Journal of Mathematics Research; Vol. 6, No. ; ISSN 96-9795 E-ISSN 96-989 Published by Canadian Center of Science and Education New Approach of Ǵ/G Expansion Method. Applications to KdV Equation Mohammad Ali Bashir & Alaaeddin Amin Moussa,3 Mathematics Department, Faculty of Science, Alnillin University, Alkortom, Alsodan Mathematics Department, Faculty of post Graduate Studies, Omdurman Islamic University, Alkortom, Alsodan 3 Department of Management Information System and Production Management, College of Business & Economics, Qassim University, Buraidah, K.S.A. Correspondence: Alaaeddin Amin Moussa, Mathematics Department, Faculty of post Graduate Studies, Omdurman Islamic University, Alkortom, Alsodan. E-mail: Alaamath8@gmail.com Received: November 3, 3 Accepted: December, 3 Online Published: December, 3 doi:.5539/jmr.v6np URL: http://dx.doi.org/.5539/jmr.v6np Abstract This paper proposes a new approach of Ǵ/G -expansion method for constructing more general exact solutions of nonlinear evolution equation. By using this method numerous new and more general exact solutions have been obtained. We will apply this new method to solve the nonlinear KdV equation. Various exact traveling wave solutions of these equations are obtained that include the exponential function solutions, the hyperbolic function solutions and the trigonometric function solutions. Further, the method is efficient to solve nonlinear evolution equations in many areas such as physics and engineering. Keywords: Ǵ/G -expansion method, KdV equation, exact solutions, travelling wave solution. Introduction Several phenomena in various fields are frequently described by nonlinear partial differential equation NLPDEs, for instance nonlinear optics, elastic media, plasma physics, fluid mechanics, chemistry, biology and many others Rogers & Shadwick, 98; Akbar, Ali, & Zayed,. For better understanding of nonlinear phenomena and its real life applications, it is more significant to establish exact traveling wave solutions. Recently, many methods have been proposed to generate analytical solutions. For instance, the Backlund transformation method Rogers & Shadwick, 98, the inverse scattering method Ablowitz & Clarkson, 99 the truncated Painleve expansion method Weiss, Tabor, & Carnevale, 98, the Weirstrass elliptic function method Kudryashov, 99, the Hirotas bilinear transformation method Hirota, 97, the Jacobi elliptic function expansion method S. K. Liu, Fu, S. D. Liu, & Zhao,, the generalized Riccati equation method Yan & Zhang, ; Naher & Abdullah,, the tanh-coth method Malfliet, 99; Wazwaz, 7, the F-expansion method Wang & Li, 5; Abdou, 7, the variational iteration method Mohyud-Din, M. Noor, K. Noor, & Hosseini,, the direct algebraic method Soliman & Abdo, 9, the homotopy perturbation method Mohyud-Din, Yildirim, & Sariaydin,, the Exp-function method He & Wu, 6; Ma & Zhu,, and others Wazwaz, ; Bagarti, Roy, Kundu, & Dev,. This work intends to generate many novel and more general exact traveling wave solutions, by proposing new approach of Ǵ/G -expansion method for investigating NLEEs. The proposed method is used to solve the KdV equation to show its efficiency. Description of New Approach of Ǵ/G Expansion Method Let us consider the following equation: Fu, u t, u x, u tt, u xt, u xx, u xxt,... = F is a polynomial in its arguments that includes nonlinear terms and the highest order derivatives. Step Use the transformation u x, t = u ; = k x k t, k, k are constant, to be determined latter. permits use reducing to an ODE for u = u in the form
Pu, k u, k u, k k u,... =, u = du d, Note that the left hand side of is a polynomial in u and its various derivatives. Step Suppose that the NLODE has the following solution: m i= α i A G G B B A G G G = G satisfies the following auxiliary ordinary differential equation: i G G μg = A, A, B, B,α i i =,,,...m,and μ are constants to be determined later, m is a positive integer. Step 3 Find the value of m by balancing the highest order derivative and nonlinear terms in 3. Step Substituting 3 into and using, and then setting all the coefficients of Ǵ G of the obtained systems numerator to zero, produces a set of over determined nonlinear algebraic equations for k, k, A, A, B, B and α i i =,,,...m. Step 5 Suppose that the value of k, k, A, A, B, B and α i i =,,,...m can be found by solving the algebraic system which are found in step. Because the solutions of are known, substituting k, k, A, A, B, B, c,α i i =,,,...m and the general solutions of into 3, we have the exact solutions of the nonlinear PDEs. 3. Exact Solutions of the Nonlinear KdV Equation In this section, the proposed methods is applied for studying the KdV equation: using the wave transformation into the 5, produces: u t δ u u x u xxx = 5 k ú δ k u ú k 3 u3 = 6 the homogeneous balance between u ú and u 3 in 6, we obtain m =. Therefore, the solution of 6 is of the form: A G G A A G B B G α G A 7 G B B G G α,α,α, A, A, B and B are arbitrary constants to be determined. Substituting 7 and into 6, and then setting all the coefficients of j Ǵ G, j =,,,... of the obtaining systems numerator to zero, yields a set of over determined nonlinear algebraic equations for α,α,α, A, A, B, B, k and k. Solving the over determined algebraic equations by Maple, we find the following results: Case α = α,α = α,α = α, A = A, B = B, B = B 8 μ B α B A α A B α α B 3 A = 8α B δ 8 α A α α B k = B δ k [ k = B α B α α 6α B B α α α B A A a μ ] 5
Substituting 8 into 7, we have: A. G G α A G G B α B A α A B α α B 8α B B B G G B α B A α A B α α B 8α B B B G G δ 8 α A α α B = B x δ k B α B α α 6α B B α α α B A A a μ t Therefor, the following three types of exact solution of 5 are obtained: Case - When μ, we obtain: α A A c sinh c cosh B B B α B A α A B α α B 8α B c sinh c cosh c sinh c cosh c sinh B B c cosh B α B A α A B α α B 8α B α,α,α, A, B, B, c and c are constants. Particularly, setting α =,α =,α =, A =, B =, B = 3,μ =, =, c =, c,δ = 6. We find: 7 coth 7 coth 3 coth 3 coth = [ ] 7 3 8 x t 6 6 See Figure. Case - When μ we obtain: α c sin A c sin A μ μ μ μ c cos μ c sin μ μ c sin c cos B B μ c sin c cos μ c sin μ μ c sin c cos B B. μ c sin 6 B α B A α A B α α B 8α B μ μ B α B A α A B α α B 8α B μ μ 9 3
α,α,α, A, B, B, c and c are constants. Particularly, setting α =,α =,α =, A =, B =, B =,μ= 3,= 3, c =, c,δ= 6. We find: 3 tan 3 9 3 3 3 tan 3 tan 3 9 3 3 tan 3 = 3 53 8 x t 3 8 See Figure. Case -3 When μ =, we obtain: A c c c A B α α B 8α B α A B B. c c c c c c B B A B α α B c c c 8α B 5 α,α,α, A, B, B, c and c are constants. In particular, setting α =,α =,α =, A =, B = 3, B =,μ=,=, c =, c =,δ= 6. We find: See Figure 3. Case 3 53 7 6 3 = 3 8 x t 8 A = A, A = A, B = B, B =, k = k,α = α 7 k μ 8A k A A k A 8μ 6μ δα A A k = A A α = α = 6A B k μ δ A A 3B k μ δ A A Substituting 7 into 7, we have: 3B k μ A α A δ A A B G G A A B B G G G G A G G k μ 8A k A A [ k = k A 8μ ] 6μ δα A A x A A t 8 9 Thus, the following three types of exact solution of 6 are obtained: 7
Case - When μ we obtain: 6A B k μ α δ A A A B A. c sinh c cosh A c sinh B. c cosh A. c sinh c cosh A c sinh B. c cosh A, A, B, k,α,δ,c and c are constants. Particularly, setting, A =, A =, B =,μ=,=, c =, c =,δ= 6, k =,α =. We find: See Figure. Case - When μ we obtain: 6 cosh sinh = x t 6A B k μ α δ A A A B μ c. sin A c cos μ μ c sin μ A μ c sin c cos μ B μ c sin μ μ c sin A c cos μ μ c sin μ A μ c sin c cos μ B μ c sin μ A, A, B, k,α,δ,c and c are constants. Particularly, setting: A = 3, A =, B =,μ= 3,= 3, c =, c =,δ= 6, k =,α =. We find: sin 3 3 9 sin 3 3 = x 77 t See Figure 5. Case -3 When μ =, we get the not important solution in the form: α α is arbitrary constant. 8
Case 3 α = α,α = α, B = B, B =, A = A, k = k 5 α = B k δa A = A B k α δa B k μ α δ A 8α δk B k = 96k B 8k B Substituting 5 into 7, we have: B [A k δa [ A G G G G AB k α δa B k ] ] AB k α δa B k 6 = k x 96k B μ α δ A 8α δk B 8k B t 7 Therefore, the three we following types of exact solution of 6 are obtained: Case 3- When μ, we obtain: B k δa A c cosh A c cosh c sinh c cosh c sinh c cosh AB k α δa B k AB k α δa B k 8 α,α, B, k, A,δ,c and c are constants. Particularly, setting α =,α =,α =, k =,μ=,=, c =,δ= 6, B =, A =. We find: See Figure 6. Case 3- When μ, we obtain: B k δa A c sin coth. coth 9 A c sin α,α, k, A, B,δ,c and c are constants. = x 9 t 3 μ μ μ μ μ c cos μ c sin μ c cos μ c sin 9 AB k α δa B k AB k α δa B k 3
Particularly, setting α =,α =, k =, A =, B =,μ=,= 3, c =,δ= 6. We find: tan tan 3 See Figure 7. Case 3-3 When μ =, we obtain: [ B A. c c c k δa = x t 33 [ A. c c c AB k α δa B k ] ] AB k α δa B k α,α, A, B,δ,k, c and c are constants. Particularly, setting α =,α =, k =, B =, A =,μ=,=, c =,δ= 6. We find: 3 7 8 35 See Figure 8.. Table Graphics = x t 36.5.5.5 - Figure. - - - 7 coth 3 coth 7 coth 3 coth 3 Figure. - - 3 tan 3 9 3 3 tan 3 - - 3 tan 3 9 3 3 tan 3-6 -.5 5-3 - - - - -.5 - - - - Figure 3. 353 7 3 Figure. 6 cosh sinh 3
.6E8.E8 8E7 E7 - - - - Figure 5. 9 sin33 sin3 E3.5E3 E3 5E9 - - Figure 6. coth coth - - 7 E3 6 5 3E3 E3 3 - - - - E3 - - - - Figure 7. tan tan Figure 8. 7 8 5. Remarks and Conclusion We note that the solution in the Case 3, a perfect fit with solutions that give them the classic Ǵ G -expansion method. However, the solution in both cases and are new, and cannot be obtained from the classic Ǵ G - expansion method. [ For example, in the case, if we substitute A = B ; B = ; α = α.wehave u = α α Ǵ G Ǵ ] α G. The proposed solutions have been tested with Maple by substituting them in Equation 5. In this article, a novel approach of Ǵ G -expansion method is proposed, and applied to find the exact solutions for the nonlinear KdV equation. These solutions comprise the trigonometric function solutions, the hyperbolic function solutions, and the rational function solutions. This work shows that, the new approach of Ǵ G -expansion method is efficient and can be applied to several other NLPDEs in mathematical physics. References Abdou, M. A. 7. The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos, Solitons, and Fract, 3, 95-. http://dx.doi.org/.6/j.chaos.5.9.3 Ablowitz, M. J., & Clarkson, P. A. 99. Cambridge Univ. Press, Cambridge. Akbar, M. A., Ali, N. H. M., & Zayed, E. M. E.. A Generalized and Improved-Expansion Method for Nonlinear Evolution Equations. Math. Prob. Engr.,, Article ID 59879. http://dx.doi.org/.55//59879 Bagarti, T., Roy, A., Kundu, K., & Dev, B. N.. A reaction diffusion model of pattern formation in clustering of adatoms on silicon surfaces. AIP Advances,,. http://dx.doi.org/.63/.75759 He, J. H., & Wu, X. H. 6. Exp-function method for nonlinear wave equations. Chaos, Solitons and Fract, 33, 7-78. http://dx.doi.org/.6/j.chaos.6.3. Hirota, R. 97. Exact Solution of the Korteweg-de Vries Equation for Multiple Collisions of Solitons. Phys. 3
Rev. Lett., 7, 9-9. Kudryashov, N. A. 99. Exact solutions of the generalized Kuramoto-Sivashinsky equation. Phys. Lett. A, 7, 87-9. http://dx.doi.org/.6/375-96999-x Liu, S. K., Fu, Z., Liu, S. D., & Zhao, Q.. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A, 89-, 69. http://dx.doi.org/.6/s375-9658- Ma, W. X., & Zhu, Z.. Solving the 3-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Compu., 8, 87-879. http://dx.doi.org/.6/j.amc..5.9 Malfliet, W. 99. Solitary wave solutions of nonlinear wave equations. Am. J. Phys., 6, 65. http://dx.doi.org/.9/.7 Mohyud-Din, S. T., Noor, M. A., Noor, K. I., & Hosseini, M. M.. Solution of singular equations by He s variational iteration method. Int. J. Nonlinear Sci. Numer. Simulat,, 8-86. Mohyud-Din, S. T., Yildirim, A., & Sariaydin, S.. Numerical soliton solution of the Kaup-Kupershmidt equation. Int. J. Numerical Meth. Heat Fluid Flow, 3, 7-8. http://dx.doi.org/.8/96553859 Naher, H., & Abdullah, F. A.. New Traveling Wave Solutions by the Extended Generalized Riccati Equation Mapping Method of the -Dimensional Evolution Equation. J. Appl. Math.,, Article ID 8658. http://dx.doi.org/.55//8658 Rogers, C., & Shadwick, W. F. 98. Bäcklund Transformations and Their Applications. New York: Academic Press. Soliman, A. A., & Abdo, H. A. 9. New exact Solutions of nonlinear variants of the RLW, the PHI-four and Boussinesq equations based on modified extended direct algebraic method. Int. J. Nonlinear Sci., 73, 7-8. Wang, M. L., & Li, X. Z. 5. Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation. Chaos, Solitons and Fract, 5, 57-68. http://dx.doi.org/.6/j.chaos..9. Wazwaz, A. M. 7. The tanhccoth method for solitons and kink solutions for nonlinear parabolic equations. Appl. Math. and Comput, 88, 67. http://dx.doi.org/.6/j.amc.6..3 Wazwaz, A. M.. A new -dimensional Korteweg-de Vries equation and its extension to a new 3- dimensional Kadomtsev-Petviashvili equation. Phys. Scr., 83. http://dx.doi.org/.88/3-899/8/3/35 Weiss, J., Tabor, M., & Carnevale, G. 983. The Painlevé property for partial differential equations. J. Math. Phys.,, 5. http://dx.doi.org/.63/.557 Yan, Z., & Zhang, H. Q.. New explicit solitary wave solutions and periodic wave solutions for Whitham- Broer-Kaup equation in shallow water. Phy. Lett. A, 585-6, 355. http://dx.doi.org/.6/s375-96376- Copyrights Copyright for this article is retained by the authors, with first publication rights granted to the journal. This is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license http://creativecommons.org/licenses/by/3./. 3