A New Approach of Bernoulli Sub-ODE Method to Solve Nonlinear PDEs
Keywords:
Modified Liouville equation, regularized long wave equation, traveling wave solutions.Abstract
In this paper, a new approach of the Bernoulli Sub-ODE method is proposed and this method is applied to solve the modified Liouville equation and the regularized long wave equation. As a result some new traveling wave solutions for them are successfully established. When the parameters are taken as special values, the solitary wave solutions are originated from these traveling wave solutions. Further, graphical representation of some solutions are given to visualize the dynamics of the equation. The results reveal that this method may be useful for solving higher order nonlinear partial differential equations.
References
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[16]. Bekir A. and Boz A., Exact solutions for nonlinear evolution equation using Exp-function method, Physics Letters A 372(2008), 1619–1625.
[17]. Islam M. E., Khan K., Akbar M. A. and Islam R., Traveling Wave Solutions of Nonlinear Evolution Equation via Enhanced (G'/G)-expansion Method. GANIT J. Bangladesh Math. Soc., (ISSN 1606-3694) Vol. 33, 2013, 83-92.
[18]. Jawad A. J. M., Petkovi´c M. D. and Biswas A., Modified simple equation method for nonlinear evolution equations, App. Math. and Compu., vol. 217, no. 2, (2010), pp. 869–877.
[19]. Wang M., Li X., Extended F-expansion and periodic wave solutions for the generalized Zakharov equations. Phys. Lett. A. 343(2005): 48-54.
[20]. Wang M., Li X., Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solutions & Fractals. 24(2005): 1257-1268.
[21]. Ben Jing. A new Bernoulli Sub ODE method for constructing travelling wave solutions for two nonlinear equations with any order. U.P.B science bulletin, series A, vol.73, issue 3, page 85-93, 2011.
[22]. Xu F., and Feng Q. A. Generalized Sub-ODE Method and Applications for Nonlinear Evolution Equation s. J. Sci. Res. Report. 2(2): 571-581, 2013.
[23]. Mahmoud A.E. Abdelrahman. Exact Traveling Wave Solutions for Fitzhugh-Nagumo(FN) Equation and Modified Liouville Equation. International Journal of Computer Applications (0975 8887), Volume 113 - No. 3, March 2015.
[24]. Salam M. A., Traveling-Wave Solution of Modified Liouville Equation by Means of Modified Simple Equation Method, ISRN Applied Mathematics, Article ID 565247,4 pages doi:10.5402/2012/565247, 2012.
[25]. Wazwaz A.M. A. sine-cosine method for handling nonlinear wave equations. Math. And Comput. Modelling. 40, 499-508, 2004.
[2]. Malfliet W., Solitory wave solutions of nonlinear wave equations, Am. J. Phys.,60(1992), 650–654.
[3]. Hereman W. and Malfliet W., The tanh method I: exact solutions of nonlinear evolution and wave equations, Phys. Scripta, 54(1996), 563–568.
[4]. Wazwaz A. M., The tanh method: solitons and periodic solutions for the Dodd-Bullough-Mikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos Soliton Fractal, 25(2005), 55–63.
[5]. Wazwaz A. M., The extended tanh method for new soliton solutions for many forms of the fifth-order KdV equations, Appli. Math. Comput. 184(2007), 1002–1014.
[6]. Chen Y., Wang Q., Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to the (1+1)- dimensional nonlinear dispersive long wave equation. Chaos, Solitons & Fractals. 24(2005): 745-757 .
[7]. Li Z., Wang M. L., A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with higher order nonlinear terms. Phys. Lett. A. 361(2007): 115-118 .
[8]. Yan Z., Abundant families of Jacobi elliptic functions of the (2+1)-dimensional integrable Davey- Stawartson-type equation via a new method. Chaos, Solitons & Fractals. 18(2003): 299-309.
[9]. Boiti M., Leon J., Pempinelli P., Spectral transform for a two spatial dimension extension of the dispersive long wave equation. Inverse Probl. 3(1987): 371-387.
[10]. Miura M. R, Backlund Transformation. Springer, Berlin, (1978)
[11]. Ren Y., Zhang H., New generalized hyperbolic functions and auto-Backlund transformation to find exact solutions of the (2+1)-dimensional NNY equation. Phys. Lett. A. 357(2006): 438-448.
[12]. Rogers C., Shadwick W. F., Backlund Transformation. Academic Press, New York, (1982)
[13]. Moussa M. H. M., ElShikh R. M., Two applications of the homogeneous balance method for solving the generalized Hirota-Satsuma coupled KdV system with variable coefficients, Inter. J. Nonlinear Sci. 7(2009): 29-38.
[14]. Vakhmenko V. O., Parkers E. J., The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method. Chaos, Solutions & Fractals. 13(2003): 1819-1826.
[15]. Abdou M. A., Soliman A. A., New applications of variational iteration method. Physica D, 211 (1–2), 2005, 1–8.
[16]. Bekir A. and Boz A., Exact solutions for nonlinear evolution equation using Exp-function method, Physics Letters A 372(2008), 1619–1625.
[17]. Islam M. E., Khan K., Akbar M. A. and Islam R., Traveling Wave Solutions of Nonlinear Evolution Equation via Enhanced (G'/G)-expansion Method. GANIT J. Bangladesh Math. Soc., (ISSN 1606-3694) Vol. 33, 2013, 83-92.
[18]. Jawad A. J. M., Petkovi´c M. D. and Biswas A., Modified simple equation method for nonlinear evolution equations, App. Math. and Compu., vol. 217, no. 2, (2010), pp. 869–877.
[19]. Wang M., Li X., Extended F-expansion and periodic wave solutions for the generalized Zakharov equations. Phys. Lett. A. 343(2005): 48-54.
[20]. Wang M., Li X., Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solutions & Fractals. 24(2005): 1257-1268.
[21]. Ben Jing. A new Bernoulli Sub ODE method for constructing travelling wave solutions for two nonlinear equations with any order. U.P.B science bulletin, series A, vol.73, issue 3, page 85-93, 2011.
[22]. Xu F., and Feng Q. A. Generalized Sub-ODE Method and Applications for Nonlinear Evolution Equation s. J. Sci. Res. Report. 2(2): 571-581, 2013.
[23]. Mahmoud A.E. Abdelrahman. Exact Traveling Wave Solutions for Fitzhugh-Nagumo(FN) Equation and Modified Liouville Equation. International Journal of Computer Applications (0975 8887), Volume 113 - No. 3, March 2015.
[24]. Salam M. A., Traveling-Wave Solution of Modified Liouville Equation by Means of Modified Simple Equation Method, ISRN Applied Mathematics, Article ID 565247,4 pages doi:10.5402/2012/565247, 2012.
[25]. Wazwaz A.M. A. sine-cosine method for handling nonlinear wave equations. Math. And Comput. Modelling. 40, 499-508, 2004.
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Published
2018-06-16
How to Cite
Salam, M. A., Islam, M. S., Islam, M. H., & Aziz, M. A. (2018). A New Approach of Bernoulli Sub-ODE Method to Solve Nonlinear PDEs. American Scientific Research Journal for Engineering, Technology, and Sciences, 44(1), 58–67. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/4118
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