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Exact Solutions of Nonlinear Partial Differential Equations with exp(-φ(ζ))- Expansion Method

Year 2017, Volume: 17 Issue: 1, 86 - 92, 24.04.2017

Filiz Taşcan Arzu Akbulut

Abstract

The aim of this paper is obtaining the exact solutions of the Bogoyavlenskii equation and the modified
KdV-Zakharov-Kuznetsev equation with the help of the exp(-φ(ζ))-expansion method. Solutions are
obtained with different forms of functions as hyperbolic, trigonometric and rational functions.
Discussed method is useful for obtaining solutions of nonlinear equations in mathematical physics and
engineering.

Keywords

Exact solutions; Symbolic computation Partial differential equation

References

  • Liu, G.T, Fan, TY., 2005. New applications of developed Jacobi elliptic function expansion methods, Physics Letters A, 345, 161-166.
  • Achab, AE., 2016. Constructing of exact solutions to the nonlinear Schrödinger equation (NLSE) with power-law nonlinearity by the Weierstrass elliptic function method. Optik, 127, 1229-1232.
  • Matveev, VB., Salle, MA., 1991. Darboux Transformation and Solitons. Berlin: Springer.
  • Tascan, F., Yakut, A., 2015. Conservation Laws and Exact Solutions with Symmetry Reduction of Nonlinear Reaction Diffusion Equations. International Journal of Nonlinear Sciences and Numerical Simulation, 16, 191-196.
  • Bluman, GW., Kumei, S.,1989. Symmetries and differential equations. New York, NY, USA: Springer Verlag.
  • Bekir, A., Cevikel, AC., 2009. Solitary wave solutions of two nonlinear physical models by tanh--coth method. Communications in Nonlinear Science and Numerical Simulation, 14, 1804-1809.
  • Kaplan, M,, Akbulut, A,, Bekir, A.,2015. Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method. Zeitschrift für Naturforschung A,70, 969-974.
  • Wazwaz, AM., 2008. The extended tanh method for the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms, Communications in Nonlinear Science and Numerical Simulation, 13, 1039-1047.
  • Taşcan, F., Bekir, A., 2009. Analytic solutions of the (2+1)-dimensional nonlinear evolution equations using the sine-cosine method, Applied Mathematics and Computation, 215, 3134-3139.
  • Eslami, M., Vajargah, BF., Mirzazadeh, M., 2014. Exact solutions of modified Zakharov-- Kuznetsov equation by the homogeneous balance method. Ain Shams Engineering Journal ,5, 221-225.
  • Ghaneai, H., Hosseini. MM.,2016. Solving differential-algebraic equations through variational iteration method with an auxiliary parameter. Applied Mathematical Modelling, 40, 3991-4001.
  • Zayed, EME.,, Abdelaziz, MAM., 2011. Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine-- cosine and the exp-function methods, Applied Mathematics and Computation, 218, 2259- 2268.
  • Khan, K., Akbar, MA., 2014. Traveling wave solutions of nonlinear evolution equations via the enhanced (G'/G )-expansion method, Journal of the Egyptian Mathematical Society, 22, 220-226.
  • Kaplan, M., Bekir, A., Akbulut, A., Aksoy, E.,2015. The modified simple equation method for nonlinear fractional differential equations. Romanian Journal of Physics ,60, 1374-1383.
  • Taghizadeh, N., Mirzazadeh, M., Farahrooz, F., 2011. Exact solutions of the nonlinear Schrödinger equation by the first integral method, Journal of Mathematical Analysis and Applications, 374, 549-553.
  • Zhang,Z., 2008. New Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation, Turkish Journal of Physics, 32, 235 - 240.
  • Khan, K., Akbar, MA., 2014. The exp(−φ (x )) - expansion method for finding travelling wave solutions of Vakhnenko-Parkes equation, International Journal of Dynamical Systems and Differential Equations, 5, 72-83.
  • Khan, K., Akbar, MA., 2013. Application of ()−−)(expxφ expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation. World Applied Sciences Journal, 24, 1373-1377.
  • Hafez, MG., Ali, MY., Akter, MT., Kauser, MA., 2014. Application of the ())(expxφ− -expansion method for finding exact solutions of the (1+1)-dimensional dispersive long wave equations. British Journal of Mathematics & Computer Science , 22, 3191-3201.
  • Abdelrahman, MAE., Zahran, HM., Khater, MA., 2015. The ())(expxφ− -expansion method and its application for solving nonlinear evolution equations. International Journal of Modern Nonlinear Theory and Application, 4, 37-47.
  • Roshid, H.O., Alam, M.N., Akbar M.A., 2015. Traveling Wave Solutions for Fifth Order (1+1)-Dimensional Kaup-Kupershmidt Equation with the Help of ())(expxφ−Expansion Method. Walailak Journal, 12, 1063-1073.
  • Khater, M.A., 2016. Exact traveling wave solutions for the generalized Hirota-Satsuma couple KdV system using the ())(expxφ−expansion method. Cogent Mathematics, 3, 1172397.
  • Zahran, E.H.M., Khater, M.A., 2015. Exact Traveling Wave Solutions For Some Nonlinear Evolution Equations by Using the ())(expxφ−Expansion Method. Asian Journal of Mathematics and Computer Research, 4, 195-207.
  • Malik, A., Kumar, H., Chand, F., Singh, S., Mischra, S.C.,2010. Exact Traveling Wave Solutions of the Bogoyavlenskii Equation. International Archive of Applied Sciences and Technology, 1, 93-98.
  • Najafi, M., Arbabi, S., Najafi, M., 2012. New Exact Solutions of (2 + 1)-Dimensional Bogoyavlenskii Equation by the Sine-Cosine Method. International Journal of Basic and Applied Sciences, 1, 490-497.
  • Naher, H., Abdullah, F.A., Akbar, M.A., 2013. Generalized and Improved (G'/G)-Expansion Method for (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation. Plos One, 8, 7 pp.
  • Hafez, MG., Akbar, MA., 2015. New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp ())(expxφ− -expansion method. Propulsion and Power Research, 4, 31-39.
  • Malik, A., Chand, F., Kumar, H., Mishra, SC., 2012. Exact solutions of the Bogoyavlenskii equation using the multiple (G'/G)-expansion method, Computers and Mathematics with Applications, 64 : 2850-2859.
  • Peng, Y., Shen, M., 2006. On exact solutions of the Bogoyavlenskii equation, Pramana-journal of physics , 67 : 449-456.
  • Islam, Md H., Khan, K., Akbar, MA., and Salam, Md A., 2014. Exact traveling wave solutions of modified KdV-Zakharov-Kuznetsov equation and viscous Burgers equation. SpringerPlus, 3, 105, 9 pp.

Year 2017, Volume: 17 Issue: 1, 86 - 92, 24.04.2017

Filiz Taşcan Arzu Akbulut

Abstract

References

  • Liu, G.T, Fan, TY., 2005. New applications of developed Jacobi elliptic function expansion methods, Physics Letters A, 345, 161-166.
  • Achab, AE., 2016. Constructing of exact solutions to the nonlinear Schrödinger equation (NLSE) with power-law nonlinearity by the Weierstrass elliptic function method. Optik, 127, 1229-1232.
  • Matveev, VB., Salle, MA., 1991. Darboux Transformation and Solitons. Berlin: Springer.
  • Tascan, F., Yakut, A., 2015. Conservation Laws and Exact Solutions with Symmetry Reduction of Nonlinear Reaction Diffusion Equations. International Journal of Nonlinear Sciences and Numerical Simulation, 16, 191-196.
  • Bluman, GW., Kumei, S.,1989. Symmetries and differential equations. New York, NY, USA: Springer Verlag.
  • Bekir, A., Cevikel, AC., 2009. Solitary wave solutions of two nonlinear physical models by tanh--coth method. Communications in Nonlinear Science and Numerical Simulation, 14, 1804-1809.
  • Kaplan, M,, Akbulut, A,, Bekir, A.,2015. Exact Travelling Wave Solutions of the Nonlinear Evolution Equations by Auxiliary Equation Method. Zeitschrift für Naturforschung A,70, 969-974.
  • Wazwaz, AM., 2008. The extended tanh method for the Zakharov-Kuznetsov (ZK) equation, the modified ZK equation, and its generalized forms, Communications in Nonlinear Science and Numerical Simulation, 13, 1039-1047.
  • Taşcan, F., Bekir, A., 2009. Analytic solutions of the (2+1)-dimensional nonlinear evolution equations using the sine-cosine method, Applied Mathematics and Computation, 215, 3134-3139.
  • Eslami, M., Vajargah, BF., Mirzazadeh, M., 2014. Exact solutions of modified Zakharov-- Kuznetsov equation by the homogeneous balance method. Ain Shams Engineering Journal ,5, 221-225.
  • Ghaneai, H., Hosseini. MM.,2016. Solving differential-algebraic equations through variational iteration method with an auxiliary parameter. Applied Mathematical Modelling, 40, 3991-4001.
  • Zayed, EME.,, Abdelaziz, MAM., 2011. Exact solutions for the nonlinear Schrödinger equation with variable coefficients using the generalized extended tanh-function, the sine-- cosine and the exp-function methods, Applied Mathematics and Computation, 218, 2259- 2268.
  • Khan, K., Akbar, MA., 2014. Traveling wave solutions of nonlinear evolution equations via the enhanced (G'/G )-expansion method, Journal of the Egyptian Mathematical Society, 22, 220-226.
  • Kaplan, M., Bekir, A., Akbulut, A., Aksoy, E.,2015. The modified simple equation method for nonlinear fractional differential equations. Romanian Journal of Physics ,60, 1374-1383.
  • Taghizadeh, N., Mirzazadeh, M., Farahrooz, F., 2011. Exact solutions of the nonlinear Schrödinger equation by the first integral method, Journal of Mathematical Analysis and Applications, 374, 549-553.
  • Zhang,Z., 2008. New Exact Traveling Wave Solutions for the Nonlinear Klein-Gordon Equation, Turkish Journal of Physics, 32, 235 - 240.
  • Khan, K., Akbar, MA., 2014. The exp(−φ (x )) - expansion method for finding travelling wave solutions of Vakhnenko-Parkes equation, International Journal of Dynamical Systems and Differential Equations, 5, 72-83.
  • Khan, K., Akbar, MA., 2013. Application of ()−−)(expxφ expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation. World Applied Sciences Journal, 24, 1373-1377.
  • Hafez, MG., Ali, MY., Akter, MT., Kauser, MA., 2014. Application of the ())(expxφ− -expansion method for finding exact solutions of the (1+1)-dimensional dispersive long wave equations. British Journal of Mathematics & Computer Science , 22, 3191-3201.
  • Abdelrahman, MAE., Zahran, HM., Khater, MA., 2015. The ())(expxφ− -expansion method and its application for solving nonlinear evolution equations. International Journal of Modern Nonlinear Theory and Application, 4, 37-47.
  • Roshid, H.O., Alam, M.N., Akbar M.A., 2015. Traveling Wave Solutions for Fifth Order (1+1)-Dimensional Kaup-Kupershmidt Equation with the Help of ())(expxφ−Expansion Method. Walailak Journal, 12, 1063-1073.
  • Khater, M.A., 2016. Exact traveling wave solutions for the generalized Hirota-Satsuma couple KdV system using the ())(expxφ−expansion method. Cogent Mathematics, 3, 1172397.
  • Zahran, E.H.M., Khater, M.A., 2015. Exact Traveling Wave Solutions For Some Nonlinear Evolution Equations by Using the ())(expxφ−Expansion Method. Asian Journal of Mathematics and Computer Research, 4, 195-207.
  • Malik, A., Kumar, H., Chand, F., Singh, S., Mischra, S.C.,2010. Exact Traveling Wave Solutions of the Bogoyavlenskii Equation. International Archive of Applied Sciences and Technology, 1, 93-98.
  • Najafi, M., Arbabi, S., Najafi, M., 2012. New Exact Solutions of (2 + 1)-Dimensional Bogoyavlenskii Equation by the Sine-Cosine Method. International Journal of Basic and Applied Sciences, 1, 490-497.
  • Naher, H., Abdullah, F.A., Akbar, M.A., 2013. Generalized and Improved (G'/G)-Expansion Method for (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation. Plos One, 8, 7 pp.
  • Hafez, MG., Akbar, MA., 2015. New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp ())(expxφ− -expansion method. Propulsion and Power Research, 4, 31-39.
  • Malik, A., Chand, F., Kumar, H., Mishra, SC., 2012. Exact solutions of the Bogoyavlenskii equation using the multiple (G'/G)-expansion method, Computers and Mathematics with Applications, 64 : 2850-2859.
  • Peng, Y., Shen, M., 2006. On exact solutions of the Bogoyavlenskii equation, Pramana-journal of physics , 67 : 449-456.
  • Islam, Md H., Khan, K., Akbar, MA., and Salam, Md A., 2014. Exact traveling wave solutions of modified KdV-Zakharov-Kuznetsov equation and viscous Burgers equation. SpringerPlus, 3, 105, 9 pp.
There are 30 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Filiz Taşcan This is me

Arzu Akbulut This is me

Publication Date April 24, 2017
Submission Date April 18, 2016
Published in Issue Year 2017 Volume: 17 Issue: 1

Cite

APA Taşcan, F., & Akbulut, A. (2017). Exact Solutions of Nonlinear Partial Differential Equations with exp(-φ(ζ))- Expansion Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 17(1), 86-92.
AMA Taşcan F, Akbulut A. Exact Solutions of Nonlinear Partial Differential Equations with exp(-φ(ζ))- Expansion Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. April 2017;17(1):86-92.
Chicago Taşcan, Filiz, and Arzu Akbulut. “Exact Solutions of Nonlinear Partial Differential Equations With Exp(-φ(ζ))- Expansion Method”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 17, no. 1 (April 2017): 86-92.
EndNote Taşcan F, Akbulut A (April 1, 2017) Exact Solutions of Nonlinear Partial Differential Equations with exp(-φ(ζ) - Expansion Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 17 1 86–92.
IEEE F. Taşcan and A. Akbulut, “Exact Solutions of Nonlinear Partial Differential Equations with exp(-φ(ζ))- Expansion Method”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 17, no. 1, pp. 86–92, 2017.
ISNAD Taşcan, Filiz - Akbulut, Arzu. “Exact Solutions of Nonlinear Partial Differential Equations With Exp(-φ(ζ))- Expansion Method”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 17/1 (April 2017), 86-92.
JAMA Taşcan F, Akbulut A. Exact Solutions of Nonlinear Partial Differential Equations with exp(-φ(ζ))- Expansion Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2017;17:86–92.
MLA Taşcan, Filiz and Arzu Akbulut. “Exact Solutions of Nonlinear Partial Differential Equations With Exp(-φ(ζ))- Expansion Method”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 17, no. 1, 2017, pp. 86-92.
Vancouver Taşcan F, Akbulut A. Exact Solutions of Nonlinear Partial Differential Equations with exp(-φ(ζ))- Expansion Method. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2017;17(1):86-92.


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