Skip to main content
SpringerLink
Log in

New hyperbolic structures for the conformable time-fractional variant bussinesq equations

  • Published:
Optical and Quantum Electronics Aims and scope Submit manuscript

Abstract

In this work, exact analytical solutions for the time fractional variant bussinesq equations are constructed in the sense of the newly devised fractional derivative called conformable fractional derivative. Using wave transformation, we converted the problem under consideration to an ordinary differential equation and then employed the modified extended tanh expansion method for hyperbolic function solutions. The Mathematica software is used throughout for the solution of the system of algebraic equations obtained along the way and also for the graphical illustrations, respectively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  • Alquran, M., Jaradat, H.M., Syam, M.I.: Analytical solution of the time-fractional Phi-4 equation using modified power series method. Nonlinear Dyn. 90, 2525–2529 (2017)

    Article  MathSciNet  Google Scholar 

  • Bakodah, H.O., Al Qarni, A.A., Banaja, M.A., Zhou, Q., Moshokoa, S.P., Biswas, A.: Bright and dark thirring optical solitons with improved Adomian decomposition method. Int. J. Light Electron Opt. 130, 1115–1123 (2017a)

    Article  Google Scholar 

  • Bakodah, H.O., Al-Zaid, N.A., Mirzazadeh, M., Zhou, Q.: Decomposition method for solving Burgers’ equation with Dirichlet and Neumann boundary conditions. Int. J. Light Electron Opt. 130, 1339–1346 (2017b)

    Article  Google Scholar 

  • Banaja, M., Al Qarni, A.A., Bakodah, H.O., Zhou, Q., Moshokoa S., Biswas, A.: Bright and dark solitons in cascaded system by improved Adomian decomposition scheme. Int. J. Light Electron Opt. (2016). https://doi.org/10.1016/j.ijleo.2016.11.125

    Google Scholar 

  • Baskonus, H.M., Sulaiman , T.A., Bulut, H.: New solitary wave solutions to the (2+1)-dimensional Calogero–Bogoyavlenskii-Schiff and the Kadomtsev-Petviashvili hierarchy equations. Indian J Phys 91, 1237–1243 (2017)

    Article  ADS  Google Scholar 

  • Bekir, A., Guner, O.: Exact solutions of nonlinear fractional differential equations by \(G^{\prime }/G\)-expansion method. China Phys. B 22, 110202 (2013)

    Article  Google Scholar 

  • Bulut, H., Sulaiman, T.A., Erdogan, F., Baskonus, H.M.: On the new hyperbolic and trigonometric structures to the simplified MCH and SRLW equations. Eur. Phys. J. Plus. 132, 350 (2017)

    Article  Google Scholar 

  • Guner, O., Bekir, A., Cevikel, A.C.: A variety of exact solutions for the time fractional Cahn–Allen equation. Eur. Phys. J. Plus. 130, 146–58 (2015a)

    Article  Google Scholar 

  • Guner, O., Bekir, A., Bilgil, H.L.: A note on exp-function method combined with complex transform method applied to fractional differential equations. Adv. Nonlinear Anal. 4, 201–208 (2015b)

    MathSciNet  MATH  Google Scholar 

  • Hosseini, K., Ayati, Z., Ansari, R.: New exact solutions of the Tzitzéica type equations arising in nonlinear optics using a modified version of the improved \(tan(\phi (\xi )/2)\)-expansion method. Opt. Quantum Electron. 49, 273 (2017)

    Article  Google Scholar 

  • Inan, I.E.: \(tan(F(\xi )/2)\)-expansion method for traveling wave solutions of the variant bussinesq equations. Acilim Metodu, Karaelmas Fen ve Muh. Derg. 7, 201–206 (2017)

    Google Scholar 

  • Inc, M., Yusuf, A., Aliyu, A.I.: Dark optical and other soliton solutions for the three different nonlinear schrodinger equations. Opti. Quantum 49, 11 (2017a)

    Article  Google Scholar 

  • Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D.: Optical soliton solutions for the higher-order dispersive cubic-quintic nonlinear Schrodinger equation. Superlattice Microstruct. 112, 164–179 (2017b)

    Article  ADS  Google Scholar 

  • Islam, M.H., Khan, K., Akbar, M.A., Salam, M.A.: Exact traveling wave solutions of modified KdV–Zakharov–Kuznetsov equation and viscous Burgers equation. SpringerPlus. 3, 105 (2014)

    Article  Google Scholar 

  • Islam, M.T., Akbar, M.A., Azad, M.A.K.: A rational (G/G)-expansion method and its application to modified KdV-Burgers equation and the (2\(+\)1)-dimensional Boussineq equation. Nonlinear Stud. 6, 4 (2015)

    MATH  Google Scholar 

  • Khalid, K.A., Nuruddeen, R.I.: Analytical treatment for the conformable space-time fractional Benney–Luke equation via two reliable methods. Int. J. Phys. Res. 5, 109–114 (2017)

    Article  Google Scholar 

  • Khalid, K.A., Raslan, K.R., Talaat, S.E.: Non-polynomial spline method for solving coupled Burger’s equations. Comput. Methods Differ. Equ. 3, 218–230 (2016)

    MathSciNet  Google Scholar 

  • Khalid, K.A., Nuruddeen, R.I., Raslan, K.R.: New structures for the space-time fractional simplified MCH and SRLW equations. Chaos, Solitons Fractals 106, 304–309 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  • Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 265, 65–701 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17(6), 2248–2253 (2012)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Lakestani, M., Manafian, J.: Application of the ITEM for the modified dispersive water-wave system. Opt. Quantum Electron. 49, 128 (2017)

    Article  Google Scholar 

  • Manafian, J.: Application of the ITEM for the system of equations for the ion sound and Langmuir waves. Opt. Quantum Electron. 49, 17 (2017)

    Article  Google Scholar 

  • Manafian, J., Lakestani, M.: Application of \(tan(\phi (\xi )/2)\)-expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearity. Optik. 127, 2040–2054 (2016a)

    Article  Google Scholar 

  • Manafian, J., Lakestani, M.: Dispersive dark optical soliton with Tzitzeica type nonlinear evolution equations arising in nonlinear optics. Opt. Quantum Electron. 48, 116 (2016b)

    Article  Google Scholar 

  • Manafian, J., Eslami, M., Kordy, M.: The functional variable method for solving the fractional Korteweg-de Vries equations and the coupled Korteweg–De Vries equations. Pramana J. Phys. 85, 583–92 (2015)

    Article  ADS  Google Scholar 

  • Manafian, J., Jalali, J., Ranjbaran, A.: Applications of IBSOM and ETEM for solving a discretion electrical lattice. Opt. Quantum Electron. 49, 406 (2017a)

    Article  Google Scholar 

  • Manafian, J., Bolghar, P., Mohammadalian, A.: Abundant soliton solutions of the resonant nonlinear Schrodinger equation with time-dependent coefficients by ITEM and He’s semi-inverse method. Opt. Quantum Electron. 49, 322 (2017b)

    Article  Google Scholar 

  • Manafian, J., Foroutan, M., Guzali, A.: Application of the ETEM for obtaining optical soliton solution for the Lakshmann–Porsezian–Daniel model. Eur. Phys. J. Plus. 132, 494 (2017c)

    Article  Google Scholar 

  • Nuruddeen, R.I.: Elzaki decomposition method and its applications in solving linear and nonlinear Schrodinger equations. Sohag J. Math. 4, 2 (2017a)

    Article  Google Scholar 

  • Nuruddeen, R.I.: Multiple soliton solutions for the (3+1) conformable space-time fractional modified Korteweg-de-Vries equations. Ocean J. Eng. Sci. (2017b). https://doi.org/10.1016/j.joes.2017.11.004

  • Nuruddeen, R.I., Nass, A.M.: Exact solutions of wave-type equations by the Aboodh decomposition method. Stoch. Model Appl. 21, 23–30 (2017)

    Google Scholar 

  • Nuruddeen, R.I., Muhammad, L., Nass, A.M., Sulaiman, T.A.: A review of the integral transforms-based decomposition methods and their applications in solving nonlinear PDEs. Palest. J. Math. 7, 262–280 (2018)

    MathSciNet  MATH  Google Scholar 

  • Raslan, K.R., Talaat, S.E., Khalid, K.A.: Collocation method with quintic B-spline method for solving Hirota-Satsuma coupled KDV equation. Int. J. Appl. Math. Res. 5, 123–31 (2016a)

    Article  MATH  Google Scholar 

  • Raslan, K.R., Talaat, S.E., Khalid, K.A.: Collocation method with quintic B-spline method for solving the Hirota equation. J. Abstr. Comput. Math. 1, 1–12 (2016b)

    Google Scholar 

  • Raslan, K.R., Khalid, K.A., Shallal, M.A.: The modified extended tanh method with the Riccati equation for solving the space-time fractional EW and MEW equations. Chaos, Solitons Fractals 103, 404–409 (2017a)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  • Raslan, K.R., Talaat, S.E., Khalid, K.A.: Exact solution of space-time fractional coupled EW and coupled MEW equations. Eur. Phys. J. Plus. 132, 319 (2017b)

    Article  MATH  Google Scholar 

  • Raslan, K.R., Talaat, S.E., Khalid, K.A.: New exact solution of coupled general equal width wave equation using sine-cosine function method. J. Egypt Math. Soc. 25, 350–354 (2017c)

    Article  MathSciNet  MATH  Google Scholar 

  • Shukri, S., Al-Khaled, K.: The extended tan method for solving systems of nonlinear wave equations. Appl. Math. Comput. 217, 5 (2010)

    Article  MATH  Google Scholar 

  • Sindi, C.T., Manafian, J.: Soliton solutions of the quantum Zakharov-Kuznetsov equation which arises in quantum magneto-plasmas. Eur. Phys. J. Plus. 132, 67 (2017)

    Article  Google Scholar 

  • Talaat, S.E., Raslan, K.R., Khalid, K.A.: Collocation method with cubic B-splines for solving the generalized long wave equation. Int. J. Num. Method Appl. 15, 39–59 (2016)

    MATH  Google Scholar 

  • Wazwaz, A.: A variety of soliton solutions for the Boussinesq-Burgers equation and the higher-order Boussinesq-Burgers equation. Filomat. 31, 831–840 (2017)

    Article  MathSciNet  Google Scholar 

  • Younis, M., Cheemaa, N., Mahmood, S.A., Rizvi, S.T.R.: On optical solitons: the chiral nonlinear Schrodinger equation with perturbation and Bohm potential. Opt. Quantum Electron. 48, 542 (2016)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Faculty of Science, Al-Azhar University, Nasr-City, Cairo, Egypt

    Khalid K. Ali & K. R. Raslan

  2. Department of Mathematics, Faculty of Science, Federal University Dutse, Dutse, Jigawa State, Nigeria

    R. I. Nuruddeen

Authors
  1. Khalid K. Ali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. I. Nuruddeen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. R. Raslan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Khalid K. Ali.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ali, K.K., Nuruddeen, R.I. & Raslan, K.R. New hyperbolic structures for the conformable time-fractional variant bussinesq equations. Opt Quant Electron 50, 61 (2018). https://doi.org/10.1007/s11082-018-1330-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-018-1330-6

Keywords

Mathematics Subject Classification

Search

Navigation