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Applications of two reliable methods for solving a nonlinear conformable time-fractional equation

Abstract

The current work presents analytical solutions of a nonlinear conformable time-fractional equation by using two different techniques. These are the modified simple equation method and the exponential rational function method. Based on the conformable fractional derivative and traveling wave transformation, the fractional partial differential equation is turned into the nonlinear non-fractional ordinary differential equation. Therefore, we implement the algorithms to this nonlinear non-fractional ordinary differential equation. To the best of our knowledge, the exact solutions obtained in this paper might be very useful in various areas of applied mathematics in interpreting some physical phenomena.

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References

  • Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  • Aksoy, E., Kaplan, M., Bekir, A.: Exponential rational function method for space–time fractional differential equations. Waves Random Complex Media 26(2), 142–151 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  • Akter, J., Akbar, M.A.: Solitary wave solutions to the ZKBBM equation and the KPBBM equation via the modified simple equation method. J. Partial Differ. Equ. 29(2), 143–160 (2016)

    MathSciNet  MATH  Google Scholar 

  • Atangana, A., Baleanu, D., Alsaedi, A.: New properties of conformable derivative. Open Math. 13, 889–898 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  • Bekir, A., Guner, O., Cevikel, A.C.: Fractional complex transform and exp-function methods for fractional differential equations. Abstr. Appl. Anal. 2013, 426462 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  • Cenesiz, Y., Baleanu, D., Kurt, A., Tasbozan, O.: New exact solutions of Burgers’ type equations with conformable derivative. Waves Random Complex Media 27(1), 103–116 (2017)

    MathSciNet  Article  Google Scholar 

  • Demiray, S., Unsal, O., Bekir, A.: New exact solutions for Boussinesq type equations by using (\(G^{\prime }/G,1/G\)) and (\(1/G^{\prime }\) )-expansion methods. Acta Phys. Pol. A 125(5), 1093–1098 (2014)

    Article  MATH  Google Scholar 

  • Demiray, S.T., Pandir, Y., Bulut, H.: New solitary wave solutions of Maccari system. Ocean Eng. 103, 153–159 (2015)

    Article  Google Scholar 

  • Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  • Ekici, M., Mirzazadeh, M., Eslami, M., Zhou, Q., Moshokoa, S.P., Biswas, A., Belic, M.: Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives. Optik 127(22), 10659–10669 (2016)

    ADS  Article  Google Scholar 

  • Eslami, M.: Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations. Appl. Math. Comput. 285, 141–148 (2016)

    MathSciNet  Google Scholar 

  • Eslami, M., Rezazadeh, H.: The first integral method for Wu–Zhang system with conformable time-fractional derivative. Calcolo 53, 475–485 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  • Eslami, M., Fathi Vajargah, B., Mirzazadeh, M., Biswas, A.: Application of first integral method to fractional partial differential equations. Indian J. Phys. 88(2), 177–184 (2014)

    ADS  Article  Google Scholar 

  • Hosseini, K., Ansari, R.: New exact solutions of nonlinear conformable time-fractional Boussinesq equations using the modified Kudryashov method. Waves. Random. Complex Media. 27(4), 628–636 (2017)

    MathSciNet  Article  Google Scholar 

  • Hosseini, K., Mayeli, P., Ansar, R.: Modified Kudryashov method for solving the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities. Optik 130, 737–742 (2017)

    ADS  Article  Google Scholar 

  • Iyiola, O.S., Tasbozan, O., Kurt, A., Çenesiz, Y.: On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion. Chaos Solitons Fractals 94, 1–7 (2017)

    ADS  MathSciNet  Article  Google Scholar 

  • Jawad, A.J.M., Petković, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217(2), 869–877 (2010)

    MathSciNet  MATH  Google Scholar 

  • Kaplan, M., Bekir, A., Akbulut, A., Aksoy, E.: Exact solutions of nonlinear fractional differential equations by modified simple equation method. Rom. J. Phys. 60(9–10), 1374–1383 (2015)

    Google Scholar 

  • Kaplan, M., Bekir, A., Ozer, M.N.: A simple technique for constructing exact solutions to nonlinear differential equations with conformable fractional derivative. Opt. Quantum Electron. 49, 266 (2017)

    Article  Google Scholar 

  • Khalil, R., Al Horani, M., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  • Lohmann, A.W., Mendlovic, D., Zalevsky, Z., Dorsch, R.G.: Some important fractional transformations for signal processing. Opt. Commun. 125(1), 18–20 (1996)

    ADS  Article  Google Scholar 

  • Metzler, R., Barkai, E., Klafter, J.: Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82(18), 3563-3567 (1999)

    ADS  Article  Google Scholar 

  • Oldham, K.B., Spanier, F.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  • Pandir, Y., Gurefe, Y., Misirli, E.: The extended trial equation method for some time fractional differential equations. Discrete Dyn. Nat. Soc. 2013, 491359 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  • Pandir, Y., Demiray, S.T., Bulut, H.: A new approach for some NLDEs with variable coefficients. Optik 127(23), 11183–11190 (2016)

    ADS  Article  Google Scholar 

  • Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  • Taghizadeh, N., Mirzazadeh, M., Rahimian, M., Akbari, M.: Application of the simplest equation method to some time-fractional partial differential equations. Ain Shams Eng. J. 4, 897–902 (2013)

    Article  Google Scholar 

  • Unal, E., Gokdogan, A.: Solution of conformable fractional ordinary differential equations via differential transform method. Optik 128, 264–273 (2017)

    ADS  Article  Google Scholar 

  • Younis, M.: A new approach for the exact solutions of nonlinear equations of fractional order via modified simple equation method. Appl. Math. 5, 1927–1932 (2014)

    Article  Google Scholar 

  • Zayed, E.M.E., Gepreel, A.K.: Some applications of the \((G^{\prime }/G)\)-expansion method to nonlinear partial differential equations. Appl. Math. Comput. 212, 1–13 (2009)

    MathSciNet  MATH  Google Scholar 

  • Zheng, B.: \((G^{\prime }/G)\)-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun. Theor. Phys. 58, 623–630 (2012)

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Melike Kaplan.

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Kaplan, M. Applications of two reliable methods for solving a nonlinear conformable time-fractional equation. Opt Quant Electron 49, 312 (2017). https://doi.org/10.1007/s11082-017-1151-z

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  • DOI: https://doi.org/10.1007/s11082-017-1151-z

Keywords

  • Conformable derivative
  • Exact solutions
  • Modified simple equation method
  • Exponential rational function method