Application of Hyperbolic Tangent Method to Classical Boussinesq System

Journal Name:

Publication Year:

Abstract (2. Language): 
Tanh method is a powerful solution method for the computation of onedimensional travelling wave solutions of evolution and wave equations. This method is based on the fact that solutions may be written as a finite power series of a hyperbolic tangent. In this work, we apply Hyperbolic Tangent (Tanh) method to solve Classical Boussinesq systems of partial differential equations.
Abstract (Original Language): 
Tanh yöntemi bir boyutlu lineer olmayan dalga ve değişimsel denklemlerinin yönlendirilmiş dalga çözümlerinde kullanılan çok güçlü bir çözüm yöntemidir. Bu yöntem çözümlerin sonlu hiperbolik tanjant kuvvet serileri şeklinde yazılabilmesi temeline dayanır. Bu çalışmada, aynı yöntem lineer olmayan Klasik Boussinesq kısmi diferansiyel denklem sistemine uygulandı.



Abdou, M.A. (2007). The extended tanh method and its applications for solving
nonlinear physical models, Applied Mathematics and Computation 190,
Ablowitz, M.J., Clarkson, P.A. (1991). Soliton, Nonlinear Evolution Equations and
Inverse Scattering,Cambridge University Press, New York.
Ablowitz, M., Kaup, D., Newell, A. , Segur, H. (1974). The inverse scattering
transform-Fourier analysis for nonlinear problems, Stud. Appl. Math.53, 249–
Bluman, G.W., Kumei, S.(1989). Symmetries and Differential Equations, Springer-
Verlag, New York.
Cariello, F., Tabor, M (1989). Physica D 39, 77.Chen, Y., Li, B., Zhang, H.Q.(2002). Commun. Theor. Phys. 38, 261.
Chen, Y., Li, B., Zhang, H.Q.(2002). J Phys A: Math. Gen. 35, 8253.
Chen, Y., Zheng, Y.(2003). Generalized extended tanh-function method to construct
new explicit exact solutions for the approximate equations for long water
waves, Int. J. Mod. Phys. C 14 (4) .
Debnath, L. (1997). Nonlinear Partial Differential Equations for Scientist and
Engineers, Birkhäuser, Boston .
Elwakil, S.A., El-labany, S.K., Zahran, M.A. and Sabry, R.(2002). Phys. Lett.299, 179.
Fan, E.(2000). Extented tanh-function method and its applications to nonlinear
equations. Phys. Lett. A.277,212.
Fan, E. (2002). Comput. Math. Appl. 43 , 671.
Fan, E., Zhang, J., Benny, Y.C.(2001). Hon Phys. Lett. A 291, 376.
Gao, Y.T., Tian, B. (2001). Comput. Phys. Commun. 133, 158.
Gu, C.H and et al, (1990). Soliton Theory and its Application, Zhejiang Science and
Technology Press,Zhejiang.
Hirota, R. (2004). The Direct Method in Soliton Theory, Cambridge University Press,
Kakutani, T. and Kawahara, T.(1970). J. Phys. Soc. Japan 29, 1068
Khater, A.H., Malfiet, W., Callebaut, D.K., and Kamel, E.S.(2002). Chaos Soliton.
Fract. 14, 513.
Li, Y., Ma, W. and Zhang Jin, E.(2000).Darboux transformation of classical
Boussinesq system and its new solutions, Phys. Lett. A, 275, 60-66.
Li Z.B, Liu Y.P. (2002). Comput Phys Commun ,148,56.
Li Z.B, Liu Y.P.(1993). J. Phys. A: Math. Gen. 26 , 6027.
Lou, S., Huang, G., Ruan, H.(1991).J. Phys. A: Math. Gen. 24 , L584
Malfliet, W. (1992). Solitary wave solutions of nonlinear wave equations, Am. J. Phys.
60, 650-654.
Malfliet, W. and Hereman, W.(1996). The tanh method: I. Exact solutions of nonlinear
evolution and wave equations, Physica Scripta 54, 563-568
Malfliet, W. and Hereman, W.(1996). The tanh method: II. Perturbation technique for
conservative systems, Physica Scripta 54, 569-575 .
Malfliet, W.(2004). The tanh method: a tool for solving certain classes of nonlinear
evolution and wave equations, J. Comp. Appl. Math 164-165, 529-541 .
Malfliet, W. and Hereman W. (2005).The Tanh Method: A Tool to Solve Nonlinear
Partial Differential Equations with Symbolic Software, 9th World
Multiconference on Systemics,Cybernetics and Informatics (WMSCI2005) ,
Orlando , Florida,July 10-13, pp.165-168.
Matveev, V.B., Salle, M.A. (1991). Darboux Transformation and Soliton,
Nuseir, A. (1994). Symbolic Computation of Exact Solitions of Nonlinear Partial
Differential Equations Using Direct Methods ”, thesis of Doctor of
Olver, P.J. (1986). Applications of Lie Groups to Differential Equations, Springer-
Verlag, New York.
Parkes, E.J., Duffy, B.R.(1996). Phys. Lett. A 214, 271.
Parkes, E.J., Duffy, B.R. (1997). Travelling solitary wave solutions to a compound
KdV-Burgers equation, Phys. Lett. A 229, 217.Tanoğlu, G. (2007). Solitary wave solution of nonlinear multi-dimensional wave
equation by bilinear transformation method, Communications in Nonlinear
Science and Numerical Simulation 12, 1195–1201.
Tian, B., Gao, Y.T. (2002). Z. Naturforsch. A 57, 39.
Wang, M.L. (1996). Application of a homogeneous balance method to exact solutions
of nonlinear equations in mathematical physics, Phys.Lett. A216,67.
Wazwaz, A.M.(2002). Partial Differential Equations: Methods and Applications,
Balkema, The Netherlands.
Wazwaz, A.M. (2004). The tanh method for travelling wave solutions of nonlinear
equations. Applied Mathematics and Computation. 154(3), 713-723.
Wazwaz, A.M. (2005). The tanh and the sine–cosine methods for compact and
noncompact solutions of the nonlinear Klein–Gordon equation, Applied
Mathematics and Computation 167, 1179–1195
Yan, Z.Y.(2001). New explicit travelling wave solutions for two new integrable
coupled nonlinear evolution equations, Phys. Lett. A 292, 100.

Thank you for copying data from http://www.arastirmax.com