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)

Application of the He’s Semi-inverse Method for (2 + 1)-Dimensional Nonlinear PDEs

Mohammad Najafi*, Malihe Najafi, Somayeh Arbabi

Medical Biology Research Center, Kermanshah University of Medical Sciences, Kermanshah, Iran

Abstract: We make use of the He’s semi-inverse method and symbolic computation to construct new exact traveling wave solutions for the (2 + 1)-dimensional Boussinesq and breaking soliton equations. Many new exact traveling wave solutions are successfully obtained, which contain soliton solutions. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.

Keywords: He’s semi-inverse method, (2+1)-dimensional Boussinesq equation, (2+1)-dimensional breaking soliton equation

Received May 07, 2013; Revised Septamber 17, 2013; Accepted September 23, 2013

Citation: Mohammad Najafi, Malihe Najafi, Somayeh Arbabi, Application of the He’s Semi-inverse Method for (2 + 1)-Dimensional Nonlinear PDEs. Journal of Mathematical Sciences and Applications, 1 (2). 29-31. January 2013. doi:10.12691/jmsa-1-2-3

1. Introduction

Nonlinear partial differential equations are very important in a variety of scientific fields, especially in fluid mechanics, solid state physics, plasma physics, plasma waves, capillary-gravity waves and chemical physics. The nonlinear wave phenomena observed in the above mentioned scientific fields, are often modeled by the bell-shaped sech solutions and the kink-shaped tanh solutions. The availability of these exact solutions, for those nonlinear equations can greatly facilitate the verification of numerical solvers on the stability analysis of the solution. Nonlinear wave phenomena of dispersion, dissipation, diﬀusion, reaction and convection are very important in nonlinear wave equations. In recent years, new exact solutions may help to ﬁnd new phenomena. Also, the explicit formulas may provide physical information and help us to understand the mechanism of related physical models.

In this paper, by means of the He’s semi-inverse method, we will obtain some Solitary solutions of the (2 + 1)-dimensional Boussinesq and breaking soliton equations.

2. Description of He’s Semi-inverse Method

We suppose that the given nonlinear partial differential equation for to be in the form

[picture] | (1)

Step 1. Seek solitary wave solutions of Eq. (3) by taking and transform Eq. (3) to the ordinary differential equation

[picture] | (2)

where prime denotes the derivative with respect to .

Step 2. If possible, integrate Eq. (3) term by term one or more times. This yields constant(s) of integration. For simplicity, the integration constant(s) can be set to zero.

Step 3. According to He’s semi-inverse method, we construct the following trial-functional

[picture] | (3)

where is an unknown function of and its derivatives.

There exist alternative approaches to the construction of the trial-functionals, see Refs [1,2].

Step 4. By the Ritz method, we can obtain different forms of solitary wave solutions, in the form

[picture] | (4)

where and are constants to be further determined.

Substituting Eq. (5) into Eq. (6) and making stationary with respect to and results in

[picture] | (5)

[picture] | (6)

Solving simultaneously Eqs. (5) and (6) we obtain and . Hence, the solitary wave solution (4) is well determined.

3. He’s Semi-inverse Method for (2 + 1)- Dimensional Boussinesq Equation

Consider the (2 + 1)-dimensional Boussinesq equation

[picture] | (7)

Using the wave variable reduces it to an ODE

[picture] | (8)

Integrating twice and setting the constants of integration to zero, we obtain

[picture] | (9)

According to Ref. [1], By He’s semi-inverse method [2], we can arrive at the following variational formulation:

[picture] | (10)

We assume the soliton solution in the following form

[picture] | (11)

where , is an unknown constant to be further determined.

By Substituting Eq. (11) into Eq. (10) we obtain

[picture] | (12)

For making stationary with respect to and results in

[picture] | (13-1)

[picture] | (13-2)

or simplifying

[picture] | (14-1)

[picture] | (14-2)

From Eqs. (14-1) and (14-2), we can easily obtain the following relations:

[picture] | (15)

So the solitary wave solution can be approximated as

[picture] | (16)

In this solution is an arbitrary complex parameter.( For , see Figure 1)

Figure 1. Peak solition solution of Eq. (7)

4. He’s Semi-inverse Method for (2 + 1)-Dimensional Breaking Soliton Equation

We now consider the (2 + 1)-dimensional breaking soliton equations

[picture] | (17)

Using the wave variable reduces it to an ODE

[picture] | (18)

Integrating the second equation in the system and neglecting constants of integration we find

[picture] | (19)

Substituting (32) into the first equation of the system and integrating we find

[picture] | (20)

According to Ref. [1], By He’s semi-inverse method [2], we can arrive at the following variational formulation:

[picture] | (21)

We assume the soliton solution in the following form

[picture] | (22)

where , is an unknown constant to be further determined.

Figure 2. Peak solition solutions of Eqs. (17)

By Substituting Eq. (22) into Eq. (21) we obtain

[picture] | (23)

For making stationary with respect to and results in

[picture] | (24-1)

[picture] | (24-2)

or simplifying

[picture] | (25-1)

[picture] | (25-2)

From Eqs. (25-1) and (25-2), we can easily obtain the following relations:

[picture] | (26)

So the solitary wave solution can be approximated as

[picture] | (27)

And

[picture] | (28)

In this solutions is an arbitrary complex parameter.( For , see Figure 2)

5. Conclusions

In this paper, by using the He’s semi-inverse method; we obtained some solitary solutions of (2+1)-dimensional Boussinesq and breaking soliton equations. He’s semi-inverse method is a very dominant instrument to find the solitary solutions for various nonlinear equations.

References

[1]   | He, J. H., “Some asymptotic methods for strongly nonlinear equations”, Internat. J. Modern Phys. B, Vol. 20, 2006, 1141-1199.
[2]   | He, J. H., “Variational principles for some nonlinear partial differential equations with variable coefficients”, Chaos, Solitons and Fractals, Vol. 19, 2004, 847-851.
[3]   | Najafi, M., Arbabi, S., Najafi, M., “New application of sine-cosine method for the generalized (2+1)-dimensional nonlinear evolution equations”, International Journal of Advanced Mathematical Sciences, Vol. 1, No. 2, 2013, 45-49.