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HİPERBOLİK TANJANT YÖNTEMİNİN KLASİK BOUSSINESQ SİSTEMİNE UYGULANMASI

Yıl 2008, Sayı: 10, 159 - 171, 01.06.2008

Öz

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.

Kaynakça

  • Abdou, M.A. (2007). The extended tanh method and its applications for solving nonlinear physical models, Applied Mathematics and Computation 190, 988–996.
  • 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– 315.
  • 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, Cambridge.
  • 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, Springer,Berlin.
  • Nuseir, A. (1994). Symbolic Computation of Exact Solitions of Nonlinear Partial Differential Equations Using Direct Methods ”, thesis of Doctor of Philosophy.
  • 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.

Application of Hyperbolic Tangent Method to Classical Boussinesq System

Yıl 2008, Sayı: 10, 159 - 171, 01.06.2008

Öz

Tanh method is a powerful solution method for the computation of one-dimensional 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.

Kaynakça

  • Abdou, M.A. (2007). The extended tanh method and its applications for solving nonlinear physical models, Applied Mathematics and Computation 190, 988–996.
  • 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– 315.
  • 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, Cambridge.
  • 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, Springer,Berlin.
  • Nuseir, A. (1994). Symbolic Computation of Exact Solitions of Nonlinear Partial Differential Equations Using Direct Methods ”, thesis of Doctor of Philosophy.
  • 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.
Toplam 39 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Bölüm Research Article
Yazarlar

Mustafa Mızrak Bu kişi benim

Yayımlanma Tarihi 1 Haziran 2008
Yayımlandığı Sayı Yıl 2008 Sayı: 10

Kaynak Göster

APA Mızrak, M. (2008). HİPERBOLİK TANJANT YÖNTEMİNİN KLASİK BOUSSINESQ SİSTEMİNE UYGULANMASI. Dicle Üniversitesi Ziya Gökalp Eğitim Fakültesi Dergisi(10), 159-171.