Investigation of solutions of the Pade-II equation by MEFM

Abstract

In this study, singular soliton, topological, nontopological and rational solutions of the Pade-II equation, which is a type of nonlinear partial differential equations, are obtained. By selecting the appropriate parameters, solutions were obtained which provided the equations. Two and three dimensional graphics of the solutions were drawn. It appears that the resulting solution functions and graphics have similar features. Hyperbolic and trigonometric functions were also seen to have the same properties in their graphs since they are periodic functions at the same time. The obtained solutions were found by using the modified expansion method. All solutions using this method are controlled by the Mathematica software program which provides the Pade-II equation. Using the modified expansion function method, the nonlinear partial differential equation with travelling wave transformation takes the form of nonlinear ordinary differential equation. According to the balancing principle, there are also the degree of the solution function containing the exponential function.

Keywords

• The modified expansion function method (MEFM),The Pade-II equation

References

1. Akturk, T., Bulut, H., \& Gurefe, Y., New function method to the (n+1)-dimensional nonlinear problems , An Int. Jour. of Opt. and Control: Theories \& App. (2017), 7(3), 234-239.
2. Akturk, T., Bulut, H., \& Gurefe, Y., An application of the new function method to the Zhiber-Shabat equation , An Int. Jour. of Opt. and Control: Theories \& App. (2017), 7(3), 271-274.
3. Baskonus, H. M., Bulut, H., \& Sulaiman, T. A. Investigation of various travelling wave solutions to the extended (2+1)-dimensional quantum ZK equation. The European Physical Journal Plus (2017), 132(11), 482.
4. Baskonus, H. M., Bulut, H., \& Atangana, A. On the complex and hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Materials and Structures (2016), 25(3), 035022.
5. Bulut, H., Akturk, T., \& Gurefe, Y., Travelling wave solutions of the (N+1)-dimensional sine-cosine-Gordon equation, AIP Conf. Proc. (2014), 5.
6. Chen, Y., \& Yan, Z. New exact solutions of (2+ 1)-dimensional Gardner equation via the new sine-Gordon equation expansion method. Chaos, Solitons \& Fractals (2005), 26(2), 399-406. Chen, Y., \& Yan, Z. New exact solutions of (2+ 1)-dimensional Gardner equation via the new sine-Gordon equation expansion method. Chaos, Solitons \& Fractals (2005), 26(2), 399-406.
7. He, J. H., \& Wu, X. H. (2006). Exp-function method for nonlinear wave equations. Chaos, Solitons \& Fractals, 30(3), 700-708.
8. Kudryashov, N. A., One method for finding exact solutions of nonlinear differential equations, Commun. Nonl. Sci. Numer. Simul. (2012), 17, 2248-2253.

9. Liu, C. S. Trial equation method and its applications to nonlinear evolution equations, Acta. Phys. Sin., 54(6) (2005), 2505-2509.
10. Naher, H., \& Abdullah, F. A. New approach of (G'/G)-expansion method and new approach of generalized (G'/G)-expansion method for nonlinear evolution equation. AIP Advances (2013), 3(3), 032116.
11. Pandir, Y., Gurefe, Y., Kadak, U., \& Misirli, E., Classification of exact solutions for some nonlinear partial differential equations with generalized evolution, Abstr. Appl. Anal. (2012), 16.
12. Shen, G., Sun, Y., \& Xiong, Y., New travelling-wave solutions for Dodd-Bullough equation, J. Appl. Math., (2013), 5.
13. Sun, Y., New travelling wave solutions for Sine-Gordon equation, J. Appl. Math. (2014), 4.
14. Shi, Y., Li, X., \& Zhang, B. G. Traveling Wave Solutions of Two Nonlinear Wave Equations by-Expansion Method. Advances in Mathematical Physics (2018).
15. Xu, F. Application of Exp-function method to symmetric regularized long wave (SRLW) equation. Physics Letters A (2008), 372(3), 252-257.