Solving Nonlinear Evolution Equations by <i>(G'/G)</i>-Expansion Method

Attia Rani; Munazza Saeed; Muhammad Ashraf; Rakshanda Zaman; Qazi Mahmood-Ul-Hassan; Kamran Ayub; Muhammad Yaqub Khan; Madiha Afzal

doi:doi:10.11648/j.optics.20180701.17

Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method

Attia Rani, Munazza Saeed, Muhammad Ashraf, Rakshanda Zaman, Qazi Mahmood-Ul-Hassan, Kamran Ayub, Muhammad Yaqub Khan, Madiha Afzal

Published in Optics (Volume 7, Issue 1)

Received: 24 June 2018 Accepted: 13 July 2018 Published: 8 August 2018

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Abstract

Nonlinear mathematical models and their solutions attain much attention in soliton theory. In this paper, main focus is to find travelling wave solutions of foam drainage equation and NLEE of fourth order. (G'/G)-expansion method is applied on these nonlinear differential equations. Wave transformation is used to convert nonlinear partial differential equation into an ordinary differential equation. It is observed that (G'/G)-expansion method is advanced and easy tool for finding solution of NLEEs in engineering, optics and mathematical physics. The proposed method is highly effective and reliable.

Published in	Optics (Volume 7, Issue 1)
DOI	10.11648/j.optics.20180701.17
Page(s)	43-53
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), 2018. Published by Science Publishing Group

Keywords

(G'/G)-Expansion Method, Nonlinear Evolution Equations, Travelling Wave Solutions, Maple 18

References

[1]	I. Aslan, Exact and explicit solutions to some nonlinear evolution equations by utilizing the (G’/G)-expansion method, App. Math. Comp., 217 (20), 8134-8139, 2011.
[2]	A. Bekir, Application of the (G’/G)-expansion method for nonlinear evolution equations. Phys. Lett. A 372, 3400–3406, 2008.
[3]	M. F. El-Sabbagh, A. T. Ali, and S. El-Ganaini, New Abundant Exact Solutions for the System of (2+1)-Dimensional Burgers Equations, App. Math. Inf. Sci., 2 (1), 31-41, 2008.
[4]	J. Feng, W. Li and Q. Wan, “Using (G′/G)−expansion method to seek traveling wave solution of Kolmogorov-Petrovskii-Piskunov equation,” App. Math. Comput., 217, 5860-5865, 2011.
[5]	K. A. Gepreel, “Exact solutions for nonlinear PDEs with the variable coefficients in mathematical physics,” J. Inf. Comput. Sci., 6 (1), 003-014, 2011.
[6]	F. Khanib, S. H. Nezhad, M. T. Darvishia, S. W. Ryu, New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method, Nonlinear Analysis, Real World App., 10, 1904–1911, 2009.
[7]	X. Liu, L. Tian, Y. Wu, Application of (G`/G)-expansion method to two nonlinear evolution equations. Appl. Math. Comput., in press, (2009).
[8]	H. Naher, F. Abdullah and M. A. Akbar, “The (G′/G)−expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation,” Mathematical Problems in Engineering, 2011, 218216, (in press), 2011.
[9]	T. Ozis and I. Aslan, “Application of the (G′/G)−expansion method to Kawahara type equations using symbolic computation,” App. Math. Comput., 216, 2360-2365, 2010.
[10]	I. Podlubny, Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation, fract. Cal. App. anal., 5, Number 4, 2002.
[11]	A. M. Wazwaz, Solitary wave solutions and periodic solutions, for higher-order nonlinear evolution equations, App. Math. Comput., 181, 1683–1692, 2006.
[12]	M. Wang, X. Li and J. Zhang, The (G′/G)−expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phy. Lett. A, 372, 417-423, 2008.
[13]	A. Yildirim and S. T. Mohyud-Din, Analytical approach to space and time fractional Burger’s equations, Chinese Phy. Lett., 27 (9), 090501, 2010.
[14]	E. M. E. Zayed, and S. Al-Joudi, “Applications of an Extended (G′/G)−Expansion Method to Find Exact Solutions of Nonlinear PDEs in Mathematical Physics,” Math, Prob. Eng., 768573, 2010.
[15]	Q. M. Hassan and S. T. Mohyud-Din, Investigating biological population model using exp-function method. Int. J. Biomath. 9 (2), 1650026, 2016.
[16]	Duranda M, Langevin D. Physicochemical approach to the theory of foam drainage. Europhys J Eng., 7:35-44, 2002.
[17]	Y. M. Zhao, Y. J. Yang and W. Li, “Application of the improved (G′/G)−expansion method for the Variant Boussinesq equations,” App. Math. Sci., 5 (58), 2855-2861, 2011.
[18]	E. M. E. Zayed and M. A. M. Abdelaziz, An extended (G′/G) − expansion method and its applications to the (2+1)-dimensional nonlinear evolution equations, Wseas Transactions on Mathematics, 11 (11), 1039-1047, 2012.
[19]	J. Chen and B. Li, Multiple (G`/G)-expansion method and its applications to nonlinear evolution equations in mathematical physics, Ind. Acad. Sci., 28 (3), 375-388, 2012.
[20]	M. Wang, J. Zhang and X. Li, Application of the (G′/G) expansion to traveling wave solutions of the Broer-Kaup and the approximate long water wave equations. Appl. Math. Comput., 206, 321-326, 2008. Oz ̈is
[21]	I. Aslan and T. Oz ̈is, Analytic study on two nonlinear evolution equations by using the (G`/G)-expansion method. Appl. Math. Comput. 209, 425-429, 2009.
[22]	X. J. Yang, A new integral transform operator for solving the heat-diffusion problem, Applied Mathematics Letters. 64 (2017) 193-197.
[23]	X. J. Yang, A New integral transform method for solving steady Heat transfer problem, Thermal Science. 20 (2016) S639-S642.
[24]	X. J. Yang, J. A. Tenreiro Machado, D. Baleanu, C. Cattani, On exact traveling wave solutions for local fractional Korteweg-de Vries equation, Chaos. 26 (8) (2016) 084312.
[25]	X. J. Yang, G. Feng, H. M. Srivastava, Exact travelling wave solutions for the local fractional two-dimensional Burgers-type equations, Computers & Mathematics with Applications. 73 (2) (2017) 203-210.
[26]	X. J. Yang, J. A. Tenreiro Machado, D. Baleanu, On exact traveling-wave solution for local fractional Boussinesq equation in fractal domain, Fractals. 25 (4) (2017) 1740006.
[27]	X. J. Yang, G. Feng, H. M. Srivastava, Non-Differentiable exact solutions for the nonlinear ODEs defined on fractal sets, Fractals. 25 (4) (2017) 1740002.
[28]	K. M. Saad, E. H. AL-Shareef, M. S. Mohamed, X. J. Yang, Optimal q-homotopy analysis method for time-space fractional gas dynamics equation, The European Physical Journal Plus. 132 (23) (2017) https://doi.org/10.1140/epjp/i2017-11303-6.
[29]	X. J. Yang, G. Feng, A New Technology for Solving Diffusion and Heat Equations, Thermal Science. 21 (1A) (2017) 133-140.
[30]	G. Feng, X. J. Yang, H. M. Srivastava, Exact Travelling wave solutions for linear and nonlinear heat transfer equations, Thermal Science. (2017) doi 10.2298/TSCI161013321G.

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APA Style

Attia Rani, Munazza Saeed, Muhammad Ashraf, Rakshanda Zaman, Qazi Mahmood-Ul-Hassan, et al. (2018). Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method. Optics, 7(1), 43-53. https://doi.org/10.11648/j.optics.20180701.17

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ACS Style

Attia Rani; Munazza Saeed; Muhammad Ashraf; Rakshanda Zaman; Qazi Mahmood-Ul-Hassan, et al. Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method. Optics. 2018, 7(1), 43-53. doi: 10.11648/j.optics.20180701.17

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AMA Style

Attia Rani, Munazza Saeed, Muhammad Ashraf, Rakshanda Zaman, Qazi Mahmood-Ul-Hassan, et al. Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method. Optics. 2018;7(1):43-53. doi: 10.11648/j.optics.20180701.17

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@article{10.11648/j.optics.20180701.17,
  author = {Attia Rani and Munazza Saeed and Muhammad Ashraf and Rakshanda Zaman and Qazi Mahmood-Ul-Hassan and Kamran Ayub and Muhammad Yaqub Khan and Madiha Afzal},
  title = {Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method},
  journal = {Optics},
  volume = {7},
  number = {1},
  pages = {43-53},
  doi = {10.11648/j.optics.20180701.17},
  url = {https://doi.org/10.11648/j.optics.20180701.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.optics.20180701.17},
  abstract = {Nonlinear mathematical models and their solutions attain much attention in soliton theory. In this paper, main focus is to find travelling wave solutions of foam drainage equation and NLEE of fourth order. (G'/G)-expansion method is applied on these nonlinear differential equations. Wave transformation is used to convert nonlinear partial differential equation into an ordinary differential equation. It is observed that (G'/G)-expansion method is advanced and easy tool for finding solution of NLEEs in engineering, optics and mathematical physics. The proposed method is highly effective and reliable.},
 year = {2018}
}

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TY  - JOUR
T1  - Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method
AU  - Attia Rani
AU  - Munazza Saeed
AU  - Muhammad Ashraf
AU  - Rakshanda Zaman
AU  - Qazi Mahmood-Ul-Hassan
AU  - Kamran Ayub
AU  - Muhammad Yaqub Khan
AU  - Madiha Afzal
Y1  - 2018/08/08
PY  - 2018
N1  - https://doi.org/10.11648/j.optics.20180701.17
DO  - 10.11648/j.optics.20180701.17
T2  - Optics
JF  - Optics
JO  - Optics
SP  - 43
EP  - 53
PB  - Science Publishing Group
SN  - 2328-7810
UR  - https://doi.org/10.11648/j.optics.20180701.17
AB  - Nonlinear mathematical models and their solutions attain much attention in soliton theory. In this paper, main focus is to find travelling wave solutions of foam drainage equation and NLEE of fourth order. (G'/G)-expansion method is applied on these nonlinear differential equations. Wave transformation is used to convert nonlinear partial differential equation into an ordinary differential equation. It is observed that (G'/G)-expansion method is advanced and easy tool for finding solution of NLEEs in engineering, optics and mathematical physics. The proposed method is highly effective and reliable.
VL  - 7
IS  - 1
ER  -

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Author Information

Attia Rani

Department of Mathematics, University of Wah, Wah Cantt., Pakistan
Munazza Saeed

Department of Mathematics, University of Wah, Wah Cantt., Pakistan
Muhammad Ashraf

Department of Mathematics, University of Wah, Wah Cantt., Pakistan
Rakshanda Zaman

Department of Mathematics, University of Wah, Wah Cantt., Pakistan
Qazi Mahmood-Ul-Hassan

Department of Mathematics, University of Wah, Wah Cantt., Pakistan
Kamran Ayub

Department of Mathematics, Riphah International University, Islamabad, Pakistan
Muhammad Yaqub Khan

Department of Mathematics, Riphah International University, Islamabad, Pakistan
Madiha Afzal

Department of Mathematics, Allama Iqbal Open University, Islamabad, Pakistan

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Plain Text BibTeX RIS

APA Style

Attia Rani, Munazza Saeed, Muhammad Ashraf, Rakshanda Zaman, Qazi Mahmood-Ul-Hassan, et al. (2018). Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method. Optics, 7(1), 43-53. https://doi.org/10.11648/j.optics.20180701.17

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ACS Style

Attia Rani; Munazza Saeed; Muhammad Ashraf; Rakshanda Zaman; Qazi Mahmood-Ul-Hassan, et al. Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method. Optics. 2018, 7(1), 43-53. doi: 10.11648/j.optics.20180701.17

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AMA Style

Attia Rani, Munazza Saeed, Muhammad Ashraf, Rakshanda Zaman, Qazi Mahmood-Ul-Hassan, et al. Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method. Optics. 2018;7(1):43-53. doi: 10.11648/j.optics.20180701.17

Copy | Download

@article{10.11648/j.optics.20180701.17,
  author = {Attia Rani and Munazza Saeed and Muhammad Ashraf and Rakshanda Zaman and Qazi Mahmood-Ul-Hassan and Kamran Ayub and Muhammad Yaqub Khan and Madiha Afzal},
  title = {Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method},
  journal = {Optics},
  volume = {7},
  number = {1},
  pages = {43-53},
  doi = {10.11648/j.optics.20180701.17},
  url = {https://doi.org/10.11648/j.optics.20180701.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.optics.20180701.17},
  abstract = {Nonlinear mathematical models and their solutions attain much attention in soliton theory. In this paper, main focus is to find travelling wave solutions of foam drainage equation and NLEE of fourth order. (G'/G)-expansion method is applied on these nonlinear differential equations. Wave transformation is used to convert nonlinear partial differential equation into an ordinary differential equation. It is observed that (G'/G)-expansion method is advanced and easy tool for finding solution of NLEEs in engineering, optics and mathematical physics. The proposed method is highly effective and reliable.},
 year = {2018}
}

Copy | Download

TY  - JOUR
T1  - Solving Nonlinear Evolution Equations by (G'/G)-Expansion Method
AU  - Attia Rani
AU  - Munazza Saeed
AU  - Muhammad Ashraf
AU  - Rakshanda Zaman
AU  - Qazi Mahmood-Ul-Hassan
AU  - Kamran Ayub
AU  - Muhammad Yaqub Khan
AU  - Madiha Afzal
Y1  - 2018/08/08
PY  - 2018
N1  - https://doi.org/10.11648/j.optics.20180701.17
DO  - 10.11648/j.optics.20180701.17
T2  - Optics
JF  - Optics
JO  - Optics
SP  - 43
EP  - 53
PB  - Science Publishing Group
SN  - 2328-7810
UR  - https://doi.org/10.11648/j.optics.20180701.17
AB  - Nonlinear mathematical models and their solutions attain much attention in soliton theory. In this paper, main focus is to find travelling wave solutions of foam drainage equation and NLEE of fourth order. (G'/G)-expansion method is applied on these nonlinear differential equations. Wave transformation is used to convert nonlinear partial differential equation into an ordinary differential equation. It is observed that (G'/G)-expansion method is advanced and easy tool for finding solution of NLEEs in engineering, optics and mathematical physics. The proposed method is highly effective and reliable.
VL  - 7
IS  - 1
ER  -

Copy | Download