Skip to main content
SpringerLink
Log in

On a new technique for solving the nonlinear conformable time-fractional differential equations

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

Abstract

Real-world phenomena often are modelled by the nonlinear fractional differential equations. In this work, a novel technique called the \(\exp \left( { - \phi \left( \varepsilon \right)} \right)\) method is employed to find the exact solutions of nonlinear FDEs. Some well-known time-fractional differential equations in the context of conformable derivative, viz. the time-fractional modified Benjamin–Bona–Mahony (BBM) equation and the time-fractional Cahn–Hilliard (CH) equation are considered to test the usefulness of the method. The utility of the \(\exp \left( { - \phi \left( \varepsilon \right)} \right)\) method in solving nonlinear FDEs is proved.

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

  • Baleanu, D., Ugurlu, Y., Inc, M., Kilic, B.: Improved (G′/G)-expansion method for the time-fractional biological population model and Cahn–Hilliard equation. J. Comput. Nonlinear Dyn. 10, 051016 (2015). doi:10.1115/1.4029254

    Article  Google Scholar 

  • Bulut, H., Pandir, Y.: Modified trial equation method to the nonlinear fractional Sharma–Tasso–Olever equation. Int. J. Model. Optim. 3, 353–357 (2013)

    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 (2013). doi:10.1155/2013/636802

    MathSciNet  Google Scholar 

  • Dehghan, M., Manafian, J.: Analytical treatment of some partial differential equations arising in mathematical physics by using the Exp-function method. Int. J. Mod. Phys. B 25, 2965–2981 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Dehghan, M., Manafian Heris, J., Saadatmandi, A.: Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics. Int. J. Numer. Methods Heat Fluid Flow 21, 736–753 (2011)

    Article  MathSciNet  Google Scholar 

  • Demiray, S.T., Bulut, H.: Soliton solutions of some nonlinear evolution problems by GKM. Neural Comput. Appl. (2017). doi:10.1007/s00521-017-2999-3

    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

    Article  MATH  MathSciNet  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)

    Article  MATH  MathSciNet  Google Scholar 

  • Guner, O., Korkmaz, A., Bekir, A.: Dark soliton solutions of space-time fractional Sharma–Tasso–Olver and potential Kadomtsev–Petviashvili equations. Commun. Theor. Phys. 67, 182–188 (2017)

    Article  ADS  MATH  Google Scholar 

  • Hafez, M.G., Sakthivel, R., Talukder, M.R.: Some new electrostatic potential functions used to analyze the ion-acoustic waves in a Thomas Fermi plasma with degenerate electrons. Chin. J. Phys. 53, 120901 (2015). doi:10.6122/CJP.20150921

    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)

    Article  MATH  MathSciNet  Google Scholar 

  • Hosseini, K., Yazdani Bejarbaneh, E., Bekir, A., Kaplan, M.: New exact solutions of some nonlinear evolution equations of pseudoparabolic type. Opt. Quantum Electron. 49, 241 (2017a). doi:10.1007/s11082-017-1070-z

    Article  Google Scholar 

  • Hosseini, K., Bekir, A., Kaplan, M.: New exact traveling wave solutions of the Tzitzéica-type evolution equations arising in non-linear optics. J. Mod. Opt. 64, 1688–1692 (2017b)

    Article  ADS  Google Scholar 

  • Hosseini, K., Xu, Y.J., Mayeli, P., Bekir, A., Yao, P., Zhou, Q., Güner, Ö.: A study on the conformable time-fractional Klein–Gordon equations with quadratic and cubic nonlinearities. Optoelectron. Adv. Mater. Rapid Commun. 11, 423–429 (2017c)

    Google Scholar 

  • Hosseini, K., Bekir, A., Ansari, R.: Exact solutions of nonlinear conformable time-fractional Boussinesq equations using the exp (−ϕ(ε))-expansion method. Opt. Quantum Electron. 49, 131 (2017d). doi:10.1007/s11082-017-0968-9

    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 (2017e)

    Article  ADS  Google Scholar 

  • Islam, S.M.R., Khan, K., Akbar, M.A.: Exact solutions of unsteady Korteweg-de Vries and time regularized long wave equations. Springer Plus 4, 124 (2015). doi:10.1186/s40064-015-0893-y

    Article  Google Scholar 

  • Jahani, M., Manafian, J.: Improvement of the Exp-function method for solving the BBM equation with time-dependent coefficients. Eur. Phys. J. Plus 131, 54 (2016). doi:10.1140/epjp/i2016-16054-2

    Article  Google Scholar 

  • Kaplan, M., Bekir, A.: A novel analytical method for time-fractional differential equations. Optik 127, 8209–8214 (2016)

    Article  ADS  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)

    Article  MATH  MathSciNet  Google Scholar 

  • Korkmaz, A.: Exact solutions to (3 + 1) conformable time fractional Jimbo–Miwa, Zakharov–Kuznetsov and modified Zakharov–Kuznetsov equations. Commun. Theor. Phys. 67, 479–482 (2017)

    Article  ADS  MATH  Google Scholar 

  • Korkmaz, A.: On the wave solutions of conformable fractional evolution equations. Communications 67, 68–79 (2018)

    Google Scholar 

  • Korkmaz, A., Hosseini, K.: Exact solutions of a nonlinear conformable time-fractional parabolic equation with exponential nonlinearity using reliable methods. Opt. Quantum Electron. 49, 278 (2017). doi:10.1007/s11082-017-1116-2

    Article  Google Scholar 

  • Manafian, J.: On the complex structures of the Biswas–Milovic equation for power, parabolic and dual parabolic law nonlinearities. Eur. Phys. J. Plus 130, 255 (2015). doi:10.1140/epjp/i2015-15255-5

    Article  Google Scholar 

  • Manafian, J.: Optical soliton solutions for Schrödinger type nonlinear evolution equations by the tan (ϕ(ξ)/2)-expansion method. Optik 127, 4222–4245 (2016)

    Article  ADS  Google Scholar 

  • Manafian, J., Lakestani, M.: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity. Eur. Phys. J. Plus 130, 61 (2015). doi:10.1140/epjp/i2015-15061-1

    Article  Google Scholar 

  • Manafian, J., Lakestani, M.: Abundant soliton solutions for the Kundu–Eckhaus equation via tan (ϕ(ξ)/2)-expansion method. Optik 127, 5543–5551 (2016)

    Article  ADS  Google Scholar 

  • Manafian, J., Lakestani, M.: A new analytical approach to solve some of the fractional-order partial differential equations. Indian J. Phys. 91, 243–258 (2017)

    Article  ADS  Google Scholar 

  • Manafian, J., Fazli Aghdaei, M., Khalilian, M., Sarbaz Jeddi, R.: Application of the generalized G′/G-expansion method for nonlinear PDEs to obtaining soliton wave solution. Optik 135, 395–406 (2017)

    Article  ADS  Google Scholar 

  • Mirzazadeh, M.: Analytical study of solitons to nonlinear time fractional parabolic equations. Nonlinear Dyn. 85, 2569–2576 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  • Roshid, H.O., Kabir, M.R., Bhowmik, R.C., Datta, B.K.: Investigation of solitary wave solutions for Vakhnenko–Parkes equation via exp-function and Exp (−ϕ(ξ))-expansion method. Springer Plus 3, 692 (2014). doi:10.1186/2193-1801-3-692

    Article  Google Scholar 

  • Saha Ray, S., Sahoo, S.: Two efficient reliable methods for solving fractional fifth order modified Sawada–Kotera equation appearing in mathematical physics. J. Ocean Eng. Sci. 1, 219–225 (2016)

    Article  Google Scholar 

  • Sahoo, S., Saha Ray, S.: New approach to find exact solutions of time-fractional Kuramoto–Sivashinsky equation. Physica A 434, 240–245 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  • Tandogan, Y.A., Bildik, N.: Exact solutions of the time-fractional Fisher equation by using modified trial equation method. AIP Conf. Proc. 1738, 290018 (2016). doi:10.1063/1.4952090

    Article  Google Scholar 

  • Taşcan, F., Akbulut, A.: Exact solutions of nonlinear partial differential equations with exp (−φ(ξ))-expansion method. Afyon Kocatepe Univ. J. Sci. Eng. 17, 86–92 (2017)

    Google Scholar 

  • Teymuri Sindi, C., Manafian, J.: Wave solutions for variants of the KdV–Burger and the K(n,n)-Burger equations by the generalized G′/G-expansion method. Math. Methods Appl. Sci. 40, 4350–4363 (2017a)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Teymuri Sindi, C., Manafian, J.: Soliton solutions of the quantum Zakharov–Kuznetsov equation which arises in quantum magneto-plasmas. Eur. Phys. J. Plus 132, 67 (2017b). doi:10.1140/epjp/i2017-11354-7

    Article  MATH  Google Scholar 

  • Topsakal, M., Guner, O., Bekir, A., Unsal, O.: Exact solutions of some fractional differential equations by various expansion methods. J. Phys. Conf. Ser. 766, 012035 (2016). doi:10.1088/1742-6596/766/1/012035

    Article  Google Scholar 

  • Yasar, E., Giresunlu, I.B.: The (G′/G, 1/G)-expansion method for solving nonlinear space-time fractional differential equations. Pramana J. Phys. 87, 17 (2016). doi:10.1007/s12043-016-1225-7

    Article  ADS  Google Scholar 

  • Younis, M.: Soliton solutions of fractional order KdV–Burger’s equation. arXiv:1309.0098v1 [math-ph] (2013)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Rasht Branch, Islamic Azad University, Rasht, Iran

    K. Hosseini

  2. Department of Mathematics and Computer, Art–Science Faculty, Eskisehir Osmangazi University, Eskisehir, Turkey

    A. Bekir & M. Kaplan

  3. Department of International Trade, Faculty of Economics and Administrative Sciences, Cankiri Karatekin University, Cankiri, Turkey

    Ö. Güner

Authors
  1. K. Hosseini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Bekir
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Kaplan
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Ö. Güner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. Hosseini.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hosseini, K., Bekir, A., Kaplan, M. et al. On a new technique for solving the nonlinear conformable time-fractional differential equations. Opt Quant Electron 49, 343 (2017). https://doi.org/10.1007/s11082-017-1178-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11082-017-1178-1

Keywords

Search

Navigation