutions:

Case 1.  then we have the hyperbolic solution

[picture] | (16)

In this case the hyperbolic solutions of Equation.(6) take the following form:

[picture] | (17)

Case 2.  then we have the trigonometric solution.

[picture] | (18)

In this case the hyperbolic solutions of Equation.(6) take the following form:

[picture] | (19)

In particular, if we set in Equation.(16), then we get

[picture] | (20)

The traveling wave solution of Equation.(6) take the following form:

[picture] | (21)

If then we deduce that:

[picture] | (22)

The traveling wave solution of Equation.(6) take the following form:

[picture] | (23)

where .The solutions (21) and (23) represent the solitary wave solutions of Equation.(6).

3.2. Example 2. The RLW Equationuation

In this subsection, we study the RLW equation [30] in the form :

[picture] | (24)

where and are real constants. The regularized-long-wave (RLW) equation was first obtained by Peregrine [3l] to describe the development of an undular bore; i.e, a smooth solitary wave that is observed to propagate in shallow water channels. Since then, the RLW equation has been used as a one-dimensional model for drift waves in plasmas . It is also appropriate to describe Rossby waves in geophysics. Let us now solve Equation.(24) by the proposed method. To this end, we see that the traveling wave variable (2) permits us converting Equation.(24) into the following ODE:

[picture] | (25)

integrating Equation.(25) with respect to once, yields

[picture] | (26)

where is an integration constant.

Considering the homogeneous balance between highest order derivatives and nonlinear terms in Equation.(26), we get .Thus,we have

[picture] | (27)

where and are arbitrary constants to be determined later. Substituting Equation.(27) with Equation.(5) into Equation.(26) , collecting all terms with the same power of and setting them to zero, we have the following system of algebraic equations:

[picture] | (28)

Solving the above algebraic equations (28) by using the Maple or Mathematica, yields.

[picture] | (29)

Substituting Equation.(29) into Equation.(27) we obtain

[picture] | (30)

where

[picture] | (31)

From Equation.(15) and Equation.(30), we deduce the following three types of traveling wave solutions:

Case 1.  then we have the hyperbolic solution

[picture] | (32)

Case 2.  then we have the trigonometric solution

[picture] | (33)

Case 3.  then we have the rational solution

[picture] | (34)

In particular if and then we deduce from Equation.(32) that:

[picture] | (35)

while, if then we deduce that:

[picture] | (36)

where . The solutions (35) and (38) represent the solitary wave solutions of Equation.(24).

4. Conclusion

In this paper, we have seen that three types of traveling wave solutions in terms of hyperbolic, trigonometric and rational functions for the equation and the RLW equation are successfully found out by using the -expansion method. From our results obtained in this paper, we conclude that the -expansion method is powerful, effective and convenient. The performance of this method is reliable, simple and gives many new solutions. The -expansion method has more advantages: It is direct and concise. Also, the solutions of the proposed nonlinear evolution equations in this paper have many potential applications in physics and engineering. Finally, this method provides a powerful mathematical tool to obtain more general exact solutions of a great many nonlinear PDEs in mathematical physics.

References

[1]   | M. J. Ablowitz and P.A. Clarkson, Soliton, Nonlinear Evolution Equationuations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.
[2]   | C. H. Gu et al., Soliton Theory and Its Application, Zhejiang Science and Technology Press, Zhejiang, 1990.
[3]   | V. B. Matveev and M.A. Salle, Darboux Transformation and Soliton, Springer, Berlin, 1991.
[4]   | R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, 2004.
[5]   | S.Y. Lou, J.Z. Lu, Special solutions from variable separation approach: Davey-Stewartson equation, J. Phys. A: Math. Gen. 29 (1996) 4209.
[6]   | E. J. Parkes and B.R. Duffy, Travelling solitary wave solutions to a compound KdV-Burgers equation, Phys. Lett. A 229 (1997) 217.
[7]   | E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212.
[8]   | Z. Y. Yan, New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A 292 (2001) 100.
[9]   | Y. Chen, Y. Zheng, Generalized extended tanh-function method to construct new explicit exact solutions for the approximate equations for long water waves, Int. J. Mod. Phys. C 14 (4) (2003) 601.
[10]   | M. L. Wang, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Phys. Lett. A 216 (1996) 67.
[11]   | G. W. Bluman and S. Kumei, Symmetries and Differential Equationuations, Springer-Verlag, New York, 1989.
[12]   | P. J. Olver, Applications of Lie Groups to Differential Equationuations, Springer-Verlag, New York, 1986.
[13]   | Z. Y. Yan, A reduction mKdV method with symbolic computation to constract new doubly- periodic solutions for nonlinear wave equations, Int. J. Mod. Phys. C, 14 (2003) 661.
[14]   | Z. Y. Yan, The new tri-function method to multiple exact solutions of nonlinear wave equations, Physica Scripta, 78 (2008) 035001.
[15]   | Z. Y. Yan, Periodic, solitary and rational wave solutions of the 3D extended quantum Zakharov– Kuznetsov equation in dense quantum plasmas, Physics Letters A, 373 (2009) 2432.
[16]   | D. C. Lu and B. J. Hong, New exact solutions for the (2+1)-dimensional Generalized Broer-Kaup system, Appl. Math. Comput, 199(2008)572.
[17]   | A. V. Porubov, Periodical solution to the nonlinear dissipative equation for surface waves in a convecting liquid, Phys. Lett. A, 221 (1996) 391.
[18]   | M. Wazwaz, The tanh and sine- cosine method for compact and noncompact solutions of nonlinear Klein Gordon equation, Appl. Math. Comput, 167 (2005)1179.
[19]   | Z. Y. Yan and H. Q. Zhang, New explicit solitary wave solutions and periodic wave solutions for Whitham–Broer–Kaup equation in shallow water, Phys. Lett. A, 285 (2001) 355.
[20]   | D. C. Lu, Jacobi elliptic functions solutions for two variant Boussinesq equations, Chaos, Solitons and Fractals, 24 (2005) 1373.
[21]   | Z. Y. Yan, Abundant families of Jacobi elliptic functions of the (2+1) dimensional integrable Davey- Stawartson-type equation via a new method, Chaos, Solitons and Fractals, 18 (2003) 299.
[22]   | C. L. Bai and H. Zhao, Generalized method to construct the solitonic solutions to (3+1)- dimensional nonlinear equation, Phys. Lett. A, 354 (2006) 428.
[23]   | F. Cariello and M. Tabor, Similarity reductions from extended Painleve’ expansions for nonintegrable evolution equations, Physica D, 53 (1991) 59.
[24]   | M. Wang and X. Li, Extended F-expansion and periodic wave solutions for the generalized Zakharov equations, Phys. Lett. A, 343 (2005) 48.
[25]   | X. Feng, Exploratory approach to explicit solution of nonlinear evolution equations, Int. J. Theor. Phys, 39 (2000) 222.
[26]   | E. M. E. Zayed and K.A.Gepreel, The (G’/G) −expansion method for finding traveling wave solutions of nonlinear PDEs in mathematical physics, J. Math. Phys., 50 (2009) 013502.
[27]   | A. Bekir, Application of the (G’/G)− expansion method for nonlinear evolution equations, Phys. Lett A, 372 (2008) 3400.
[28]   | M. Wang, X. Li and J.Zhang, The(G’/G)−expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett A, 372 (2008) 417.
[29]   | Q. Ding, The NLS⁻equation and its SL(2,R) structure, J. Phys. A: Math, 33 (2000) L325.
[30]   | A. Bekir, New Exact TravellingWave Solutions for Regularized Long-wave, Phi-Four and Drinfeld-Sokolov Equationuations, International Journal of Nonlinear Science,6(2008)46.
[31]   | D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech, 25 (1966) 321.