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New exact solutions of some nonlinear evolution equations of pseudoparabolic type

Abstract

This paper aims to conduct an analytical study into some nonlinear models of pseudoparabolic type, including the Oskolkov, Oskolkov–Benjamin–Bona–Mahony–Burgers, and Benjamin–Bona–Mahony–Peregrine–Burgers equations. A number of new exact solutions for these pseudoparabolic type equations have been derived based on the modified Kudryashov method that its calculations are performed in a symbolic computation system known as Maple.

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References

  • Akcagil, S., Aydemir, T., Gozukizil, O.F.: Exact travelling wave solutions of nonlinear pseudoparabolic equations by using the G′/G expansion method. New Trends Math. Sci. 4, 51–66 (2016)

    MathSciNet  Article  Google Scholar 

  • Ayati, Z., Hosseini, K., Mirzazadeh, M.: Application of Kudryashov and functional variable methods to the strain wave equation in microstructured solids. Nonlinear Eng. (2016). doi:10.1515/nleng-2016-0020

    Google Scholar 

  • Bekir, A., Güner, Ö., Bhrawy, A.H., Biswas, A.: Solving nonlinear fractional differential equations using exp-function and G′/G expansion methods. Rom. J. Phys. 60, 360–378 (2015)

    Google Scholar 

  • Biswas, A., Mirzazadeh, M.: Dark optical solitons with power law nonlinearity using (G′/G)-expansion. Optik 125, 4603–4608 (2014)

    ADS  Article  Google Scholar 

  • Bulut, H., Baskonus, H.M., Pandir, Y.: The modified trial equation method for fractional wave equation and time fractional generalized Burgers equation. Abstr. Appl. Anal. 2013, 636802 (2013a)

    MathSciNet  Google Scholar 

  • Bulut, H., Pandir, Y., Baskonus, H.M.: Symmetrical hyperbolic Fibonacci function solutions of generalized Fisher equation with fractional order. AIP Conf. Proc. 1558, 1914 (2013b)

    ADS  Article  Google Scholar 

  • Bulut, H., Sulaiman, T.A., Baskonus, H.M.: New solitary and optical wave structures to the Korteweg–de Vries equation with dual-power law nonlinearity. Opt. Quant. Electron. 48, 564 (2016). doi:10.1007/s11082-016-0831-4

    Article  Google Scholar 

  • Çenesiz, Y., Baleanu, D., Kurt, A., Tasbozan, O.: New exact solutions of Burgers’ type equations with conformable derivative. Waves Random Complex Media (2016). doi:10.1080/17455030.2016.1205237

    Google Scholar 

  • Demiray, S.T., Pandir, Y., Bulut, H.: Generalized Kudryashov method for time-fractional differential equations. Abstr. Appl. Anal. 2014, 901540 (2014)

    MathSciNet  Google Scholar 

  • Ege, S.M., Misirli, E.: The modified Kudryashov method for solving some fractional-order nonlinear equations. Adv. Differ. Equ. 2014, 135 (2014). doi:10.1186/1687-1847-2014-135

    MathSciNet  Article  MATH  Google Scholar 

  • Ekici, M., Mirzazadeh, M., Zhou, Q., Moshokoa, S.P., Biswas, A., Belic, M.: Solitons in optical metamaterials with fractional temporal evolution. Optik 127, 10879–10897 (2016)

    ADS  Article  Google Scholar 

  • Eslami, M., Mirzazadeh, M.: Exact solutions for fifth-order KdV-type equations with time-dependent coefficients using the Kudryashov method. Eur. Phys. J. Plus 129, 192 (2014). doi:10.1140/epjp/i2014-14192-1

    Article  Google Scholar 

  • Gözükizil, O.F., Akçağil, Ş.: The tanh-coth method for some nonlinear pseudoparabolic equations with exact solutions. Adv. Differ. Equ. 2013, 143 (2013). doi:10.1186/1687-1847-2013-143

    MathSciNet  Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Guner, O., Aksoy, E., Bekir, A., Cevikel, A.C.: Different methods for (3 + 1)-dimensional space-time fractional modified KdV–Zakharov–Kuznetsov equation. Comput. Math Appl. 71, 1259–1269 (2016)

    MathSciNet  Article  Google Scholar 

  • Hosseini, K., Ayati, Z.: Exact solutions of space-time fractional EW and modified EW equations using Kudryashov method. Nonlinear Sci. Lett. A 7, 58–66 (2016)

    Google Scholar 

  • Hosseini, K., Gholamin, P.: Feng’s first integral method for analytic treatment of two higher dimensional nonlinear partial differential equations. Differ. Equ. Dyn. Syst. 23, 317–325 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  • Hosseini, K., Ansari, R., Gholamin, P.: Exact solutions of some nonlinear systems of partial differential equations by using the first integral method. J. Math. Anal. Appl. 387, 807–814 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  • Hosseini, K., Sadeghi, F., Ansari, R.: First integral method for solving nonlinear physical systems of partial differential equations. J. Nat. Sci. Sustain. Tech. 8, 391–400 (2014)

    Google Scholar 

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

    ADS  Article  Google Scholar 

  • Hosseini, K., Bekir, A., Ansari, R.: New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method. Optik 132, 203–209 (2017b)

    ADS  Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Kaplan, M., Bekir, A.: The modified simple equation method for solving some fractional-order nonlinear equations. Pramana J. Phys. 87, 15 (2016). doi:10.1007/s12043-016-1205-y

    ADS  Article  Google Scholar 

  • Kaplan, M., Bekir, A., Akbulut, A., Aksoy, E.: The modified simple equation method for nonlinear fractional differential equations. Rom. J. Phys. 60, 1374–1383 (2015)

    Google Scholar 

  • Khan, K., Ali Akbar, M., Rashidi, M.M., Zamanpour, I.: Exact traveling wave solutions of an autonomous system via the enhanced (G′/G)-expansion method. Waves Random Complex Media 25, 644–655 (2015)

    ADS  MathSciNet  Article  Google Scholar 

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

    ADS  MathSciNet  Article  MATH  Google Scholar 

  • Manafian, J.: Application of the ITEM for the system of equations for the ion sound and Langmuir waves. Opt. Quant. Electron. 49, 17 (2017). doi:10.1007/s11082-016-0860-z

    Article  Google Scholar 

  • Mirzazadeh, M., Ekici, M., Zhou, Q., Sonmezoglu, A.: Analytical study of solitons to the generalized resonant dispersive nonlinear Schrödinger’s equation with power law nonlinearity. Superlattices Microstruct. (2016). doi:10.1016/j.spmi.2016.12.003

    Google Scholar 

  • Odabasi, M., Misirli, E.: On the solutions of the nonlinear fractional differential equations via the modified trial equation method. Math. Methods Appl. Sci. (2015). doi:10.1002/mma.3533

    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 

  • Ryabov, P.N.: Exact solutions of the Kudryashov–Sinelshchikov equation. Appl. Math. Comput. 217, 3585–3590 (2010)

    MathSciNet  MATH  Google Scholar 

  • Saha, R.S.: New analytical exact solutions of time fractional KdV–KZK equation by Kudryashov methods. Chin. Phys. B 25, 040204 (2016)

    ADS  Article  Google Scholar 

  • Sahoo, S., Saha, S.: Ray, Solitary wave solutions for time fractional third order modified KdV equation using two reliable techniques (G′/G)-expansion method and improved (G′/G)-expansion method. Phys. A 448, 265–282 (2016)

    MathSciNet  Article  Google Scholar 

  • Taghizadeh, N., Zhou, Q., Ekici, M., Mirzazadeh, M.: Soliton solutions for Davydov solitons in α-helix proteins. Superlattices Microstruct. 102, 323–341 (2017)

    ADS  Article  Google Scholar 

  • Tariq, H., Akram, G.: New approach for exact solutions of time fractional Cahn–Allen equation and time fractional Phi-4 equation. Phys. A (2017). doi:10.1016/j.physa.2016.12.081

    MathSciNet  Google Scholar 

  • Younis, M.: The first integral method for time-space fractional differential equations. J. Adv. Phys. 2, 220–223 (2013)

    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 

  • Younis, M., Rizvi, S.T.R.: Dispersive dark optical soliton in (2 + 1)-dimensions by (G′/G)-expansion with dual-power law nonlinearity. Optik 126, 5812–5814 (2015)

    ADS  Article  Google Scholar 

  • Younis, M., Zafar, A.: Exact solution to nonlinear differential equations of fractional order via (G/G′)-expansion method. Appl. Math. 5, 1–6 (2014)

    Article  Google Scholar 

  • Zayed, E.M.E., Alurrfi, K.A.E.: The modified Kudryashov method for solving some seventh order nonlinear PDEs in mathematical physics. World J. Model. Simul. 11, 308–319 (2015)

    Google Scholar 

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Correspondence to A. Bekir.

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Hosseini, K., Yazdani Bejarbaneh, E., Bekir, A. et al. New exact solutions of some nonlinear evolution equations of pseudoparabolic type. Opt Quant Electron 49, 241 (2017). https://doi.org/10.1007/s11082-017-1070-z

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  • DOI: https://doi.org/10.1007/s11082-017-1070-z

Keywords

  • Pseudoparabolic type equations
  • Oskolkov equation
  • Oskolkov–Benjamin–Bona–Mahony–Burgers equation
  • Benjamin–Bona–Mahony–Peregrine–Burgers equation
  • Modified Kudryashov method
  • New exact solutions