**September 20, 2016**. Read More

# Modified Simple Equation Method and its Applications for some Nonlinear Evolution Equations in Mathematical Physics

10.5120/11402-6718 |

Elsayed M E Zayed and Hoda Ibrahim S A.. Article: Modified Simple Equation Method and its Applications for some Nonlinear Evolution Equations in Mathematical Physics. *International Journal of Computer Applications* 67(6):39-44, April 2013. Full text available. BibTeX

@article{key:article, author = {Elsayed M. E. Zayed and Hoda Ibrahim S. A.}, title = {Article: Modified Simple Equation Method and its Applications for some Nonlinear Evolution Equations in Mathematical Physics}, journal = {International Journal of Computer Applications}, year = {2013}, volume = {67}, number = {6}, pages = {39-44}, month = {April}, note = {Full text available} }

### Abstract

In this paper, we employ the modified simple equation method to find the exact traveling wave solutions involving parameters of nonlinear evolution equations via the (1+1)-dimensional generalized shallow water-wave equation and the(2+1)-dimensional KdV-Burgers equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the proposed method provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear evolution equations.

### References

- Ablowitz M. . J. and Clarkson P. A. , Solitons, Nonlinear Evolution Equation and Inverse Scattering, Cambridge University press, New York, 1991.
- Hirota, R. Exact solutions of the KdV equation and multiple collisions of solitons , Phys. Rev. Lett. ,27 (1971) 1192-1194.
- Weiss J. , Tabor M. and Carnvalle G. , The Painleve property for PDEs. , J. Math. Phys. 24 (1983) 522-526.
- Kudryashov N. A. , Exact soliton solutions of the generalized evolution equation of wave dynamics, J. Appl. Math. Mech. 52 (1988) 361-365.
- Kudryashov N. A. , Exact soliton solutions of the generalized Kuramoto-Sivashinsky equation, Phys. Lett. A 147 (1990) 287-291.
- Kudryashov N. A. , On types nonlinear nonintegrable differential equations with exact solutions, Phys. Lett. A 155 (1991) 269-275.
- Miura M. R. , Backlund transformation. Springer, Berline, 1978.
- Rogers C. and Shadwick W. F. , Backlund transformation, Academic Press, New York, 1982.
- He J. H. and Wu X. H. , Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30 (2006) 700-708.
- Yusufoglu E. , New solitary solutions for the MBBM equations using the exp- function method. Phys. Lett. A 372 (2008) 442-446.
- Zhang S. , Application of the exp- function method to high dimensional evolution equations, Chaos, Solitons & Fractals, 38 (2008) 270- 276.
- Bekir A. , The exp-function method for Ostrovsky equation, Int. J. Nonlinear Sci. Num. Simul. 10 (2009) 735-739.
- Bekir A. , Application of the exp-function method for nonlinear differential-difference equations, Appl. Math. . Comput. , 215 (2010) 4049-4053.
- Abdou M. A. , The extended tanth-method and its applications for solving nonlinear physical models, Appl. Math. Comput. 190 (2007) 988-996.
- Fan E. G. , Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212-218.
- Zhang S. and Xia T. C. , A further improved tanh- function method exactly solving (2+1)-dimensional dispersive long wave equations, Appl. Math. E-Notes, 8 (2008) 58-66.
- Yusufoglu E. and Bekir A. , Exact solutions of coupled nonlinear Klein- Gordon equations, Math. Comput. Modeling, 48 (2008) 1694-1700.
- Chen Y. and Wang Q. , Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to the(1+1)-dimensional nonlinear dispersive long wave equation. Chaos, Solitons & Fractals. 24 (2005) 745-757.
- Liu S. , Fu Z. , Liu S. D. and Zhao Q. , Jacobi elliptic function expansion method and periodic wave solutions of nonlinear equations. Phys. Lett. A. 289 (2001) 69-74.
- Lu D. , Jacobi elliptic function solutions for two variant variant Boussineq equation, Chaos, Solitons. & Fractals, 24 (2005) 1373-1375.
- Wang M. L. , Li X. and Zhang J. , The (G'/G) –expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A 372, 4 (2008) 417-423.
- Zhang S. , Tong J. L. and Wang W. , A generalized (G'/G) –expansion method for the KdV equation with variable coefficients, Phys. Lett. A 372 (2008) 2254-2257.
- Zayed E. M. E. and Gepreel K. A. The (G'/G) –expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. J. Math. Phys. 50 (2009) 013502- 013513.
- Zayed E. M. E. , The (G'/G) –expansion method and its applications to some nonlinear evolution equations in mathematical physics. J. Appl. Math. Computing, 30 (2009) 89-103.
- Bekir A. , Application of the (G'/G) –expansion method for nonlinear evolution equations, Phys. Lett. A 372 (2008) 3400-3406.
- Ayhan B. and Bekir A. ,The (G'/G) –expansion method for nonlinear lattice equations, Commu. Nonlinear Sci. Numer. Simula. 17 (2012) 3490-3498
- Kudryashov N. A. , A note on the (G'/G) –expansion method, Appl. Math. Comput. 217 (2010) 1755-1758.
- Aslan I. and Ozis T. , Analytic study on two nonlinear evolution equations by using the (G'/G) –expansion method, Appl. Math. Comput. 209 (2009) 425-429.
- Kudryashov N. A. , Meromorphic solutions of nonlinear ordinary differential equations, Comm. Nonlinear Sci. Numer. Simula. 15 (2010) 2778-2790.
- Li X. L. , Li Q. E. and Wang L. W. , The(G'/G)–expansion method and its application to traveling wave solutions of Zakharov equations, Appl. Math. J. Chinese Univ. 25 (2010) 454-462.
- Zayed E. M. E. and Abdelaziz M. A. M. , The two variable (G'/G, 1/G) –expansion method for solving the nonlinear KdV-mKdV equation, Math. Prob. Eng. Vol. 2012, Article ID 725061, 14 pages.
- Tascan F. and Bekier A. , Applications of the first integral method to the nonlinear evolution equations, Chinese Phys. B 19 (2010) 080201-4.
- Zayed E. M. E. and Abdel Rahman H. M. , On solving the KdV-Burgers equation and the Wu- Zhang equations using the modified variational iteration method, Int. J. Nonlinear Sci. Numer. Simula. 10 (2009) 1093-1103.
- Zayed E. M. E. and Abdel Rahman H. M. , The variational iteration method and the variational homotopy perterbation method for solving the KdV-Burgers equation and the Sharma-Tasso- Olver equation, Z. Naturforsch, 65a (2010) 25-33.
- Jawad A. T. M. , Patkovic M. D. and Biswas A. , Modified simple equation method for nonlinear evolution equation, Appl. Math. Compu. , 217 (2010) 869-877.
- Zayed E. M. E. , A note on the modified simple equation method applied to Sharma-Tasso- Olver equation, Appl. Math. Comput. , 218 (2011) 3962-3964.
- Zayed E. M. E. and Hoda Ibrahim S. A. , Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method, Chinese Phys, Lett. 29 (2012) 060201-060204.
- Zayed E. M. E. and Hoda Ibrahim S. A. , Exact solutions of Kolmogorov-Petrovskii-Piskunov equation using the modified simple equation method, Acta Math. Appl. Sinica (English Series), Accepted.
- Zayed E. M. E. and Arnous A. H. , Exact solutions of the nonlinear ZK-MEW and the Potential YTSF equations using the modified simple equation method, AIP Conference Proceedings 1479 (2012) 2044-2048.
- Inc M. and Ergu M. , Periodic wave solutions for the generalized shallow water-wave equation by the improved Jacobi- elliptic function method, Appl. Math. E-Notes, 5 (2005) 89-96.
- Whitham G. B. , Linear and nonlinear waves, Wiley, New York, 1974.