The new generalized (G'/G)-expansion method is one of the powerful and competent methods that appear in recent time for establishing exact solutions to nonlinear evolution equations (NLEEs). We apply the new generalized (G'/G)-expansion method to solve exact solutions of the new coupled Konno-Oono equation and construct exact solutions expressed in terms of hyperbolic functions, trigonometric functions, and rational functions with arbitrary parameters. The significance of obtained solutions gives credence to the explanation and understanding of related physical phenomena. As a newly developed mathematical tool, this method efficiency for finding exact solutions has been demonstrated through showing its straightforward nature and establishing its ability to handle nonlinearities prototyped by the NLEEs whether in applied mathematics, physics, or engineering contexts.
Ablowitz, M.J. and Clarkson, P.A. (1991) Soliton, Nonlinear Evolution Equations and Inverse Scattering Method. Cambridge University Press, New York. http://dx.doi.org/10.1017/CBO9780511623998
Yang, Y.J. (2013) New Application of the (G’/G,1/G)-Expansion Method to KP Equation. Applied Mathematical Sciences, 7, 959-967.
Li, L.X., Li, E.Q. and Wang, M.L. (2010) The (G’/G,1/G)-Expansion Method and Its Application to Traveling Wave Solutions of the Zakharov Equations. Applied Mathematics, 25, 454-462. http://dx.doi.org/10.1007/s11766-010-2128-x
Jawad, A.J.M., Petkovic, M.D. and Biswas, A. (2010) Modified Simple Equation Method for Nonlinear Evolution Equations. Applied Mathematics and Computation, 217, 869-877. http://dx.doi.org/10.1016/j.amc.2010.06.030
Khan, K., Akbar, M.A. and Alam, M.N. (2013) Traveling Wave Solutions of the Nonlinear Drinfel’d-Sokolov-Wilson Equation and Modified Benjamin-Bona-Mahony Equations. Journal of the Egyptian Mathematical Society, 21, 233-240. http://dx.doi.org/10.1016/j.joems.2013.04.010
Belgacem, F.B.M., Karaballi, A.A. and Kalla, S.L. (2003) Analytical Investigations of the Sumudu Transform and Applications to Integral Production Equations. Mathematical Problems in Engineering, 2003, 103-118.
Belgacem, F.B.M. and Karaballi, A.A. (2006) Sumudu Transform Fundamental Properties Investigations and Applications. Journal of Applied Mathematics and Stochastic Analysis, 2006, Article ID: 91083.
Zayed, E.M.E., Zedan, H.A. and Gepreel, K.A. (2004) On the Solitary Wave Solutions for Nonlinear Hirota-Sasuma Coupled KDV Equations. Chaos, Solitons and Fractals, 22, 285-303. http://dx.doi.org/10.1016/j.chaos.2003.12.045
Wang, M.L. (1996) Exact Solutions for a Compound KdV-Burgers Equation. Physics Letters A, 213, 279-287. http://dx.doi.org/10.1016/0375-9601(96)00103-X
Matveev, V.B. and Salle, M.A. (1991) Darboux Transformation and Solitons. Springer, Berlin. http://dx.doi.org/10.1007/978-3-662-00922-2
Zayed, E.M.E., Abourabia, A.M., Gepreel, K.A. and Horbaty, M.M. (2006) On the Rational Solitary Wave Solutions for the Nonlinear HirotaCSatsuma Coupled KdV System. Journal of Applied Analysis, 85, 751-768. http://dx.doi.org/10.1080/00036810600604789
Chow, K.W. (1995) A Class of Exact Periodic Solutions of Nonlinear Envelope Equation. Journal of Mathematical Physics, 36, 4125-4137. http://dx.doi.org/10.1063/1.530951
Wang, M., Li, X. and Zhang, J. (2008) The (G’/G)-Expansion Method and Traveling Wave Solutions of Nonlinear Evolution Equations in Mathematical Physics. Physics Letters A, 372, 417-423. http://dx.doi.org/10.1016/j.physleta.2007.07.051
Alam, M.N. and Akbar M.A. (2015) A Novel (G’/G)-Expansion Method for Solving the (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsov Equation in Mathematical Physics. International Journal of Computing Science and Mathematics, 6, 404-415. http://dx.doi.org/10.1504/IJCSM.2015.071812
Feng, J., Li, W. and Wan, Q. (2011) Using (G’/G)-Expansion Method to Seek the Traveling Wave Solution of Kolmogorov-Petrovskii-Piskunov Equation. Applied Mathematics and Computation, 217, 5860-5865. http://dx.doi.org/10.1016/j.amc.2010.12.071
Alam, M.N. (2015) Exact Solutions to the Foam Drainage Equation by Using the New Generalized -Expansion Method. Results in Physics, 5, 168-177. http://dx.doi.org/10.1016/j.rinp.2015.07.001
Guo, S. and Zhou, Y. (2010) The Extended (G’/G)-Expansion Method and Its Applications to the Whitham-Broer-Like Equations and Coupled Hirota-Satsuma KdV Equations. Applied Mathematics and Computation, 215, 3214-3221. http://dx.doi.org/10.1016/j.amc.2009.10.008
Zhu, S.D. (2008) The Generalizing Riccati Equation Mapping Method in Nonlinear Evolution Equation: Application to (2+1)-Dimensional Boiti-Leon-Pempinelle Equation. Chaos, Solitons & Fractals, 37, 1335-1342. http://dx.doi.org/10.1016/j.chaos.2006.10.015
Alam, M.N., Akbar, M.A. and Mohyud-Din, S.T. (2014) A Novel (G’/G)-Expansion Method and Its Application to the Boussinesq Equation. Chinese Physics B, 23, Article ID: 020203. http://dx.doi.org/10.1088/1674-1056/23/2/020203
Shakeel, M., Ul-Hassan, Q.M. and Ahmad, J. (2014) Applications of the Novel (G’/G)-Expansion Method for a Time Fractional Simplified Modified MCH Equation. Abstract and Applied Analysis, 2014, Article ID: 601961.
Hafez, M.G., Alam, M.N. and Akbar, M.A. (2014) Exact Traveling Wave Solutions to the Klein-Gordon Equation Using the Novel (G’/G)-Expansion Method. Results in Physics, 4, 177-184. http://dx.doi.org/10.1016/j.rinp.2014.09.001
Alam, M.N. and Akbar, M.A. (2014) A New (G’/G)-Expansion Method and Its Application to the Burgers Equation. Walailak Journal of Science and Technology, 11, 643-658.
Zayed, E.M.E. (2009) New Traveling Wave Solutions for Higher Dimensional Nonlinear Evolution Equations Using a Generalized (G’/G)-Expansion Method. Journal of Physics A: Mathematical and Theoretical, 42, Article ID: 195202. http://dx.doi.org/10.1088/1751-8113/42/19/195202
Zhang, J., Jiang, F. and Zhao, X. (2010) An Improved (G’/G)-Expansion Method for Solving Nonlinear Evolution Equations. International Journal of Mathematics, 87, 1716-1725. http://dx.doi.org/10.1080/00207160802450166
Saliman, A.A. (2004) Collocation Solution of the Kortewegde Vries Equation Using Septic Splines. International Journal of Computer Mathematics, 81, 325-331. http://dx.doi.org/10.1080/00207160410001660817
Soliman, A.A. and Hussein, M.H. (2005) Collocation Solution for RLW Equation with Septic Spline. Applied Mathematics and Computation, 161, 623-636. http://dx.doi.org/10.1016/j.amc.2003.12.053
Bluman, G.W. and Kumei, S. (1989) Symmetries and Differential Equations. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4757-4307-4
Olver, P.J. (1986) Applications of Lie Groups to Differential Equations. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-1-4684-0274-2
Domairry, G., Mohsenzadeh, A. and Famouri, M. (2009) The Application of Homotopy Analysis Method to Solve Nonlinear Differential Equation Governing Jeffery-Hamel Flow. Communications in Nonlinear Science and Numerical Simulation, 14, 85-95. http://dx.doi.org/10.1016/j.cnsns.2007.07.009
Joneidi, A.A., Domairry, G. and Babaelahi, M. (2010) Three Analytical Methods Applied to Jeffery-Hamel Flow. Communications in Nonlinear Science and Numerical Simulation, 15, 3423-3434. http://dx.doi.org/10.1016/j.cnsns.2009.12.023
Motsa, S.S., Sibanda, P., Awad, F.G. and Shateyi, S. (2010) A New Spectral-Homotopy Analysis Method for the MHD Jeffery-Hamel Problem. Computers & Fluids, 39, 1219-1225. http://dx.doi.org/10.1016/j.compfluid.2010.03.004
Makukula, Z.G., Sibanda, P. and Motsa, S.S. (2010) A Note on the Solution of the von Kármán Equations Using Series and Chebyshev Spectral Methods. Boundary Value Problems, 2010, Article ID: 471793.
Sibanda, P., Makukula, Z.G. and Motsa, S.S. (2010) A Novel Numerical Technique for Two-Dimensional Laminar Flow between Two Moving Porous Walls. Mathematical Problems in Engineering, 2010, Article ID: 528956.
Makinde, O.D. and Mhone, P.Y. (2006) Hermite-Padé Approximation Approach to MHD Jeffery-Hamel Flows. Applied Mathematics and Computation, 181, 966-972. http://dx.doi.org/10.1016/j.amc.2006.02.018
Naher, H. and Abdullah, F.A. (2013) New Approach of (G’/G)-Expansion Method and New Approach of Generalized (G’/G)-Expansion Method for Nonlinear Evolution Equation. AIP Advances, 3, Article ID: 032116. http://dx.doi.org/10.1063/1.4794947
Alam, M.N. and Akbar, M.A. (2014) The New Approach of Generalized (G’/G)-Expansion Method for Nonlinear Evolution Equations. Ain Shams Engineering Journal, 5, 595-603. http://dx.doi.org/10.1016/j.asej.2013.12.008
Alam, M.N. and Akbar, M.A. (2013) Exact Traveling Wave Solutions of the KP-BBM Equation by Using the New Approach of Generalized (G’/G)-Expansion Method. Springer Plus, 2, 617. http://dx.doi.org/10.1186/2193-1801-2-617
Alam, M.N., Akbar, M.A. and Mohyud-Din, S.T. (2014) General Traveling Wave Solutions of the Strain Wave Equation in Microstructured Solids via the New Approach of Generalized (G’/G)-Expansion Method. Alexandria Engineering Journal, 53, 233-241. http://dx.doi.org/10.1016/j.aej.2014.01.002
Alam, M.N., Akbar, M.A. and Hoque, M.F. (2014) Exact Traveling Wave Solutions of the (3+1)-Dimensional mKdV-ZK Equation and the (1+1)-Dimensional Compound KdVB Equation Using New Approach of the Generalized (G’/G)-Expansion Method. Pramana—Journal of Physics, 83, 317-329. http://dx.doi.org/10.1007/s12043-014-0776-8
Konno, K. and Oono, H. (1994) New Coupled Integrable Dispersion-Less Equations. Journal of the Phusical Society of Japan, 63, 377-378. http://dx.doi.org/10.1143/JPSJ.63.377
Khan, K. and Akbar, M.A. (2013) Traveling Wave Solutions of Some Coupled Nonlinear Evolution Equations. Mathematical Physics, 2013, Article ID: 685736.