Skip to main content
SpringerLink
Log in

A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The current work contributes to find a supportive analytical solution of Caputo time-fractional nonlinear equations of the form

$$\begin{aligned}&{\mathcal {D}}^{\alpha }_{t}\big [u(x,t)\big ]+\left( \sum _{k=1}^{N_1}\lambda _{k}u^k(x,t)\right) u_{x}(x,t)\nonumber \\&\quad +\sum _{k=2}^{N_2}\delta _{k}\frac{\partial ^ku(x,t)}{\partial x^k}=0,\ \ \ \ 0<\alpha \le 1. \end{aligned}$$

We proposed a modified version of the generalized Taylor power series method to extract a reliable approximate solution of this problem. Theorems regarding the convergence of the obtained solution are provided. The method is tested on two models of the proposed equation, namely the Caputo time-fractional Gardner and Kawahara equations. Finally, both graphical explanations and tabular analysis are performed to study the accuracy of our suggested scheme, and to study the effects of \(\alpha \) on the behavior of the obtained solution.

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
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Caputo, M., Mainardi, F.: A new dissipation model based on memory mechanism. Pure Appl. Geophys. 91, 134–147 (1971)

    Article  MATH  Google Scholar 

  2. Odibat, Z., Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear Sci. Numer. Simul. 7, 27–34 (2006)

    Article  MathSciNet  Google Scholar 

  3. Jafaria, J., Nazarib, M., Baleanuc, D., Khalique, C.M.: A new approach for solving a system of fractional partial differential equations. Comput. Math. Appl. 66, 838–843 (2013)

    Article  MathSciNet  Google Scholar 

  4. Saha, Ray S., Bera, R.K.: Analytical solution of a fractional diffusion equation by Adomian decomposition method. Appl. Math. Comput. 174, 329–336 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial differential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. 26, 448–479 (2010)

    MathSciNet  MATH  Google Scholar 

  6. Ganjiani, M.: Solution of nonlinear fractional differential equations using Homotopy analysis method. Appl. Math. Model. 34, 1634–1641 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. He, J.H.: Homotopy perturbation method: a new nonlinear analytical technique. Appl. Math. Comput. 135, 73–79 (2003)

    MathSciNet  MATH  Google Scholar 

  8. Pandey, R.K., Singh, O.P., Baranwal, V.K.: An analytic algorithm for the space-time fractional advection–dispersion equation. Comput. Phys. Commun. 182, 1134–1144 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Alquran, M., Al-Khaled, K., Sarda, T., Chattopadhyay, J.: Revisited Fisher’s equation in a new outlook: a fractional derivative approach. Phys. A Stat. Mech. Appl. 438, 81–93 (2015)

    Article  MathSciNet  Google Scholar 

  10. Alquran, M., Al-Khaled, K., Chattopadhyay, J.: Analytical solutions of fractional population diffusion model: residual power series. Nonlinear Stud. 22(1), 31–39 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Alquran, M.: Analytical solution of time-fractional two-component evolutionary system of order 2 by residual power series method. J. Appl. Anal. Comput. 5(4), 589–599 (2015)

    MathSciNet  Google Scholar 

  12. Jaradat, H.M., Al-Shara, S., Khan, Q.J.A., Alquran, M., Al-Khaled, K.: Analytical solution of time-fractional Drinfeld–Sokolov–Wilson system using residual power series method. IAENG Int. J. Appl. Math. 46(1), 64–70 (2016)

    MathSciNet  Google Scholar 

  13. Jaradat, H.M., Jaradat, I., Alquran, M., Jaradat, M.M.M., Mustafa, Z., Abohassan, K., Abdelkarim, R.: Approximate solutions to the generalized time-fractional Ito system. Ital. J. Pure Appl. Math. 37, 699–710 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Alquran, M., Al-Khaled, K., Sivasundaram, S., Jaradat, H.M.: Mathematical and numerical study of existence of bifurcations of the generalized fractional Burgers–Huxley equation. Nonlinear Stud. 24(1), 235–244 (2017)

    MathSciNet  MATH  Google Scholar 

  15. Abu, Arqub O.: Series solution of fuzzy differential equations under strongly generalized differentiability. J. Adv. Res. Appl. Math. 5, 31–52 (2013)

    Article  MathSciNet  Google Scholar 

  16. Abu, Arqub O., El-Ajou, A., Bataineh, A., Hashim, I.: A representation of the exact solution of generalized Lane Emden equations using a new analytical method. In: Abstract and Applied Analysis. (2013), Article ID 378593

  17. El-Ajou, A., Abu, Arqub O., Al Zhour, Z., Momani, S.: New results on fractional power series: theories and applications. Entropy. 15, 5305–5323 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kumar, S., Kumar, A., Baleanu, D.: Two analytical methods for time-fractional nonlinear coupled Boussinesq–Burger’s equations arise in propagation of shallow water waves. Nonlinear Dyn. 85(2), 699–715 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mahmood, B.A., Yousif, M.A.: A novel analytical solution for the modified Kawahara equation using the residual power series method. Nonlinear Dyn. 89(2), 1233–1238 (2017)

    Article  MathSciNet  Google Scholar 

  20. Chen, L., Zhao, T., Li, W., et al.: Bifurcation control of bounded noise excited Duffing oscillator by a weakly fractional-order \(PI^{\lambda } D^{\mu }\) feedback controller. Nonlinear Dyn. 83(1–2), 529–539 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Deng, W.: Short memory principle and a predictor–corrector approach for fractional differential equations. J. Comput. Appl. Math. 206(1), 174–188 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Pandir, Y., Duzgun, H.H.: New exact solutions of time fractional Gardner equation by using new version of F-expansion method. Commun. Theor. Phys. 67(1), 9 (2017)

    Article  MATH  Google Scholar 

  24. Iyiola, O.S., Olayinka, O.G.: Analytical solutions of time-fractional models for homogeneous Gardner equation and non-homogeneous differential equations. Ain Shams Eng. J. 5(3), 999–1004 (2014)

    Article  Google Scholar 

  25. Guo, S., Mei, L., Zhang, Z.: Time-fractional Gardner equation for ion-acoustic waves in negative-ion-beam plasma with negative ions and nonthermal nonextensive electrons. Phys. Plasmas. 22, 052306 (2015). https://doi.org/10.1063/1.4919264

    Article  Google Scholar 

  26. Ray, S.S., Sahoo, S.: New exact solutions of time fractional modified Kawahara equations in modelling surface tension in shallow-water and capillary gravity water waves. Eur. Phys. J. Plus. 132, 9 (2017). https://doi.org/10.1140/epjp/i2017-11276-4

    Article  Google Scholar 

  27. Yaslan, H.: New analytic solutions of the conformable space-time fractional Kawahara equation. Opt. Int. J. Light Electron Opt. 140, 123–126 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the reviewers for the valuable comments.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O. Box: 3030, Irbid, 22110, Jordan

    Marwan Alquran & Imad Jaradat

Authors
  1. Marwan Alquran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Imad Jaradat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marwan Alquran.

Ethics declarations

Conflicts of interest

The authors declares that there is no conflict of interests regarding the publication of the paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Alquran, M., Jaradat, I. A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application. Nonlinear Dyn 91, 2389–2395 (2018). https://doi.org/10.1007/s11071-017-4019-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-4019-7

Keywords

Mathematics Subject Classification

Search

Navigation