Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation

Mohannad H. Eljaily; Tarig M. Elzaki

doi:doi:10.11648/j.pamj.20150406.17

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Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation

Mohannad H. Eljaily, Tarig M. Elzaki

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 6)

Received: 12 October 2015 Accepted: 21 October 2015 Published: 3 November 2015

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Abstract

In this paper, a combined form of the Laplace transforms method with the homotopy perturbation method is proposed to solve Korteweg-DeVries (KDV) Equation. This method is called the homotopy perturbation transform method (HPTM). The (HPTM) finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The results reveal that the proposed method is very efficient, simple and can be applied to other nonlinear problems.

Published in	Pure and Applied Mathematics Journal (Volume 4, Issue 6)
DOI	10.11648/j.pamj.20150406.17
Page(s)	264-268
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Laplace Transform, Homotopy Perturbation Method, Korteweg-DeVries (KDV) Equation

References

[1]	J.H. He, Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178(1999):257-262.
[2]	N.H. Sweilam and M.M. Khader, Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method, Computers & Mathematics with Applications, 58(2009):2134 2141.
[3]	A.M. Wazwas, A study on linear and non-linear Schrodinger equations by the variational iteration method, Chaos, Solitions and Fractals, 37(4) (2008):1136 1142.
[4]	B. Jazbi and M. Moini, Application of He’s homotopy perturbation method for Schrodinger equation, Iranian Journalof Mathematical Sciences and Informatics, 3(2)(2008):13-19.
[5]	J.H. He, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computa- tion, 135(2003):73-79.
[6]	J.H. He, Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation, 156(2004):527539.
[7]	J.H. He, The homotopy perturbation method for nonlinear oscillators with discontinuities, Applied Mathematics and Computation, 151(2004):287292.
[8]	J.H. He, Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation, 6(2005):207-208.
[9]	J.H. He, Some asymptotic methods for strongly nonlinear equation, International Journal of Modern Physics, 20(2006):1144-1199.
[10]	J.H. He, Homotopy perturbation method for solving boundary value problems, Physics Letters A, 350(2006):87-88.
[11]	Rafei and D.D. Ganji, Explicit solutions of helmhotz equation and fifth order KdV equation using homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 7(2006):321-328.
[12]	A.M. Siddiqui, R. Mahmood and Q.K. Ghori, Thin film flow of a third grade fluid on a moving belt by He’s homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 7(2006):7-14.
[13]	D.D. Ganji, The applications of He’s homotopy perturbation method to nonlinear equation arising in heat transfer, Physics Letters A, 335(2006):337 341.
[14]	L. Xu, He’s homotopy perturbation method for a boundary layer equation in unbounded domain, Computers &Math- ematics with Applications, 54(2007):1067-1070.
[15]	J.H. He, An elementary introduction of recently developed asymptotic methods and nanomechanics in textile engi- neering, International Journal of Modern Physics, 22(2008):3487-3578.
[16]	J.H. He, Recent developments of the homotopy perturbation method, Topological Methods in Nonlinear Analysis, 31(2008): 205-209.
[17]	E. Hesameddini and H. Latifizadeh, An optimal choice of initial solutions in the homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10(2009):1389-1398.
[18]	E. Hesameddini and H. Latifizadeh, A new vision of the He’s homotopy perturbation method, International Journal of Nonlinear Sciences and Numerical Simulation, 10(2009):1415-1424.
[19]	A. Ghorbani and J. Saberi-Nadjafi, He’s homotopy perturbation method for calculating adomian polynomials, Inter-national Journal of Nonlinear Sciences and Numerical Simulation, 8(2007):229-232.
[20]	A. Ghorbani, Beyondadomian’s polynomials: He polynomials, Chaos Solitons Fractals, 39(2009):1486-1492.
[21]	J. Biazar, M. Gholami Porshokuhi and B. Ghanbari, Extracting a general iterative method from an adomiandecom- position method and comparing it to the variational iteration method, Computers & Mathematics with Applications, 59(2010): 622-628.
[22]	M. Madani, M. Fathizadeh, Homotopy perturbation algorithm using Laplace transformation, Nonlinear Science Letters A 1 (2010) 263–267.
[23]	S.T. Mohyud-Din, M.A. Noor and K.I. Noor, Traveling wave solutions of seventh-order generalized KdV equation using He’s polynomials, International Journal of Nonlinear Sciences and Numerical Simulation, 10(2009):227-233.
[24]	S.A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics 1 (2001) 141–155.
[25]	E. Yusufoglu, Numerical solution of Duffing equation by the Laplace decomposition algorithm, Applied Mathematics and Computation 177 (2006) 572–580.
[26]	Yasir Khan, An effective modification of the Laplace decomposition method for nonlinear equations, International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009) 1373–1376.
[27]	Yasir Khan, Naeem Faraz, A new approach to differential difference equations, Journal of Advanced Research in Differential Equations 2 (2010) 1–12.
[28]	S. Islam, Y. Khan, N. Faraz, F. Austin, Numerical solution of logistic differential equations by using the Laplace decomposition method, World Applied Sciences Journal 8 (2010) 1100–1105.
[29]	Dogan Kaya, Mohammed Aassila, Application for a generalized KdV equation by the decomposition method, Physics Letters A 299 (2002) 201–206.
[30]	P.G. Drazin, R.S. Johnson, Solutions: An Introduction, Cambridge University Press, Cambridge, 1989.
[31]	P. Saucez, A.V. Wouwer, W.E. Schiesser, An adaptive method of lines solution of the Korteweg–de Vries equation, Computers & Mathematics with Applications 35 (12) (1998) 13–25.
[32]	T.A. Abassy, Magdy A. El-Tawil, H. El-Zoheiry, Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms andthePad’e technique, Computers and Mathematics with Applications, doi:10.1016/j.camwa.2006.12.067.
[33]	F. Kangalgil, F. Ayaz, Solitary wave solutions for the KdV and KdV equations by differential transform method, Chaos, Solitons and Fractals, doi:10.1016/j.chaos.2008.02.009.
[34]	Sumit Gupta, Devendra Kumar, Jagdev Singh, Analytical solutions of convection–diffusion problems by combining Laplace transform method and homotopy perturbation method, Alexandria Engineering Journal (2015) 54, 645–651.
[35]	E. Hesameddini and N. Abdollahy, Homotopy perturbation and Elzaki transform for solving Sine-Gorden and Klein-Gorden equations, Iranian J. of Numerical Analysis and Optimization Vol 3, No. 2, (2013), pp 33-46.
[36]	Yin-shanYun,Chaolu Temuer, Application of the homotopy perturbation method for the large deflection problem of a circular plate, Volume 39, Issues 3–4,(2015), Pages 1308–1316.

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APA Style

Mohannad H. Eljaily, Tarig M. Elzaki. (2015). Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation. Pure and Applied Mathematics Journal, 4(6), 264-268. https://doi.org/10.11648/j.pamj.20150406.17

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ACS Style

Mohannad H. Eljaily; Tarig M. Elzaki. Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation. Pure Appl. Math. J. 2015, 4(6), 264-268. doi: 10.11648/j.pamj.20150406.17

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AMA Style

Mohannad H. Eljaily, Tarig M. Elzaki. Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation. Pure Appl Math J. 2015;4(6):264-268. doi: 10.11648/j.pamj.20150406.17

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@article{10.11648/j.pamj.20150406.17,
  author = {Mohannad H. Eljaily and Tarig M. Elzaki},
  title = {Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {6},
  pages = {264-268},
  doi = {10.11648/j.pamj.20150406.17},
  url = {https://doi.org/10.11648/j.pamj.20150406.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150406.17},
  abstract = {In this paper, a combined form of the Laplace transforms method with the homotopy perturbation method is proposed to solve Korteweg-DeVries (KDV) Equation. This method is called the homotopy perturbation transform method (HPTM). The (HPTM) finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The results reveal that the proposed method is very efficient, simple and can be applied to other nonlinear problems.},
 year = {2015}
}

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TY  - JOUR
T1  - Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation
AU  - Mohannad H. Eljaily
AU  - Tarig M. Elzaki
Y1  - 2015/11/03
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150406.17
DO  - 10.11648/j.pamj.20150406.17
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 264
EP  - 268
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150406.17
AB  - In this paper, a combined form of the Laplace transforms method with the homotopy perturbation method is proposed to solve Korteweg-DeVries (KDV) Equation. This method is called the homotopy perturbation transform method (HPTM). The (HPTM) finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The results reveal that the proposed method is very efficient, simple and can be applied to other nonlinear problems.
VL  - 4
IS  - 6
ER  -

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Author Information

Mohannad H. Eljaily

Department of Mathematic, Faculty of Sciences, Sudan University of Sciences and Technology, Khartoum, Sudan
Tarig M. Elzaki

Mathematics Department, Faculty of Sciences and Arts-Alkamil, University of Jeddah, Jeddah, Saudi Arabia

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APA Style

Mohannad H. Eljaily, Tarig M. Elzaki. (2015). Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation. Pure and Applied Mathematics Journal, 4(6), 264-268. https://doi.org/10.11648/j.pamj.20150406.17

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ACS Style

Mohannad H. Eljaily; Tarig M. Elzaki. Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation. Pure Appl. Math. J. 2015, 4(6), 264-268. doi: 10.11648/j.pamj.20150406.17

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AMA Style

Mohannad H. Eljaily, Tarig M. Elzaki. Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation. Pure Appl Math J. 2015;4(6):264-268. doi: 10.11648/j.pamj.20150406.17

Copy | Download

@article{10.11648/j.pamj.20150406.17,
  author = {Mohannad H. Eljaily and Tarig M. Elzaki},
  title = {Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation},
  journal = {Pure and Applied Mathematics Journal},
  volume = {4},
  number = {6},
  pages = {264-268},
  doi = {10.11648/j.pamj.20150406.17},
  url = {https://doi.org/10.11648/j.pamj.20150406.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20150406.17},
  abstract = {In this paper, a combined form of the Laplace transforms method with the homotopy perturbation method is proposed to solve Korteweg-DeVries (KDV) Equation. This method is called the homotopy perturbation transform method (HPTM). The (HPTM) finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The results reveal that the proposed method is very efficient, simple and can be applied to other nonlinear problems.},
 year = {2015}
}

Copy | Download

TY  - JOUR
T1  - Homotopy Perturbation Transform Method for Solving Korteweg-DeVries (KDV) Equation
AU  - Mohannad H. Eljaily
AU  - Tarig M. Elzaki
Y1  - 2015/11/03
PY  - 2015
N1  - https://doi.org/10.11648/j.pamj.20150406.17
DO  - 10.11648/j.pamj.20150406.17
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 264
EP  - 268
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20150406.17
AB  - In this paper, a combined form of the Laplace transforms method with the homotopy perturbation method is proposed to solve Korteweg-DeVries (KDV) Equation. This method is called the homotopy perturbation transform method (HPTM). The (HPTM) finds the solution without any discretization or restrictive assumptions and avoids the round-off errors. The results reveal that the proposed method is very efficient, simple and can be applied to other nonlinear problems.
VL  - 4
IS  - 6
ER  -

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