Volume 36 | Number 1 | Year 2016 | Article Id. IJMTT-V36P507 | DOI : https://doi.org/10.14445/22315373/IJMTT-V36P507

In this paper, we apply theextended - expansion methodfor solving the Burger's equation, the Korteweg-de Vries-Burgers (KdV) equation and the Lax' fifth-order (Lax5)equation. With the aid of themathematical software Maple, some exact solutions for withequations are successfully.

[1] M. M. El-borai and R. M. Al-masroub, “EXACT SOLUTIONS FOR SOME NONLINEAR FRACTIONAL PARABOLIC EQUATIONS,” no. 10, pp. 106–122, 2015.

[2] Mahmoud M. El-Borai a , Afaf A. Zaghrout b and E. a Department of Mathematics, Faculty of science, Alexandria University, “using the extended tanh- method,” vol. 1, no. 3, 2011.

[3] A. Wazwaz, “The tanh method : exact solutions of the sine- Gordon and the sinh-Gordon equations,” vol. 167, pp. 1196– 1210, 2005.

[4] A. Wazwaz, “The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations,” vol. 184, pp. 1002–1014, 2007.

[5] A. Bekir, E. Aksoy, A. C. Cevikel, and C. H. Ammari, “Exact solutions of nonlinear time fractional partial differential equations by sub-equation method,” no. September 2014, 2015.

[6] A. El-borai, M. M., El-sayed, W. G and R. M. Al-masroub, “Exact Solutions Of Some Nonlinear complex fractional Partial Differential Equations,” Int. J. Math. Trends Technol., vol. 32, no. 1, pp. 2231– 5373.

[7] B. Li, Y. Chen, and H. Zhang, “acklund Transformations and Exact Solutions for the Generalized Two-Dimensional Korteweg-de Vries-Burgers-type Equations and Burgerstype Equations,” pp. 464–472, 2003.

[8] M. M. El-borai, W. G. El-sayed, and R. M. Al-masroub, “EXACT SOLUTIONS FOR TIME FRACTIONAL COUPLED WHITHAM- BROER-KAUP EQUATIONS VIA EXP-FUNCTION METHOD,” pp. 307–315, 2015.

[9] M. Aslam, S. T. Mohyud-din, A. Waheed, and E. A. Al-said, “Exp-function method for traveling wave solutions of nonlinear evolution equations,” Appl. Math. Comput., vol. 216, no. 2, pp. 477–483, 2010.

[10] F. Html and B. Zheng, “Exp-Function Method for Solving Fractional Partial Differential Equations,” vol. 2013, pp. 1– 10, 2013.

[11] K. A. A. Zahra and M. A. A. Hussain, “Exp-function Method for Solving Nonlinear Beam Equation,” vol. 6, no. 48, pp. 2349–2359, 2011.

[12] Y. Li, K. Li, and C. Lin, “Exp-function method for solving the generalized-Zakharov equations q,” Appl. Math. Comput., vol. 205, no. 1, pp. 197–201, 2008.

[13] W. Zhang, “The Extended Tanh Method and the Exp Function Method to Solve a Kind of Nonlinear Heat Equation,” vol. 2010, 2010.

[14] A. Bekir and A. C. Cevikel, “Using A Complex Transformation with Exp-Function Method to get an Exact Solutions for Fractional Differential Equation,” vol. 1, no. 1, pp. 35–44, 2014.

[15] J. He, “Exp-function Method for Fractional Differential Equations,” vol. 14, no. 6, pp. 363–366, 2013.

[16] A. Ebaid, “Journal of Mathematical Analysis and An improvement on the Exp-function method when balancing the highest order linear and nonlinear terms,” J. Math. Anal. Appl., vol. 392, no. 1, pp. 1–5, 2012.

[17] S. Zhang, “Exp-function method for solving Maccari ‟ s system,” vol. 371, pp. 65–71, 2007.

[18] F. Html, K. Khan, M. A. Akbar, N. Hj, and M. Ali, “The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation : The GZK-BBM Equation and Right-Handed Noncommutative Burgers Equations,” vol. 2013, no. 2013, pp. 1–6, 2015.

[19] C. M. Khalique, “Exact solutions and conservation laws of a coupled integrable dispersionless system,” vol. 5, no. January, pp. 957–964, 2012.

[20] M. M. El-borai, A. A. Zaghrout, and A. M. Elshaer, “EXACT SOLUTIONS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS BY USING THE EXTENDED MULTIPLE RICCATI,” vol. 9, no. December, pp. 370–381, 2011.

[21] E. Inan, “Generalized Jacobi Elliptic Function Method for Traveling Wave Solutions of (2+1)-Dimensional Breaking Soliton Equation,” vol. 7, no. 1, pp. 39–50, 2010.

[22] D. Zhang, “Doubly periodic solutions of the modified Kawahara equation,” vol. 25, pp. 1155–1160, 2005.

[23] C. Huai-tang, “New double periodic and multiple soliton solutions of the generalized ( 2 + 1 ) -dimensional Boussinesq equation,” vol. 20, pp. 765–769, 2004.

[24] Y. Yang and Y. H. A. F. Feng, “New Jacobi Elliptic Function Solutions for Coupled KdV-mKdV Equation,” vol. II, no. 7, pp. 22–24, 2014.

[25] Q. Liu and J. Zhu, “Exact Jacobian elliptic function solutions and hyperbolic function solutions for Sawada – Kotere equation with variable coefficient,” vol. 352, pp. 233–238, 2006.

[26] A. H. A. Ali and A. A. Soliman, “New Exact Solutions of Some Nonlinear Partial Differential Equations Modified extended tanh-function method Applications MKdV equation,” Int. J. Nonlinear Sci., vol. 5, no. 1, pp. 79–88, 2008.

[27] N. Taghizadeh, “The Modified Extended Tanh Method with the Riccati Equation for Solving Nonlinear Partial Differential Equations Modified extended tanh method with the Riccati equation,” vol. 2, no. 2, pp. 145–153, 2012.

[28] A. A. Soliman, “The modified extended tanh-function method for solving Burgers-type equations,” Phys. A Stat. Mech. its Appl., vol. 361, no. 2, pp. 394–404, 2006.

[29] A. G. Cesar, “ijpam.eu,” vol. 101, no. 2, pp. 133–140, 2015. [30] M. Physics, G. W. E. I. Wang, and T. Z. Xu, “(c) RRP 66(No.,” vol. 66, no. c, pp. 595–602, 2014.

[31] E. Fan and Y. C. Hon, “Applications of extended tanh method to „special‟ types of nonlinear equations,” Appl. Math. Comput., vol. 141, no. 2–3, pp. 351–358, 2003.

[32] X. D. Zheng, Y. Chen, B. Li, and H. Q. Zhang, “A new generalization of extended tanh-function method for solving nonlinear evolution equations,” Commun. Theor. Phys., vol. 39, no. 6, pp. 647–652, 2003.

[33] S. Zhang and H. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Phys. Lett. A, vol. 375, no. 7, pp. 1069–1073, 2011.

[34] J. F. Alzaidy, “Fractional Sub-Equation Method and its Applications to the Space – Time Fractional Differential Equations in Mathematical Physics,” vol. 3, no. 2, pp. 153– 163, 2013.

[35] H. Jafari, H. Tajadodi, D. Baleanu, A. A. Al-zahrani, Y. A. Alhamed, and A. H. Zahid, “Fractional sub-equation method for the fractional generalized reaction Duffing model and nonlinear fractional Sharma-Tasso-Olver equation ear fractional PDEs,” 2013.

[36] S. Guo, L. Mei, Y. Li, and Y. Sun, “The improved fractional sub-equation method and its applications to the space – time fractional differential equations in fluid mechanics ✩,” Phys. Lett. A, vol. 376, no. 4, pp. 407–411, 2012.

[37] M. M. El-borai, A. A. Zaghrout, and A. M. Elshaer, “EXACT SOLUTIONS FOR NONLINEAR PARTIAL DIFFERENTIAL,” vol. 9, no. December, pp. 361–369, 2011.

[38] C. Science, “A S i n e - C o s i n e M e t h o d for H a n d l i n g Nonlinear Wave Equations,” no. 2, 2004.

[39] R. Arora and A. Kumar, “Soliton Solution for the BBM and MRLW Equations by Cosine-function Method,” vol. 1, no. 2, pp. 59–61, 2011.

[40] A. Bekir, “Exact solutions of nonlinear fractional differential equations by,” vol. 22, no. 11, pp. 1–6, 2013.

[41] R. Arora and S. Yadav, “The -Expansion Method for Traveling Wave Solutions of Burgers ‟ Kdv and Generalization of Huxley Equations,” vol. 2, no. 3, pp. 119– 123, 2012.

[42] S. Guo and Y. Zhou, “-expansion method and its applications to the Whitham – Broer – Kaup – Like equations and coupled Hirota – Satsuma KdV equations,” Appl. Math. Comput., vol. 215, no. 9, pp. 3214–3221, 2010.

[43] H. Kheiri, M. R. Moghaddam, and V. Vafaei, “Application of the -expansion method for the Burgers , Burgers – Huxley and modified Burgers – KdV equations,” vol. 76, no. 6, pp. 831–842, 2011.

[44] J. Manafianheris, “Exact Solutions of the BBM and MBBM Equations by the Generalized expansion Method Equations,” vol. 19, no. 12, pp. 1789–1796, 2012.

[45] A. Iftikhar, A. Ghafoor, T. Zubair, S. Firdous, and S. T. Mohyud-din, “ -expansion method for traveling wave solutions of ( 2 + 1 ) dimensional generalized KdV , Sin Gordon and Landau-Ginzburg-Higgs Equations,” vol. 8, no. 28, pp. 1349–1359, 2013.

[46] I. E. Inan and Y. Ugurlu, “ ” vol. 128, no. 3, 2015.

[47] L. I. Ling-xiao and L. I. Er-qiang, “The -expansion method and its application to travelling wave solutions of the Zakharov equations,” vol. 25, no. 4, pp. 454–462, 2010.

[48] K. Khan, M. A. Akbar, N. Hj, and M. Ali, “The Modified Simple Equation Method for Exact and Solitary Wave Solutions of Nonlinear Evolution Equation : The GZKBBM Equation and Right-Handed Noncommutative Burgers Equations,” vol. 2013, no. 1, 2013.

[49] A. Biswas, “Modified simple equation method for nonlinear evolution equations,” vol. 217, pp. 869–877, 2010.

[50] M. M. A. Khater, “The Modified Simple Equation Method and Its Applications in Mathematical Physics and Biology,” Am. J. Comput. Math. 2015, 5, 1-17, 2015.

[51] S. M. Ege and E. Misirli, “The modified Kudryashov method for solving some fractional-order nonlinear equations,” pp. 1–13, 2014.

[52] Y. Pandir, Y. Gurefe, and E. Misirli, “Full Length Research Paper A new approach to K udryashov ‟ s method for solving some nonlinear physical models,” vol. 7, no. 21, pp. 2860–2866, 2012.

[53] D. I. Sinelshchikov, M. B. Kochanov, and S. I. Aug, “Application of the Kudryashov method for finding exact solutions of the high order nonlinear evolution equations,” pp. 1–12.

[54] Y. Chen and B. Li, “General projective Riccati equation method and exact solutions for generalized KdV-type and KdV–Burgers-type equations with nonlinear terms of any order,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 977– 984, 2004.

[55] A. Wazwaz, “The tanh – coth method for solitons and kink solutions for nonlinear parabolic equations,” vol. 188, pp. 1467–1475, 2007.

[56] H. Panahipour, “Application of Extended Tanh Method to Generalized Burgers-type Equations,” vol. 2012, pp. 1–14,2012.

[57] H. O. Bakodah, “Modified Adomain Decomposition Method for the Generalized Fifth Order KdV Equations,” vol. 2013, no. March, pp. 53–58, 2013.

[58] A. G. Cesar, “Exact solutions for the general fifth order KdV equation by the extended tanh method,” pp. 1–9, 2008.

[59] J. Hietarinta, “Solitons observations Soliton solutions Soliton equations,” vol. 15, no. part I, pp. 31–37, 2005.

Mahmoud M.El-Borai, Wagdy G. El-sayed, Ragab M. Al-Masroub, "Exact Solutions of Some Nonlinear Partial Differential Equations via Extended (G'/G)-Expansion Method Distribution," *International Journal of Mathematics Trends and Technology (IJMTT)*, vol. 36, no. 1, pp. 60-71, 2016. *Crossref*, https://doi.org/10.14445/22315373/IJMTT-V36P507