Burgers Benzeri Denklemin Bazı Seyahat Eden Dalga Çözümleri

Year 2021, Volume: 3 Issue: 1, 14 - 28, 31.03.2021

İbrahim Enam İnan

https://doi.org/10.46740/alku.837443

Abstract

Bu makalede, Burgers benzeri denklemin bazı seyahat eden dalga çözümlerini bulmak için Ba ̈cklund Dönüşümü, Benzerlik indirgeme ve Adomian Ayrıştırma yöntemleri denkleme uygulanmıştır. Yukarıdaki yöntemlerin denkleme uygulanması sonucunda denklemin rasyonel, hiperbolik ve trigonometrik çözümleri elde edilmiştir. Daha sonra Mathematica 11.2 programını kullanarak bu çözümlerin, denklemi sağladığı görülmüştür.

Keywords

Burgers benzeri denklem , Ba ̈cklund dönüşümü , Benzerlik indirgeme

References

• [1] Y. Shang, Backlund transformation, Lax pairs and explicit exact solutions for the shallow water waves equation, Appl.Math.Comput. 187 (2007) 1286-1297.
• [2] T.L. Bock, M.D. Kruskal, A two-parameter Miura transformation of the Benjamin-Ono equation, Phys. Lett. A 74 (1979) 173-176.
• [3] V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991.
• [4] A.M. Abourabia, M.M. El Horbaty, On solitary wave solutions for the two-dimensional nonlinear modified Kortweg-de Vries-Burger equation, Chaos Solitons Fractals 29 (2006) 354-364.
• [5] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992) 650-654.
• [6] Y. Chuntao, A simple transformation for nonlinear waves, Phys. Lett. A 224 (1996) 77-84.
• [7] F. Cariello, M. Tabor, Painleve expansions for nonintegrable evolution equations, Physica D 39 (1989) 77-94.
• [8] M.L. Wang, Exact Solutions for a Compound KdV-Burgers equations, Phys. Lett. A 213 (1996) 279-287.
• [9] E. Fan, Two new application of the homogeneous balance method, Phys. Lett. A 265 (2000) 353-357.
• [10] G. Adomian, A Review of the Decomposition Method and some Recent Results for Nonlinear Equations, Math. Comp. Model., 13 (1990) 17-43.
• [11] P.A. Clarkson, New similarity solutions for the modified boussinesq equation, J. Phys. A: Math. Gen. 22 (1989) 2355-2367.
• [12] W. Malfliet, Solitary wave solutions of nonlinear wave equations, Am. J. Phys. 60 (1992) 650-654.
• [13] E. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A 277 (2000) 212-218.
• [14] S. A. Elwakil, S.K. El-labany, M.A. Zahran, R. Sabry, Modified extended tanh-function method for solving nonlinear partial differential equations, Phys. Lett. A 299 (2002) 179-188.
• [15] H. Chen, H. Zhang, New multiple soliton solutions to the general Burgers-Fisher equation and the Kuramoto-Sivashinsky equation, Chaos Soliton Fract 19 (2004) 71-76.
• [16] Z. Fu, S. Liu, Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations, Phys. Lett. A 290 (2001) 72-76.
• [17] S. Shen, Z. Pan, A note on the Jacobi elliptic function expansion method, Phys. Let. A 308 (2003) 143-148.
• [18] H. T. Chen, Z. Hong-Qing, New double periodic and multiple soliton solutions of the generalized (2+1)-dimensional Boussinesq equation, Chaos Soliton Fract 20 (2004) 765-769.
• [19] Y. Chen, Q. Wang, B. Li, Jacobi elliptic function rational expansion method with symbolic computation to construct new doubly periodic solutions of nonlinear evolution equations, Z. Naturforsch. A 59 (2004) 529-536.
• [20] Y. Chen, Z. Yan, The Weierstrass elliptic function expansion method and its applications in nonlinear wave equations, Chaos Soliton Fract 29 (2006) 948-964.
• [21] M. Wang, X. Li, J. Zhang, The (G^'/G) -expansion method and travelling wave solutions of nonlinear evolutions equations in mathematical physics, Phys. Lett. A 372 (2008) 417-423.
• [22] S.Guo, Y. Zhou, The extended (G^'/G) -expansion method and its applications to the Whitham-Broer-Kaup-like equations and coupled Hirota-Satsuma KdV equations, Appl.Math.Comput. 215 (2010) 3214-3221.
• [23] H. L. Lü, X. Q. Liu, L. Niu, A generalized (G^'/G) -expansion method and its applications to nonlinear evolution equations, Appl. Math. Comput. 215 (2010) 3811-3816.
• [24] L. Li, E. Li, M. Wang, The (G^'/G,1/G) - expansion method and its application to travelling wave solutions of the Zakharov equations, Appl. Math-A J. Chin. U 25 (2010) 454-462.
• [25] J. Manafian, Optical soliton solutions for Schrödinger type nonlinear evolution equations by the tan(ϕ(φ)/2) – expansion Method, Optik 127 (2016) 4222-4245.
• [26] Mostafa M.A Khater, Emad H.M. Zahran, Soliton Soltuions of Nonlinear Evolutions Equation by using the Extended exp(-φ (ξ))-expansion method, International Journal of Computer Applications, 145, 3 (2016) 1-5..
• [27] J.H. He, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals 30 (2006) 700–708
• [28] Mostafa M.A Khater, “Extended exp(-φ (ξ))-Expansion Method for Solving the Generalized Hirota-Satsuma Coupled KdV System”, Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, 15, 7, Version 1.0 Year 2015.
• [29] Mostafa M.A Khater and Emad H.M. Zahran, “Modified extended tanh function method and its applications to the Bogoyavlenskii equation”, Applied Mathematical Modelling,40, 1769-1775, 2016.
• [30] İ.E.İnan. Kısmi Diferensiyel Denklemler için Bazı yaklaşım Metotları ve Uygulamaları. Doktora Tezi, Fırat Üniversitesi, Elazığ, Türkiye, 2004.
• [31] G. Ebadi, A. Biswas, “Application of the (G^'/G) -expansion method for nonlinear diffusion equations with nonlinear source”, Journal of the Franklin Institute, 347, 7, 1391–1398, 2010.
• [32] Zheyna Yan, “New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations”, Physics Letters A, 292, 100-106, 2001.
• [33] M.M.A. Khater, A.R. Seadawy and d. Lu, “Dispersive solitary wave solutions of new coupled Konno-Oono, Higgs field and Maccari equations and their applications”, Journal of King Saud University, 30, 417–423, 2018.
• [34] X. Zhao, L. Wang, W. Sun, “The repeated homogeneous balance method and its applications to nonlinear partial differential equations”, Chaos Solitons and Fractals, 28, 448–453, 2006.
• [35] A. Biswas, E. Topkara, S. Johnson, E. Zerrad, S. Konar, “Quasi-stationary optical solitons in non-Kerr law media with full nonlinearity”, Journal of Nonlinear Optical Physics & Materials, 20, 309–325, 2011.
• [36] A. Biswas, A.B. Aceves,” Dynamics of solitons in optical fibers”, Journal of Modern Optics, 48, 1135–1150, 2001.
• [37] A.M. Wazwaz, “Burgers hierarchy: Multiple kink solutions and multiple singular kink solutions”, Journal of the Franklin Institute, 347, 618–626, 2010.
• [38] E.M.E. Zayed, M.A.M. Abdelaziz, “The two variables (G^'/G,1/G^' )-expansion method for solving the nonlinear KdV-mKdV equation”, Mathematical Problems in Engineering, article ID 725061, 14 pp, 2012.
• [39] Mostafa M.A Khater and Emad H.M. Zahran,” New solitary wave solution of the generalized Hirota-Satsuma couple KdV system”, International Journal of Scientific &Engineering Research, 6, 1324-1331, 2015.
• [40] J. Manafian Heris, M. Lakestani, “Solitary wave and periodic wave solutions for variants of the KdV-Burger and the K(n, n)-Burger equations by the generalized tanh-coth method”, Communications in Numerical Analysis, 1–18, 2013.
• [41] A.M.Wazwaz, H. Triki,” Multiple soliton solutions for the sixth-order Ramani equation and a coupled Ramani equation”, Applied Mathematics and Computation, 216, 332–336, 2010.
• [42] A.M. Wazwaz, “The tanh–coth method for solitons and kink solutions for nonlinear parabolic equations”, Applied Mathematics and Computation, 188 (2), 1467–1475, 2007.
• [43] J. Manafian Heris, I. Zamanpour,” Analytical treatment of the coupled Higgs equation and the Maccari system via Exp-function method”, Acta Universitatis Apulensis, 33, 203–216, 2013.