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)

Dynamics of Kinks by Using Adomian Decomposition Method

Arash Ghahraman

Department of Physics, Faculty of Basic Science, Chabahar Maritime University, Chabahar, Iran

Abstract: This paper studies nonlinear waves in the presence of weak external perturbation. Dynamical behavior of kink is examined. Adomian decomposition method is employed to study the kink-impurity interaction. As a result, an analytical approximate solution is derived. Then some of the first terms of the series solution are considered to show the kink behavior in the presence of impurity.

Keywords: Soliton, phi4 Kinks, Delta function, Adomian Decomposition method

Citation: Arash Ghahraman, Dynamics of Kinks by Using Adomian Decomposition Method. American Journal of Numerical Analysis, 4 (1). 8-10. January 2016. doi:10.12691/ajna-4-1-2

1. Introduction

Nonlinear problems are of interest to physicists and other scientists because most systems are inherently nonlinear in nature. It is well known that investigation of the wave propagation through inhomogeneous and disordered media plays an important role in the study of nonlinear physical phenomena. To gain a deeper understanding of the behavior of nonlinear excitations in disordered systems, one has to investigate the interactions of soliton with separate localized inhomogeneities (impurities) [1]. An external potential can be added to the equation of motion as perturbative terms [1]. The interaction of solitary waves with spatial inhomogeneous is of considerable importance for physical applications. Impurities and/or defects are present even in the purest of material samples and their effect on the motion of solitary waves must be considered when the dynamics of such solutions are important in the problem at hand [2,3].

As is well-known, when waves scatter on a potential they can be partly reflected and partly transmitted. For the fields of soliton solutions the situation is more complicated as soliton can’t split and thus must either bounce, pass through or become trapped inside the potential. This dynamics is very sensitive to the value of all the parameters of the model and to the initial conditions as well [1,4].

In this paper we study Adomian Decomposition Method (ADM) [5,6,7,8] for investigating the effect of impurity on the motion of solitary-wave solutions on the nonlinear equation. We illustrate the method by examining he motion of soliton solution in the presence of Delta function as an impurity [1].

2. Theory

The model we consider is theory in (1+1) dimensions. This model is a nonlinear, Lorentz invariant, scalar field theory which is not integrable. In this paper an analytical model for the interaction of field with an external potential is presented. The general form of the action in an arbitrary metric is

[picture] | (1)

Where is the determinant of the metric [4]. Energy density of the field-potential can be found by varying both the field and the metric [1,3,4]. For the Lagrangian of the form

[picture] | (2)

With the potential

[picture] | (3)

where . This potential parameter has a constant value in the region where the soliton is located and far away from the position of the soliton is zero as . After varying the action, the equation of motion is

[picture] | (4)

The kink solutions of the non-perturbative equation is defined by

[picture] | (5)

3. Adomian Decomposition Method

In this paper ADM is used as approximate analytical method for solving nonlinear equation (4). It is known that the Adomian decomposition method is a powerful and effective method for solving a wide class of linear and nonlinear differential (ODEs and PDEs) and integral equations [3,8,9]. The Adomian decomposition method was first proposed by G. Adomian in 1980. In general, applying the ADM (and its improvements), we construct an approximate analytic solution as an infinite series, which may converge to an exact solution. We consider exact solutions and truncated series solutions for numerical calculations, visualizations, and comparisons. For the nonlinear PDEs, the idea of the Adomian decomposition method consists in decomposition of the unknown function of an equation into an infinite series

[picture] | (6)

Where the component are determined recursively. The nonlinear term is represented as an infinite series of which is called Adomian polynomials

[picture] | (7)

There exist several schemes for calculating Adomian polynomials. Here the following scheme considered for calculation of Adomian polynomials for the nonlinear term

[picture] | (8)

In an operator form equation (4) becomes

[picture] | (9)

where, and are defined by

[picture] | (10)

The inverse operator is defined by

[picture] | (11)

Applying to both sides of equation (9) and using initial conditions we obtain

[picture] | (12)

Substituting

[picture] | (13)

and the nonlinear term by

[picture] | (14)

into equation (12) gives the recursive relations

[picture] | (15)

[picture] | (16)

The initial conditions are

[picture] | (17)

[picture] | (18)

Figure 1 shows single soliton solution at . The first few components are given by

[picture] | (19)

Figure 2, and Figure 3 show soliton – impurity interaction with and respectively.

Figure 1. single soliton solution at μ(x,0)

Figure 2. Soilton-impurity interaction with v=0.6057, at (a)t=0.68. (b) t=80. (c)t=2.50. (d)t=10.0. (e)t=20. (f) t=25

Figure 3. Soilton-impurity interaction with v=0.0057, at (a) t=0.30. (b) t=0.45. (c) t=0.80. (d)t=1.25. (e)t=3.0. (f) t=5.0

4. Conclusion

We have studied the kink scattering by an impurity. Adomian decomposition method (ADM) employed directly in the treatment of this model to drive a series of approximations to the solution. The Dirac Delta function was considered as an impurity and kink–impurity interaction was investigated analytically. The analytic approximation to the solution provide us the ability to study physical properties of the problem easily. The computational program with MAPLE enable us to have a series term easily and there is no need to do that much computations.

References

[1]   | A. Ghahraman and K. Javidan, “Analytical Formulation for phi4 Field Potential Dynamics,” Braz J Phys, 2011.
[2]   | M. B. Fogel, S. Trullinger, A. R. Bishop and J. A. Krumhansl, “Dynamics of sine-Gordon soliton in the pesence of perturbation,” physical review B, vol. 15, no. 3, pp. 1578-1592, 1977.
[3]   | “Interaction of noncommutative solitons with defects,” BRAZILIAN JOURNAL OF PHYSICS, vol. 38, no. 4, pp. 610-614, 2008.
[4]   | A. Ghahraman and S. Eskandari, “A New Model for Dynamics of Sine-Gordon solitons in The Presence of Perturbation,” Advances in Natural and Applied Sciences, vol. 8, no. 15, pp. 21-24, 2014.
[5]   | J. Biazar and R. Islam, “Solution of wave equation by Adomian decomposition method and the restrictions of the method,” Applied Mathematics and Computation, no. 149, p. 807-814, 2004.
[6]   | J. Biazar and M. Pourabd, “A Maple Program for Solving Systems of Linear and Nonlinear Integral Equations by Adomian Decomposition Method,” INTERNATIONAL JOURNAL OF COMPUTERS FOR MATHEMATICAL LEARNING, vol. 2, no. 29, pp. 1425-1432, 2007.
[7]   | H. Bulut and B. H. M., “Geometrical Interpretation and A Comparative Study Between Three Different Methods For Solving The Non-Linear Kdv Equation,” Journal of Advanced Research in Scientific Computing, vol. 2, no. 1, pp. 69-76, 2010.
[8]   | H.M.Baskonus, H.Bulut and Y.Pandir, “The natural transform decomposition method for linear and nonlinear partial differantial equations,” Mathematics In Engineering, Science And Aerospace (MESA), vol. 5, no. 1, pp. 111-126, 2014.
[9]   | H. Bulut, F.B.M.Belgacem and H.M.Baskonus, “Partial fractional differential equation systems solutions by Adomian decomposition method implementation,” in 4th International Conference On Mathematical and Computational Applications, Manisa, Turkey, 11-13 June 2013.
[10]   | T. M. R. Filho and A. Figueiredo, “a Maple package for the symmetry analysis of differential equations,” Computer Physics Communications, no. 182, p. 467-476, 2011.
[11]   | M. Younis, S. T. R. Rizvi and S. Ali, “Analytical and soliton solutions: Nonlinear model of nanobioelectronics transmission lines,” Applied Mathematics and Computation, no. 265, p. 994-1002, 2015.