Solitary and Periodic Wave Solutions of the Fourth Order Boussinesq Equation Through the Novel Exponential Expansion Method
American Journal of Applied Mathematics
Volume 7, Issue 2, April 2019, Pages: 49-57
Received: Apr. 21, 2019;
Accepted: Jun. 13, 2019;
Published: Jun. 27, 2019
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Ayrin Aktar, Department of Mathematics, Begum Rokeya University, Rangpur, Bangladesh
Md Mashiur Rahhman, Department of Mathematics, Begum Rokeya University, Rangpur, Bangladesh
Kamalesh Chandra Roy, Department of Mathematics, Begum Rokeya University, Rangpur, Bangladesh
This article presents the new exact traveling wave solutions of fourth order (1+1)-dimensional Boussinesq equation. We proposed a new exponential expansion method and apply to undertake this study. The analytical solutions are defined by various types of mathematical functions. This study further shows some solitary and periodic waves graphically. This paper also shows that the novel exponential expansion method is easily applicable and powerful mathematical tool in the symbolic computational approach in the field of mathematical physics and engineering. The exact solutions of this equation play a vital role for describing different types of wave propagation in any varied natural instances, especially in water wave dynamics.
Md Mashiur Rahhman,
Kamalesh Chandra Roy,
Solitary and Periodic Wave Solutions of the Fourth Order Boussinesq Equation Through the Novel Exponential Expansion Method, American Journal of Applied Mathematics.
Vol. 7, No. 2,
2019, pp. 49-57.
M. L. Wang, X. Z. Li, J. Zhang, The (GG)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372 (2008) 417.
H. Kim, R. Sakthivel, New Exact Traveling Wave Solutions of Some Nonlinear Higher-Dimensional Physical Models. Reports Math. Phys. 70 (1) (2012) 39.
H. Naher, F. A. Abdullah, New generalized and improved (G′/G)-expansion method for nonlinear evolution equations in mathematical physics. AIP Advan. 3 (2013) 032116.
M. N. Alam, M. A. Akbar, S. T. Mohyud-Din, A novel (G'/G)-expansion method and its application to the Boussinesq equation. Chin. Phys. B 23 (2) (2014) 020202.
M. N. Alam, M. A. Akbar and H. O. Rohid, Traveling wave solutions of the Boussinesq equation via the new approach of generalized (G′/G)-expansion method. SpringerPlus 3 (2014); 43.
W. Malfliet, Hereman W. The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Physica. Scr. 1996; 54; 563-568.
Wazwaz AM. The extended tanh method for new compact and noncompact solutions for the KP-BBM and the ZK-BBM equations. Chaos, Solitons Fract. 2008; 38 (5); 1505–1516.
Abdou MA, Soliman AA. Modified extended tanh-function method and its application on nonlinear physical equations. Phys. Lett. A 2006; 353 (6); 487–492.
El-Wakil SA, Abdou MA. New exact travelling wave solutions using modified extended tanh-function method. Chaos, Solitons Fract. 2007; 31 (4); 840–852.
Liu S, Fu Z, Liu SD, Zhao Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A 2001; 289; 69-74.
Chen Y, Wang Q. Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions solutions to (1+1)-dimensional dispersive long wave equation. Chaos, Solitons Fract. 2005; 24; 745-757.
Zhao X, Tang D. A new note on a homogeneous balance method. Phys. Lett. A 2002; 297 (1–2); 59–67.
Zhao X, Wang L, Sun W. The repeated homogeneous balance method and its applications to nonlinear partial differential equations. Chaos, Solitons Fract. 2006; 28 (2); 448–453.
Zhaosheng F. Comment on “On the extended applications of homogeneous balance method. Appl. Math. Comput. 2004; 158 (2); 593–596.
Hirota R. Exact solution of the KdV equation for multiple collisions of solutions. Phys. Rev. Lett. 1971; 27; 1192-1194.
Chun C, Sakthivel R. Homotopy perturbation technique for solving two points boundary value problems –camparison with other mathods. Computer Phy. Commu. 2010; 181 (6); 1021-1024.
Wang, Zhang HQ. Further improved F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equation. Chaos Solitons Fract. 2005; 25; 601-610.
Wazwaz AM. A sine-cosine method for handle nonlinear wave equations. App. Math. Comp. Model. 2004; 40; 499-508.
Jawad AJM, Petkovic MD and Biswas A. Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput., 2010; 217; 869-877.
He JH, Wu XH. Exp-function method for nonlinear wave equations. Chaos, Solitons Fract. 2006; 30 (3); 700–708.
Noor MA, Mohyud-Din ST, Waheed A. Exp-function method for travelling wave solutions of nonlinear evolution equations. Appl. Math. Comput. 2010; 216; 477-483.
Hafez MG, Ali MY, Kauser MA, Akter MT. Application of the exp (-Φ (ξ))-expansion method to find exact solutions of the (1+1)-dimensional dispersive long wave equations. British J. Math. Com. Sci. 2014; 4 (22); 3191-3201. Doi: 10.9734/BJMCS/2014/13129.
Hafez MG, Alam MN, Akbar MA. Traveling wave solutions for some important coupled nonlinear physical models via the coupled Higgs equation and the Maccari system. Journal of Kink Saud University.-Science (in press) 2014.
Akbar MA, Ali NHM. Solitary wave solutions of the fourth order Boussinesq equation through the exp (-Φ (ξ))-expansion method. Springer Plus 2014; 3: 343. doi: 10.1186/2193-1801-3-344
Hafez MG, Kauser MA, Akter MT. Some new exact traveliing wave solutions of the cubic nonlinear Schrodinger equation using the exp (-Φ (ξ))-expansion method. International J. Sci. Eng. Tech. 2014; 3 (7); 848-851.
Hafez MG, Kauser MA, Akter MT. Some new exact traveling wave solutions for the Zhiber-Shabat equation. British J. Math. Com. Sci. 2014; 4 (18); 2582-2593.
Lai S, Wu YH, Zhou Y. Some physical structures for the (2+1)-dimensional Boussinesq water equation with positive and negative exponents. Comput Math Appl 2008; 56: 339–345.
Yildirm A, Mohyud-Din ST. A variational approach for soliton solution of good Boussinesq equation. J. King Saud Uni. (Sci.) 2010; 22; 205–208.
Neyrame A, Roozi A, Hosseini S S and Shafiof S M. Exact traveling wave solutions for some nonlinear partial differential equations. J. King Saud Univ. (Sci.) 2012; 22: 275-278.