ISSN (Print): 2372-2118

ISSN (Online): 2372-2126

Editor-in-Chief: Emanuele Galligani

Website: http://www.sciepub.com/journal/AJNA

   

Article

Dynamics of Kinks by Using Adomian Decomposition Method

1Department of Physics, Faculty of Basic Science, Chabahar Maritime University, Chabahar, Iran


American Journal of Numerical Analysis. 2016, 4(1), 8-10
doi: 10.12691/ajna-4-1-2
Copyright © 2016 Science and Education Publishing

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

Correspondence to: Arash  Ghahraman, Department of Physics, Faculty of Basic Science, Chabahar Maritime University, Chabahar, Iran. Email: arash.ghgood@gmail.com

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

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.
 
Show More References
[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.
 
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Article

New Constraction Seven Degree Spline Function to Solve Sencond Order Initial Value Problem

1Mathematics Department, School of Science, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq

2Mathematics Department, College of Science, University of Garmian, Kalar, Kurdistan Region, Iraq


American Journal of Numerical Analysis. 2016, 4(1), 11-20
doi: 10.12691/ajna-4-1-3
Copyright © 2016 Science and Education Publishing

Cite this paper:
Karwan H.F. Jwamer, Najim Abdullah I.. New Constraction Seven Degree Spline Function to Solve Sencond Order Initial Value Problem. American Journal of Numerical Analysis. 2016; 4(1):11-20. doi: 10.12691/ajna-4-1-3.

Correspondence to: Karwan  H.F. Jwamer, Mathematics Department, School of Science, University of Sulaimani, Sulaimani, Kurdistan Region, Iraq. Email: karwan.jwamer@univsul.edu.iq

Abstract

Our paper dedicated to find approximate solution of second order initial value problem by seven degree lacunary spline function of type (0, 1, 6). The convergence analysis of given method has studied. Numerical illustrations have given with example for calculating absolute error between spline functions and exact solution of second order initial value problem with their derivatives.

Keywords

References

[1]  Abbas Y. Al Bayaty, Rostam K. Saeed, Faraidun K. Hama Salh “The Existence, Uniqueness and Error Bound of Approximation Spline Interpolation for Solving Second-Order Initial Value Problems” Journal of Mathematics and Statistics, Vol 5(2), pp.123-129, (2009).
 
[2]  Aryan Ali Muhammad ((Spline solution and Asymptotic Behaviors of Eigen values and Eigen function for some Types of Boundary value problem)) , A Thesis submitted to The Council of Faculty of science and science Education , school of science, University of Sulaimani in Pan till Fullfillment of the Requirements of the Degree of Doctor of Philosophy of science in Mathematics; January2013.
 
[3]  C. De Boor, A Practical Guide to Splines. Springer Verlag 1978.
 
[4]  Gianluca Frasso, Splines, Differential Equations and optimal Smoothing, April 2013.
 
[5]  Fazal-i-Haq “Numerical Solution of Boundary- Value and Initial-Boundary Value Problem Using Spline Function” May 2009.
 
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[6]  J. Karwan Hama Faraj “Approximation Solution of second order Initial value problem by spline Function of degree seven”. Int. J. Contemp. Math. Science, Vol.5, 2010, no 46, 2293-2309.
 
[7]  Jwamer, K. H.-F. and R. G. Karim, 2010On Sixtic Lacunary Spline Solution of fourth Order Initial Value Problem. Asian Journal of Mathematics and Statistic, Pakistan,3 (3): 119-129.
 
[8]  Karwan H.F. Jawmer and Aryan Ali M. ((Second order Initial Value Problem and its Eight Degree spline solution)). world Applied science Journal17(12): 1694-1712, 2012.
 
[9]  L. L. Schumaker, Spline Function, Basic Theory, John Wiley, New York, 1981.
 
[10]  M. K. Jain (1984): Numerical Solution of Differential Equation. Wiley Eastern New Delhi.
 
[11]  R. K. Saeed,(0, 1, 3, 5) Lacunary Interpolation by Deficient Spline, J. Sci. Nat., 2(1993),No. 1, 28-30.
 
[12]  R. K. Saeed, and K. H.-F. Jwamer, Lacunary interpolation by Spline Function (0, 1, 4) case, J. Dohuk Univ., Printed in Kurdistan Region, 4(2001), 193-196.
 
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Article

Numerical Solution of Singularly Perturbed Delay Reaction-Diffusion Equations with Layer or Oscillatory Behaviour

1Department of Mathematics, Jimma University, Jimma, P. O. Box 378, Ethiopia

2Department of Mathematics, National Institute of Technology, Warangal-506 004, India


American Journal of Numerical Analysis. 2017, 5(1), 1-10
doi: 10.12691/ajna-5-1-1
Copyright © 2017 Science and Education Publishing

Cite this paper:
Gemechis File, Gashu Gadisa, Tesfaye Aga, Y. N. Reddy. Numerical Solution of Singularly Perturbed Delay Reaction-Diffusion Equations with Layer or Oscillatory Behaviour. American Journal of Numerical Analysis. 2017; 5(1):1-10. doi: 10.12691/ajna-5-1-1.

Correspondence to: Y.  N. Reddy, Department of Mathematics, National Institute of Technology, Warangal-506 004, India. Email: ynreddy@nitw.ac.in

Abstract

In this paper, we presented numerical method for solving singularly perturbed delay differential equations with layer or oscillatory behaviour for which a small shift (δ) is in the reaction term. First, the given singularly perturbed delay reaction-diffusion equation is converted into an asymptotically equivalent singularly perturbed two point boundary value problem and then solved by using fourth order finite difference method. The stability and convergence of the method has been investigated. The numerical results have been tabulated and further to examine the effect of delay on the boundary layer and oscillatory behavior of the solution, graphs have been given for different values of δ. Both theoretical and numerical rate of convergence have been established and are observed to be in agreement for the present method. Briefly, the present method improves the findings of some existing numerical methods in the literature.

Keywords

References

[1]  Gemechis File and Y.N. Reddy, Numerical integration of a class of singularly perturbed delay differential equations with small shift, International Journal of Differential Equation, 572723 (2012), 1-12.
 
[2]  C. G. Lange and R. M. Miura, Singular perturbation analysis of boundary-value problems for differential-difference equations very small shifts with layer behavior, SIAM J. Appl. Math., 54 (1994), 249-272.
 
[3]  V.P. Ramesh and M.K. Kadalbajoo, Numerical Algorithm for Singularly Perturbed Delay Differential Equations with Layer and Oscillatory Behavior, Neural, Parallel, and Scientific Computations, 19 (2011), 21-34.
 
[4]  D. K. Swamy, Quantative analysis of delay differential equations with layer, Advance Research and Innovations in Mechanical, Material Science, Industrial Engineering and Management - ICARMMIEM- (2014), 145-150.
 
[5]  G. BSL. Soujanya and Y.N. Reddy, Computational method for singularly perturbed delay differential equations with layer or oscillatory behaviour, Appl. Math. Inf. Sci. 10 (2), (2016), 527-536.
 
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[6]  E.R. Doolan, J.J.H. Miller and W.H.A. Schilders, Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole Press, Dublin, 1980.
 
[7]  M.K. Kadalbajoo and Y.N. Reddy, Asymptotic and numerical analysis of singular perturbation problems, Applied Mathematics and Computation, 30 (1989), 223-259.
 
[8]  H.G. Roos, M. Stynes and L. Tobiska, Numerical Methods for Singularly Perturbed Differential Eqns, Convection-Diffusion and Flow Problems, Springer Verlag, Berlin, 1996.
 
[9]  M.K. Kadalbajoo and V.P. Ramesh, Numerical methods on Shishkin mesh for singularly perturbed delay differential equations with a grid adaptation strategy, Applied Mathematics and Computation, 188 (2007), 1816-1831.
 
[10]  R. Pratima and K.K. Sharma, Numerical analysis of singularly perturbed delay differential turning point problem, Applied Mathematics and Computation, 218 (2011), 3483-3498.
 
[11]  L.E. Elsgolt’s and S.B. Norkin, Introduction to the Theory and Applications of Differential Equations with Deviating Arguments, Academic Press, New York, 1973.
 
[12]  M. K. Kadalbajoo and Y.N. Reddy, A non-asymptotic method for general singular perturbation problems, Journal of Optimization Theory and Applications, 55 (1986), 256 269.
 
[13]  R.S. Varga, Matrix Iterative Analysis. New Jersey: Prentice-Hall, Englewood Cliffs, 1962.
 
[14]  D.M. Young, Iterative Solution of Large Linear Systems, Academic Press, Inc. New York, 1971.
 
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