Research Article
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New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations

Year 2021, Issue: 37, 8 - 25, 31.12.2021
https://doi.org/10.53570/jnt.1018383

Abstract

The new exact solutions of the conformable time-fractional Bad and Good modified Boussinesq equations are obtained using the Exp-function method, which is different from previous literature works. These equations play a significant role in mathematical physics, engineering sciences and applied mathematics. Plentiful exact solutions with arbitrary parameters are effectively obtained by the method. The obtained solutions are shown graphically. It is shown that the Exp-function method provides a simpler but more effective mathematical tool for constructing exact solutions of non-linear evolution equations.

References

  • J. H. He, X. H. Wu, Exp-Function Method for Nonlinear Wave Equations, Chaos, Soliton and Fractals 30(3) (2006) 700–708.
  • E. M. E. Zayed, M. A. M. Abdelaziz, The Tanh- Function Method Using a Generalized Wave Transformation for Nonlinear Equations, International Journal of Nonlinear Sciences and Numerical Simulation 11(8) (2010) 595–601.
  • E. Fan, H. Zhang, A Note on the Homogeneous Balance Method, Physics Letters A 246(5) (1998) 403–406.
  • Y. Gurefe, A. Sönmezoğlu, E. Mrsırlı, Application of The Trial Equation Method for Solving Some Nonlinear Evolution Equations Arising in Mathematical Physics, Pramana 77(6) (2011) 1023–1029.
  • N. A. Kudryashov, A Note on the (G^'/G)-expansion Method, Applied Mathematics and Computation 217(4) (2010) 1755–1758.
  • P. N. Ryabov, D. I. Sinelshchikov, M. B. Kochanov, Application of the Kudryashov Method for Finding Exact Solutions of the High Order Nonlinear Evolution Equations, Applied Mathematics and Computation 218(7) (2011) 3965–3972.
  • J. H. He, Exp-function Method for Fractional Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation 14(6) (2013) 363–366.
  • Z. Öztürk, H. Bilgil, Exact Solutions of Rosenzweig-Macarthur (RM) Model Equations by Using Exp Function Method, European Journal of Science and Technology (16) (2019) 427–433.
  • A. Ebaid, Application of the Exp-Function Method for Solving Some Evolution Equations with Nonlinear Terms of Any Orders, Zeitschrift fur Naturforschung A 65(12) (2010) 1039–1044.
  • S. Zhang, Exp-Function Method for Riccati Equation and New Exact Solutions with Two Arbitrary Functions of (2 + 1)-Dimensional Konopelchenko-Dubrovsky Equations, Applied Mathematics and Computation 216(5) (2010) 1546–1552.
  • R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics 264 (2014) 65–70.
  • I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
  • A. Atangana, I. Koca, Chaos in A Simple Nonlinear System with Atangana–Baleanu Derivatives with Fractional Order, Chaos, Solitons and Fractals 89 (2016) 447–454.
  • N. S. Kottakkaran, O. I Ilhan, S. T. Abdulazeez, J. Manafian, S. A. Mohammed, M. S. Osman, Novel Multiple Soliton Solutions for Some Nonlinear Pdes via Multiple Exp-Function Method, Results in Physics 21 (2021) 103769.
  • K. A. Gepreel, E. M. E. Zayed, Multiple Wave Solutions for Nonlinear Burgers Equations Using the Multiple Exp-Function Method, International Journal of Modern Physics C (2021) 2150149.
  • H. M. Ahmed, M. M. A. El-Sheikh, A. H. Arnous, W. B. Rabiel, Solitons and Other Solutions to (N+1)-Dimensional Modified Zakharov–Kuznetsov Equation by Exp-Function Method, SeMA Journal 78 (2021) 1–13.
  • A. G. Hossam, H. Abd-Allah, M. Zakarya, Exact Solutions of Stochastic Fractional Korteweg De–Vries Equation with Conformable Derivatives, Chinese Physics B 29(3) (2020) 30203.
  • H. C. Yaslan, A. Girgin, Exp-Function Method for The Conformable Space-Time Fractional STO, ZKBBM and Coupled Boussinesq Equations, Arab Journal of Basic and Applied Sciences 26(1) (2019) 163–170.
  • L. K. Ravi, S. S. Ray, S. Sahoo, New Exact Solutions of Coupled Boussinesq–Burgers Equations by Exp-Function Method, Journal of Ocean Engineering and Science 2(1) (2017) 34–46.
  • E. M. Eskandari, N. Taghizadeh, Exact Solutions of Two Nonlinear Space-time Fractional Differential Equations by Application of Exp-function Method, Applications and Applied Mathematics: An International Journal 15(2) (2020) 970–977.
  • H. Durur, O. Taşbozan, A. Kurt, New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations, Applied Mathematics and Nonlinear Sciences 5(1) (2020) 447–454.
  • A. Ebaid, An Improvement on The Exp-Function Method When Balancing the Highest Order Linear and Nonlinear Terms, Journal of Mathematical Analysis and Applications 392(1) (2012) 1–5.
  • J. Boussinesq, Thorie Des Ondes Et Des Remous Qui Se Propagent Le Long D’un Canal Rectangulaire Horizantal,En Communiquant Au Liquide Contenu Dans Ce Canal Des Vitesses Sensiblement Pareilles De La Surface Au Fond, Journal de Mathmmatiques Pures at Appliques (1872) 55–108.
Year 2021, Issue: 37, 8 - 25, 31.12.2021
https://doi.org/10.53570/jnt.1018383

Abstract

References

  • J. H. He, X. H. Wu, Exp-Function Method for Nonlinear Wave Equations, Chaos, Soliton and Fractals 30(3) (2006) 700–708.
  • E. M. E. Zayed, M. A. M. Abdelaziz, The Tanh- Function Method Using a Generalized Wave Transformation for Nonlinear Equations, International Journal of Nonlinear Sciences and Numerical Simulation 11(8) (2010) 595–601.
  • E. Fan, H. Zhang, A Note on the Homogeneous Balance Method, Physics Letters A 246(5) (1998) 403–406.
  • Y. Gurefe, A. Sönmezoğlu, E. Mrsırlı, Application of The Trial Equation Method for Solving Some Nonlinear Evolution Equations Arising in Mathematical Physics, Pramana 77(6) (2011) 1023–1029.
  • N. A. Kudryashov, A Note on the (G^'/G)-expansion Method, Applied Mathematics and Computation 217(4) (2010) 1755–1758.
  • P. N. Ryabov, D. I. Sinelshchikov, M. B. Kochanov, Application of the Kudryashov Method for Finding Exact Solutions of the High Order Nonlinear Evolution Equations, Applied Mathematics and Computation 218(7) (2011) 3965–3972.
  • J. H. He, Exp-function Method for Fractional Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation 14(6) (2013) 363–366.
  • Z. Öztürk, H. Bilgil, Exact Solutions of Rosenzweig-Macarthur (RM) Model Equations by Using Exp Function Method, European Journal of Science and Technology (16) (2019) 427–433.
  • A. Ebaid, Application of the Exp-Function Method for Solving Some Evolution Equations with Nonlinear Terms of Any Orders, Zeitschrift fur Naturforschung A 65(12) (2010) 1039–1044.
  • S. Zhang, Exp-Function Method for Riccati Equation and New Exact Solutions with Two Arbitrary Functions of (2 + 1)-Dimensional Konopelchenko-Dubrovsky Equations, Applied Mathematics and Computation 216(5) (2010) 1546–1552.
  • R. Khalil, M. A. Horani, A. Yousef, M. Sababheh, A New Definition of Fractional Derivative, Journal of Computational and Applied Mathematics 264 (2014) 65–70.
  • I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Elsevier, 1998.
  • A. Atangana, I. Koca, Chaos in A Simple Nonlinear System with Atangana–Baleanu Derivatives with Fractional Order, Chaos, Solitons and Fractals 89 (2016) 447–454.
  • N. S. Kottakkaran, O. I Ilhan, S. T. Abdulazeez, J. Manafian, S. A. Mohammed, M. S. Osman, Novel Multiple Soliton Solutions for Some Nonlinear Pdes via Multiple Exp-Function Method, Results in Physics 21 (2021) 103769.
  • K. A. Gepreel, E. M. E. Zayed, Multiple Wave Solutions for Nonlinear Burgers Equations Using the Multiple Exp-Function Method, International Journal of Modern Physics C (2021) 2150149.
  • H. M. Ahmed, M. M. A. El-Sheikh, A. H. Arnous, W. B. Rabiel, Solitons and Other Solutions to (N+1)-Dimensional Modified Zakharov–Kuznetsov Equation by Exp-Function Method, SeMA Journal 78 (2021) 1–13.
  • A. G. Hossam, H. Abd-Allah, M. Zakarya, Exact Solutions of Stochastic Fractional Korteweg De–Vries Equation with Conformable Derivatives, Chinese Physics B 29(3) (2020) 30203.
  • H. C. Yaslan, A. Girgin, Exp-Function Method for The Conformable Space-Time Fractional STO, ZKBBM and Coupled Boussinesq Equations, Arab Journal of Basic and Applied Sciences 26(1) (2019) 163–170.
  • L. K. Ravi, S. S. Ray, S. Sahoo, New Exact Solutions of Coupled Boussinesq–Burgers Equations by Exp-Function Method, Journal of Ocean Engineering and Science 2(1) (2017) 34–46.
  • E. M. Eskandari, N. Taghizadeh, Exact Solutions of Two Nonlinear Space-time Fractional Differential Equations by Application of Exp-function Method, Applications and Applied Mathematics: An International Journal 15(2) (2020) 970–977.
  • H. Durur, O. Taşbozan, A. Kurt, New Analytical Solutions of Conformable Time Fractional Bad and Good Modified Boussinesq Equations, Applied Mathematics and Nonlinear Sciences 5(1) (2020) 447–454.
  • A. Ebaid, An Improvement on The Exp-Function Method When Balancing the Highest Order Linear and Nonlinear Terms, Journal of Mathematical Analysis and Applications 392(1) (2012) 1–5.
  • J. Boussinesq, Thorie Des Ondes Et Des Remous Qui Se Propagent Le Long D’un Canal Rectangulaire Horizantal,En Communiquant Au Liquide Contenu Dans Ce Canal Des Vitesses Sensiblement Pareilles De La Surface Au Fond, Journal de Mathmmatiques Pures at Appliques (1872) 55–108.
There are 23 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Article
Authors

Zafer Öztürk 0000-0001-5662-4670

Sezer Sorgun 0000-0001-8708-1226

Halis Bilgil 0000-0002-8329-5806

Ümmügülsüm Erdinç 0000-0002-4504-3675

Publication Date December 31, 2021
Submission Date November 3, 2021
Published in Issue Year 2021 Issue: 37

Cite

APA Öztürk, Z., Sorgun, S., Bilgil, H., Erdinç, Ü. (2021). New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations. Journal of New Theory(37), 8-25. https://doi.org/10.53570/jnt.1018383
AMA Öztürk Z, Sorgun S, Bilgil H, Erdinç Ü. New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations. JNT. December 2021;(37):8-25. doi:10.53570/jnt.1018383
Chicago Öztürk, Zafer, Sezer Sorgun, Halis Bilgil, and Ümmügülsüm Erdinç. “New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations”. Journal of New Theory, no. 37 (December 2021): 8-25. https://doi.org/10.53570/jnt.1018383.
EndNote Öztürk Z, Sorgun S, Bilgil H, Erdinç Ü (December 1, 2021) New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations. Journal of New Theory 37 8–25.
IEEE Z. Öztürk, S. Sorgun, H. Bilgil, and Ü. Erdinç, “New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations”, JNT, no. 37, pp. 8–25, December 2021, doi: 10.53570/jnt.1018383.
ISNAD Öztürk, Zafer et al. “New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations”. Journal of New Theory 37 (December 2021), 8-25. https://doi.org/10.53570/jnt.1018383.
JAMA Öztürk Z, Sorgun S, Bilgil H, Erdinç Ü. New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations. JNT. 2021;:8–25.
MLA Öztürk, Zafer et al. “New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations”. Journal of New Theory, no. 37, 2021, pp. 8-25, doi:10.53570/jnt.1018383.
Vancouver Öztürk Z, Sorgun S, Bilgil H, Erdinç Ü. New Exact Solutions of Conformable Time-Fractional Bad and Good Modified Boussinesq Equations. JNT. 2021(37):8-25.


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