Regarding New Complex Analytical Solutions for the Nonlinear Partial Vakhnenko-Parkes Differential Equation via Bernoulli Sub-Equation Function Method
Volume 1, Issue 1, June 2015, Pages: 1-9
Received: Jun. 6, 2015;
Accepted: Jun. 18, 2015;
Published: Jun. 19, 2015
Views 3867 Downloads 225
Haci Mehmet Baskonus, Department of Computer Engineering, Tunceli University, Tunceli, Turkey
Hasan Bulut, Department of Mathematics, University of Firat, Elazig, Turkey
Dilara Gizem Emir, Department of Mathematics, University of Firat, Elazig, Turkey
In this research, a structure of the Bernoulli sub-equation function method is proposed. The nonlinear partial Vakhnenko-Parkes differential equation which is another name the reduced Ostrovsky equation has been taken into consideration. Then, analytical solutions such as rational function solution, exponential function solution, hyperbolic function solution, complex trigonometric function solution and periodic wave solution have been obtained by the same method. All necessary calculations while obtaining the analytical solutions have been accomplished through using commercial wolfram software Mathematica 9.
Haci Mehmet Baskonus,
Dilara Gizem Emir,
Regarding New Complex Analytical Solutions for the Nonlinear Partial Vakhnenko-Parkes Differential Equation via Bernoulli Sub-Equation Function Method, Mathematics Letters.
Vol. 1, No. 1,
2015, pp. 1-9.
J.H. He, Variational iteration method—some recent results and new interpretations, Journal of Computational and Applied Mathematics, 207(1), 3–17, 2007.
X.M Qian, SY Lou, XB Hu. Variable Separation Approach for a Differential difference Asymmetric Nizhnik– Novikov –Veselov Equation, Z Naturforsch A, 59, 645–58, 2004.
H. Bulut, Classification of exact solutions for generalized form of K(m,n) equation, Abstract and Applied Analysis, 2013, 1-11 pages, 2013.
X.M Qian, SY Lou, XB Hu. Variable separation approach for a differential–difference system: special Toda equation, Journal of Physics A: Mathematical and General, 37, 2401–24011, 2004.
C.S. Liu, Applications of complete discrimination system for polynomial for classiﬁcations of traveling wave solutions to nonlinear differential equations, Computer Physics Communications, 181, 317-324, 2010.
C.S. Liu, Trial equation method and its applications to nonlinear evolution equations, Acta Physica Sinica, 54, 2505-2509, 2005.
C.S. Liu, A new trial equation method and its applications, Communications in Theoretical Physics, 45, 395-397, 2006.
C.S. Liu, Trial Equation Method to Nonlinear Evolution Equations with Rank In homogeneous: Mathematical Discussions and Its Applications, Communications in Theoretical Physics, 45, 219-223, 2006.
A.Bekir, O. Guner and B. Ayhan, Exact solutions of some systems of fractional differential-difference equations, Mathematical Methods in the Applied Sciences, DOI: 10.1002/mma.3318, 2014.
H. Bulut, H. M. Baskonus, Y. Pandir, The modified trial equation method for fractional wave equation and time-fractional generalized Burgers equation, Abstract and Applied Analysis, 2013, 13 pages, 2013.
B. Zheng, Application of A Generalized Bernoulli Sub-ODE Method For Finding Traveling Solutions of Some Nonlinear Equations, WSEAS Transactions on Mathematics, 7(11), 618-626, 2012.
E. Yusufoğlu, A. Bekir, The tanh and the sine-cosine methods for exact solutions of the MBBM and the Vakhnenko equations, Chaos, Solitons and Fractals, 38, 1126–1133, 2008.
H. Roshid , R. Kabir, Rajandra Chadra Bhowmik and Bimal Kumar Datta, Investigation of Solitary wave solutions for Vakhnenko-Parkes equation via exp-function and Exp(−ϕ(ξ))-expansion method, SpringerPlus, 3(692), 1-10, 2014.
V. O., Vakhnenko, E. J., Parkes, A. J., Morrison, A Böcklund transformation and the inverse scattering transform method for the generalized Vakhnenko equation, Chaos Soliton Fractals, 17(4), 683-692, 2003.
Y. Ye, J. Song, S. Shen, Y. Di, New coherent structures of the Vakhnenko–Parkes equation, Results in Physics, 2, 170–174, 2012.
VO. Vakhnenko High-frequency soliton-like waves in a relaxing medium, Journal of Mathematical Physics, 40, 2011–2020, 1999.
V. O. Vakhnenko and E. J. Parkes, The two loop soliton solution of the Vakhnenko equation, Nonlinearity, 11(6), 1457–1464, 1998.
F., Kangalgil, F., Ayaz, New exact travelling wave solutions for the Ostrovsky equation, Physics Letters A., 372, 1831-1835, 2008.
L. A. Ostrovsky, Nonlinear internal waves in a rotating ocean, Oceanology, 18, 119–125, 1978.
E. Yasar, New travelling wave solutions to the Ostrovsky equation, Applied Mathematics and Computation, 216(11), 3191–3194, 2010.
E. Yusufoglu and A. Bekir, A travelling wave solution to the Ostrovsky equation, Applied Mathematics and Computation, 186(1), 256–260, 2007.
AJ, Morrison, E.J., Parkes, V.O., Vakhnenko, The n loop soliton solution of the Vakhnenko equation, Nonlinearity,12, 427–1437, 1999.
V.O.,Vakhnenko, E.J., Parkes, A.V., Michtchenko The Vakhnenko equation from the viewpoint of the inverse scattering method for the KdV equation, International Journal of Differential Equations and Applications, 1, 429–49, 2000.
V.O., Vakhnenko, E.J., Parkes, The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method. Chaos, Solitons & Fractals, 13, 1819–1826, 2002.