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NEW APPROACH TO THE SOLUTIONS OF THE PIB EQUATION

Year 2011, Volume 01, Issue 1, 35 - 40, 01.06.2011

Abstract

In this paper, based on the Exp-function method and mathematical derivation, we obtain several explicit and exact traveling wave solutions for the PIB equation.

References

  • Ablowitz, M.J. and Segur, H., (1981), Soliton and the inverse scattering transformation, SIAM, Philadelphia, PA.
  • He, J.H., (1999), Variational iteration method-a kind of non-linear analytical technique: some examples. Int. J. Non-linear Mech. 34 (4), 699-708.
  • He, J.H., (2006), New interpretation of homotopy perturbation method. Int. J. Mod. Phys. B 20 (18), 2561-2568.
  • He, J.H., (2005), Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals 26(3), 695-700.
  • He, J.H., (2005), Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlin- ear Sci. Numer. Simul. 6 (2), 207-208.
  • Wadati, M., (1975), Wave propagation in nonlinear lattice, I, J. Phys. Soc. Jpn. 38, 673-680.
  • Wang, D.S. and Zhang, H.Q., (2005), Auto-Backlund transformation and new exact solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equation, Int. J. Mod. Phys. C 16(3), 393.
  • Wazwaz, A.M., (2005), The tanh method: solitons and periodic solutions for the Dodd-Bullough- Tzikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos, Solitons and Fractals 25, 55-63.
  • Franz, P. and Hongyou, W., (1995), Discretizing constant curvature surfaces via loop group factoriza- tions: the discrete sine- and sinh-Gordon equations, J. Geomet. Phys. 17 (3), 245-260.
  • Xiqiang, Z., Limin, W. and Weijun, S., (2006), The repeated homogeneous balance method and its applications to nonlinear partial differential equations, Chaos, Solitons and Fractals 28(2), 448-453.
  • Fan, E. and Jian, Z., (2002), Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A 305 (6), 383-392.
  • Novikov, D.P., (1999), Algebraic geometric solutions of the Harry Dym equations, Math. J. 40, 136.
  • He, J.H., (2006), Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B 20 (10), 1141-1199.
  • He, J.H., (2006), X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals 30 (3), 700-708.
  • He, J.H., (2006), Non-perturbative method for strongly nonlinear problems. Berlin: dissertation. De- Verlag im internet GmbH.
  • El-Wakil, S.A., Madkour, M.A. and Abdou, M.A., (2007), Application of Exp-function method for nonlinear evolution equations with variable coefficients, Phys. Lett. A 369, 62-69.
  • He, J.H. and Wu, X.H., (2006), Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals 29, 108-113.
  • Wu, X.H. and He, J.H., EXP-function method and its application to nonlinear equations, Chaos, Solitons and Fractals (in press), doi:10.1016/j.chaos.2007.01.024.
  • He, J.H. and Abdou, M.A., (2007), New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos, Solitons and Fractals, 34, 1421-1429.
  • Zhu, S.D., (2007), Exp-function method for the discrete mKdV lattice, International Journal of Non- linear Sciences and Numerical Simulation 8(3), 465-468.
  • Wang, Q., Chen, Y. and Zhang, H., (2005), A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation. Chaos, Solitons and Fractals , 25, 101928.
  • Hong, K.Z. et al., (2003), Painleve analysis and some solutions of (2 + 1)-dimensional generalized Burgers equations. Commun Theor Phys, 39, 3934.
  • Tang, X.Y. and Lou, S., (2003), Variable separation solutions for the (2 + 1)-dimensional Burgers equations. Chin Phys Lett., 20(3), 335.

Year 2011, Volume 01, Issue 1, 35 - 40, 01.06.2011

Abstract

References

  • Ablowitz, M.J. and Segur, H., (1981), Soliton and the inverse scattering transformation, SIAM, Philadelphia, PA.
  • He, J.H., (1999), Variational iteration method-a kind of non-linear analytical technique: some examples. Int. J. Non-linear Mech. 34 (4), 699-708.
  • He, J.H., (2006), New interpretation of homotopy perturbation method. Int. J. Mod. Phys. B 20 (18), 2561-2568.
  • He, J.H., (2005), Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals 26(3), 695-700.
  • He, J.H., (2005), Homotopy perturbation method for bifurcation of nonlinear problems. Int. J. Nonlin- ear Sci. Numer. Simul. 6 (2), 207-208.
  • Wadati, M., (1975), Wave propagation in nonlinear lattice, I, J. Phys. Soc. Jpn. 38, 673-680.
  • Wang, D.S. and Zhang, H.Q., (2005), Auto-Backlund transformation and new exact solutions of the (2+1)-dimensional Nizhnik-Novikov-Veselov equation, Int. J. Mod. Phys. C 16(3), 393.
  • Wazwaz, A.M., (2005), The tanh method: solitons and periodic solutions for the Dodd-Bullough- Tzikhailov and the Tzitzeica-Dodd-Bullough equations, Chaos, Solitons and Fractals 25, 55-63.
  • Franz, P. and Hongyou, W., (1995), Discretizing constant curvature surfaces via loop group factoriza- tions: the discrete sine- and sinh-Gordon equations, J. Geomet. Phys. 17 (3), 245-260.
  • Xiqiang, Z., Limin, W. and Weijun, S., (2006), The repeated homogeneous balance method and its applications to nonlinear partial differential equations, Chaos, Solitons and Fractals 28(2), 448-453.
  • Fan, E. and Jian, Z., (2002), Applications of the Jacobi elliptic function method to special-type nonlinear equations, Phys. Lett. A 305 (6), 383-392.
  • Novikov, D.P., (1999), Algebraic geometric solutions of the Harry Dym equations, Math. J. 40, 136.
  • He, J.H., (2006), Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B 20 (10), 1141-1199.
  • He, J.H., (2006), X.H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons and Fractals 30 (3), 700-708.
  • He, J.H., (2006), Non-perturbative method for strongly nonlinear problems. Berlin: dissertation. De- Verlag im internet GmbH.
  • El-Wakil, S.A., Madkour, M.A. and Abdou, M.A., (2007), Application of Exp-function method for nonlinear evolution equations with variable coefficients, Phys. Lett. A 369, 62-69.
  • He, J.H. and Wu, X.H., (2006), Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals 29, 108-113.
  • Wu, X.H. and He, J.H., EXP-function method and its application to nonlinear equations, Chaos, Solitons and Fractals (in press), doi:10.1016/j.chaos.2007.01.024.
  • He, J.H. and Abdou, M.A., (2007), New periodic solutions for nonlinear evolution equations using Exp-function method, Chaos, Solitons and Fractals, 34, 1421-1429.
  • Zhu, S.D., (2007), Exp-function method for the discrete mKdV lattice, International Journal of Non- linear Sciences and Numerical Simulation 8(3), 465-468.
  • Wang, Q., Chen, Y. and Zhang, H., (2005), A new Riccati equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation. Chaos, Solitons and Fractals , 25, 101928.
  • Hong, K.Z. et al., (2003), Painleve analysis and some solutions of (2 + 1)-dimensional generalized Burgers equations. Commun Theor Phys, 39, 3934.
  • Tang, X.Y. and Lou, S., (2003), Variable separation solutions for the (2 + 1)-dimensional Burgers equations. Chin Phys Lett., 20(3), 335.

Details

Primary Language English
Journal Section Research Article
Authors

Jalil RASHİDİNİA This is me
School of Mathematics, Iran University of Science and Technology, P. O. Box, 16846-13114, Tehran, Iran.


Ali BARATİ This is me
School of Mathematics, Iran University of Science and Technology, P. O. Box, 16846-13114, Tehran, Iran.

Publication Date June 1, 2011
Published in Issue Year 2011, Volume 01, Issue 1

Cite

Bibtex @ { twmsjaem761806, journal = {TWMS Journal of Applied and Engineering Mathematics}, issn = {2146-1147}, eissn = {2587-1013}, address = {Işık University ŞİLE KAMPÜSÜ Meşrutiyet Mahallesi, Üniversite Sokak No:2 Şile / İstanbul}, publisher = {Turkic World Mathematical Society}, year = {2011}, volume = {01}, number = {1}, pages = {35 - 40}, title = {NEW APPROACH TO THE SOLUTIONS OF THE PIB EQUATION}, key = {cite}, author = {Rashidinia, Jalil and Barati, Ali} }