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On an Efficient Technique for Travelling Wave Solutions of Nonlinear Differential Equations
Current Issue
Volume 4, 2017
Issue 6 (November)
Pages: 100-104   |   Vol. 4, No. 6, November 2017   |   Follow on         
Paper in PDF Downloads: 78   Since Oct. 25, 2017 Views: 1225   Since Oct. 25, 2017
Muhammad Hashim, National College of Business Administration and Economics, Gujrat, Pakistan.
Madiha Afzal, Department of Mathematics, Allama Iqbal Open University, Islamabad, Pakistan.
Kamran Ayub, Department of Mathematics, Riphah International University, Islamabad, Pakistan.
Qazi Mahmood Ul-Hassan, Department of Mathematics, University of Wah, Wah Cantt., Pakistan.
Munaza Saeed, Department of Mathematics, University of Wah, Wah Cantt., Pakistan.
This paper obtains soliton solutions to the governing nonlinear (2+1) – dimensional differential equation. The algorithm that is employed in this paper is the exp-function method. This leads to travelling wave solutions that are valuable in the field of mathematical physics. The soliton solutions appear with all necessary constraints that are deemed necessary for them to exist. The proposed algorithm is very consistent and may be extended to other nonlinear differential equations. The competence of this algorithm reconfirm by graphical results and computational work.
Exp-function Method, Soliton Solutions, Nonlinear Differential Equation, Maple 18
John Scott Russell, Reports on Waves, Report of the fourteenth meeting of the British Assosiation for Advancement of Sciences, York, September 1844 (London 1845), pp 311-390, Plates XLVII-LVII.
C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Method for Solving the Korteweg-deVries Equation. Phys. Rev. Lett., 19 (1967) 1095-1099.
C. Rogers and W. F. Shadwick, Backlund Transformations, Academic Press, New York, (1982).
S. T. Mohyud-Din, Variational iteration method for solving fifth-order boundary value problems using He’s polynomials. Math. Prob. Engr. Article ID 954794 (2008), doi: 10.1155/2008/954794.
M. A. Abdou, The extended tanh-method and its applications for solving nonlinear physical models. Appl. Math. Comput. 190 (2007) 988-996.
S. A. El-Wakil, M. A. Abdou, E. K. El-Shewy and A. Hendi, exp(-(φ))-expansion method equivalent to the extended tanh-function method. Phys. Script., 81 (2010) 35011- 35014.
S. Zhang and T. C. Xia, A further improved tanh-function method exactly solving the (2+1)-dimensional dispersive long wave equations. Appl. Math. E-Notes, 8 (2008) 58-66.
Y. B. Zhou, M. L. Wang and Y. M. Wang, Periodic wave solutions to coupled KdV equations with variable coefficients. Phys. Lett. A, 308 (2003) 31–36.
J. H. He and X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons & Fractals 30 (3) (2006), 700--708.
J. H. He, An elementary introduction of recently developed asymptotic methods and nanomechanics in textile engineering, Int. J. Mod. Phys. B 22 (21) (2008), 3487-4578.578.
J. H. He and M. A. Abdou, New periodic solutions for nonlinear evolution equation using exp-method, Chaos, Solitons & Fractals, 34 (2007), 1421-1429.
S. T. Mohyud-Din, M. A. Noor and A. Waheed, Exp-function method for generalized travelling solutions of good Boussinesq equations, J. Appl. Math. Computg. 30 (2009), 439-445, DOI 10.1007/s12190-008-0183-8.
S. T. Mohyud-Din, M. A. Noor and K. I. Noor, Some relatively new techniques for nonlinear problems, Mathematical Problems in Engineering, Hindawi, 2009 (2009); Article ID 234849, 25 pages, doi: 10.1155/2009/234849.
M. A. Noor, S. T. Mohyud-Din and A. Waheed, Exp-function method for solving Kuramoto-Sivashinsky and Boussinesq equations, J. Appl. Math. Computg. 29 (2008), 1-13, DOI: 10.1007/s12190-008-0083-y.
M. A. Noor, S. T. Mohyud-Din and A. Waheed, Exp-function method for generalized traveling solutions of master partial differential equations, Acta Applicandae Mathematicae, 104 (2) (2008), 131-137, DOI: 10.1007/s10440-008-9245-z.
T. Ozis, C. Koroglu, A novel approach for solving the Fisher's equation using Exp-function method, Phys Lett. A 372 (2008) 3836-3840
X. H. Wu and J. H. He, Exp-function method and its applications to nonlinear equations, Chaos, Solitons & Fractals, (2007), in press.
X. H. Wu and J. H. He, Solitary solutions, periodic solutions and compacton like solutions using the exp-function method, Comput. Math. Appl. 54 (2007), 966-986.
E. Yusufoglu, New solitonary solutions for the MBBN equations using exp-function method, Phys. Lett. A. 372 (2008), 442-446.
S. Zhang, Application of exp-function method to high-dimensional nonlinear evolution equation, Chaos, Solitons & Fractals, 365 (2007), 448-455.
S. D. Zhu, Exp-function method for the Hybrid-Lattice system, Inter. J. Nonlin. Sci. Num. Simulation, 8 (2007), 461-464.
S. D. Zhu, Exp-function method for the discrete m KdV lattice, Inter. J. Nonlin. Sci. Num. Simulation, 8 (2007), 465-468.
N. A. Kudryashov, Exact soliton solutions of the generalized evolution equation of wave dynamics, J. Appl Math and Mech, 52 (3), (1988), 361.
S. Momani, An explicit and numerical solutions of the fractional KdV equation, Math. Comput. Simul. 70 (2) (2005) 110-118.
A. Ebaid An improvement on the Exp-function method when balancing the highest order linear and nonlinear terms J. Math. Anal. Appl. 392 (2012) 1-5.
F. Tascan and A. Bekir, Analytic solutions of the (2 + 1)-dimensional nonlinear evolution equations using the sine-cosine method. Applied Mathematics and Computation. 215, 3134-3139. (2009).
Mahmoud A. E. Abdelrahman and Mostafa M. A. Khater, Exact Traveling Wave Solutions for Fitzhugh-Nagumo (FN) Equation and Modified Liouville Equation. International Journal of Computer Applications, 113 (3) (2015) 1-7.
Emad H. M. Zahran & Mostafa M. A. Khater, Exact Traveling Wave Solutions For Nonlinear Dynamics of Microtubles -A New Model and The Kundu- Eckhaus Equation. Asian Journal of Mathematics and Computer Research, 4 (4) (2015) 195- 206.
Mostafa M. A. Khater, Emad H. M. Zahran, Maha S. M. Shehata, Solitary wave solution of the generalized Hirota-Satsuma coupled KdV system. Journal of the Egyptian Mathematical Society, 25 (1) (2016) 8-12.
D. S. Wang, S. Y. Tian, Y. Liu, Integrability and bright soliton solutions to the coupled nonlinear Schrödinger equation with higher-order effects. Appl. Math. Comput, 229 (2014) 296-309.
D. S. Wang, X. Wei, Integrability and exact solutions of a two-component Korteweg–de Vries system. Appl. Math. Lett., 51 (2016) 60-67.
D. S. Wang, D. J. Zhang, J. Yang, Integrable properties of the general coupled nonlinear Schrödinger equations. Journal of Mathematical Physics, 51 (2010) 023510.
W. X. Ma, B. Fuchssteiner, Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation. International Journal of Non-Linear Mechanics, 31 (1996) 329-338.
W. X. Ma, A refined invariant subspace method and applications to evolution equations. Science China Mathematics, 55 (2012) 1778-1796.
W. X. Ma, Lump solutions to the Kadomtsev–Petviashvili equation. Physics Letters A, 379 (2015) 1975–1978.
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