New exact solutions of the combined and double combined sinh-cosh-Gordon equations via modified Kudryashov method

 
 
 
  • Abstract
  • Keywords
  • References
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  • Abstract


    The combined and double combined sinh-cosh-Gordon equations are very important to a wide range of various scientific applications that ranges from chemical reactions to water surface gravity waves. In this article, with the assistance of a function transform and Painlevè property, the nonlinear combined and double combined sinh-cosh-Gordon equations turn into ordinary differential equations. Later on, modified Kudryashov method is adopted for investigating new analytical solution of the studied equations. As a consequence, a series of new analytical solutions are acquired and we demonstrated the actual behavior of the achieved solutions of the mentioned equations with the aid of 3D and 2D MATLAB graphs. Finally, we also validate the effectiveness of the modified Kudryashov method for the problem of extracting new exact solutions of the combined and double combined sinh-cosh-Gordon equations with the aid of Maple package program. It is shown that the implemented method is capable to extract new solutions and it can also use to other nonlinear partial differential equation (NLPDE's) arising in mathematical physics or other applied field.


  • Keywords


    Painlevè property; Combined sinh-cosh-Gordon equation; Double combined sinh-cosh-Gordon equation; Modified Kudryashov method; New exact solutions.

  • References


      [1] M. Wang, Y. Zhou, Z. Li, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216 (1-5) (1996) 67-75.

      [2] E.M.E. Zayed, K.A. Gepreel, The (G'/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics, Journal of Mathematical Physics, 50(1) (2009) 013502.

      [3] F. Hawlader, D. Kumar, A variety of exact analytical solutions of extended shallow water wave equations via improved (G'/G)-expansion method, International Journal of Physical Research, 5(1) (2017) 21-27.

      [4] M.M.A. Khater, D. Kumar, Implementation of three reliable methods for finding the exact solutions of (2+1) dimensional generalized fractional evolution equations, Optical and Quantum Electronics, 49(12), (2017) 427.

      [5] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Physics Letters A, 277(4) (2000) 212-218.

      [6] M.A. Abdou, The extended F-expansion method and its application for a class of nonlinear evolution equations, Chaos, Solitons & Fractals, 31(1) (2007) 95-104.

      [7] S. Liu, Z. Fu, S. Liu, Q. Zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Physics Letters A, 289(1) (2001) 69-74.

      [8] W.-X. Ma, J.-H. Lee, A transformed rational function method and exact solutions to the (3+1) dimensional Jimbo-Miwa equation, Chaos, Solitons & Fractals, 42(3) (2009) 1356-1363.

      [9]Y. Chen, Z. Yan, The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations, Chaos, Solitons & Fractals 29(4) (2006) 948-964.

      [10] A.-M. Wazwaz, Multiple-soliton solutions for the KP equation by Hirota's bilinear method and by the tanh-coth method, Applied Mathematics and Computation 190(1) (2007) 633-640.

      [11] M.M.A. Khater, D. Kumar, New exact solutions for the time fractional coupled Boussinesq–Burger equation and approximate long water wave equation in shallow water, Journal of Ocean Engineering and Science, 2(3) (2017) 223-228.

      [12] C.-S. Liu, Trial equation method to nonlinear evolution equations with rank inhomogeneous: mathematical discussions and its applications, Communications in Theoretical Physics, 45 (2006) 219-223.

      [13] N.A. Kudryashov, M.B. Soukharev, Popular Ansatz methods and Solitary wave solutions of the Kuramoto-Sivashinsky equation, Regular and Chaotic Dynamics, 14(3) (2009) 407-419.

      [14] A. Akbulut, M. Kaplan, Auxiliary equation method for time-fractional differential equations with conformable derivative. Computers & Mathematics withApplications,(2017),

      https://doi.org/10.1016/j.camwa.2017.10.016

      [15] K. Hosseini, F. Samadani, D. Kumar, M. Faridi, New optical solitons of cubic-quartic nonlinear

      Schrdinger equation, Optik, 157 (2018) 1101-1105.

      [16] D. Kumar, A.R. Seadawy, A. K. Joardar, Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese Journal of Physics, 56(1)(2018) 75-85.

      [17] K. Hosseini, D. Kumar, M. Kaplan, E.Y. Bejarbaneh, New Exact Traveling Wave Solutions of the Unstable Nonlinear Schrödinger Equations, Communications in Theoretical Physics, 68(6) (2017) 761.

      [18] D. Kumar, K. Hosseini, F. Samadani, The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics, Optik, 149 (2017) 439-446.

      [19] F. Xie, Z. Yan, H. Zhang, Explicit and exact traveling wave solutions of Whitham-Broer-Kaup shallow water equations, Physics Letters A, 285 (2001) 76-80.

      [20] A.H. Salas, J.E. Castillo H, New exact solutions to sinh-cosh-Gordon equation by using techniques based on projective Riccati equations, Computers & Mathematics with Applications, 61 (2) (2011) 470-481.

      [21] C. A. Gὸmez, A.H. Salas, New exact solutions for the combined sinh-cosh-Gordon equation, Lecturas Matemàticas 27 (2006) 87-93.

      [22] C. M. Khalique, G. Magalakwe, Combined sinh-cosh-Gordon equation: symmetry reductions, exact solutions and conservation laws, Quaestiones Mathematicae, 37(2) (2014) 199-214.

      [23] L. Wei, Exact solutions to a combined sinh-cosh-Gordon equation, Communications in Theoretical Physics, 54(4) (2010) 599.

      [24] M. Ramirez, A. Magnolia, C. de Indias, V.P.J. Jaramillo-Camacho, R.D.O. -Ortiz, Solutions for the Combined sinh-cosh-Gordon Equation, International Journal of Mathematical Analysis, 9(24) (2015) 1159-1163.

      [25] G. Magalakwe, B. Muatjetjeja, C.M. Khalique, Exact solutions and conservation laws for a generalized double combined sinh-cosh-Gordon equation, Mediterranean Journal of Mathematics, 13(5) (2016) 3221-3233.

      [26] H. Kheiri, A. Jabbari, The (G'/G)-expansion method for solving the combined and the double combined sinh-cosh-Gordon Equations, Acta universitatis apulensis, 22 (2010) 185-194.

      [27] A. Irshad, S.T. Mohyud-Din, Tanh-Coth method for nonlinear differential equations, Studies in Nonlinear Sciences, 3(1) (2012) 24-48.

      [28] A.-M. Wazwaz, Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE method, Computers & Mathematics with Applications, 50(10-12) (2005) 1685-1696.

      [29] Y. Sun, New exact traveling wave solutions for double sine-Gordon equation, Applied Mathematics and Computation, 258 (2015) 100-104.

      [30] G. Magalakwe, C.M. Khalique, New exact solutions for a generalized double sinh-Gordon equation, Abstract and Applied Analysis, (2013).


 

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Article ID: 9261
 
DOI: 10.14419/ijpr.v6i1.9261




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