Skip to main content
SpringerLink
Log in

Separation method of semi-fixed variables together with dynamical system method for solving nonlinear time-fractional PDEs with higher-order terms

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

Abstract

It is well known that methods for solving fractional-order PDEs are grossly inadequate compared with integer-order PDEs. In this paper, a new approach combined with the separation method of semi-fixed variables and dynamical system method is introduced. As an example, a time-fractional reaction-diffusion equation with higher-order terms is studied under two different kinds of fractional-order differential operators. In different parametric regions, phase portraits of systems derived from the reaction-diffusion equation are presented. Existence and dynamic properties of solutions of this nonlinear time-fractional model are investigated. In some special parametric conditions, some exact solutions of this time-fractional models are obtained. The dynamical properties of some exact solutions are discussed, and the graphs of them are illustrated.

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
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data Availability Statement

The author assures that all data present within the text of the manuscript are available and reliable.

References

  1. Daftardar-Gejji, V., Jafari, H.: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 301(2), 508–518 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bakkyaraj, T., Sahadevan, R.: An approximate solution to some classes of fractional nonlinear partial differential difference equation using Adomian decomposition method. J. Fract. Calc. Appl. 5(1), 37–52 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Bakkyaraj, T., Sahadevan, R.: Approximate analytical solution of two coupled time fractional nonlinear Schrodinger equations. Int. J. Appl. Comput. Math. 2(1), 113–135 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bakkyaraj, T., Sahadevan, R.: On solutions of two coupled fractional time derivative Hirota equations. Nonlinear Dyn. 77(4), 1309–1322 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sahadevan, R., Bakkyaraj, T.: Invariant analysis of time fractional generalized Burgers and Korteweg–de Vries equations. J. Math. Anal. Appl. 393(2), 341–347 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bakkyaraj, T., Sahadevan, R.: Invariant analysis of nonlinear fractional ordinary differential equations with Riemann–Liouville derivative. Nonlinear Dyn. 80(1–2), 447–455 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Odibat, Z.M., Shaher, M.: The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput. Math. Appl. 58, 2199–2208 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Wu, G., Lee, E.W.M.: Fractional variational iteration method and its application. Phys. Lett. A 374(25), 2506–2509 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Momani, S., Zaid, O.: Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations. Comput. Math. Appl. 54(7–8), 910–919 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Sahadevan, R., Bakkyaraj, T.: Invariant subspace method and exact solutions of certain nonlinear time fractional partial differential equations. Fract. Calc. Appl. Anal. 18(1), 146–162 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Harris, P.A., Garra, R.: Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method. Nonlinear Stud. 20(4), 471–481 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Sahadevan, R., Prakash, P.: Exact solution of certain time fractional nonlinear partial differential equations. Nonlinear Dyn. 85(1), 659–673 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Elsayed, M.E.Z., Yasser, A.A., Reham, M.A.S.: The fractional complex transformation for nonlinear fractional partial differential equations in the mathematical physics. J. Assoc. Arab. Univ. Basic Appl. Sci. 19, 59–69 (2016)

    Google Scholar 

  14. Li, Z.B., Zhu, W.H., He, J.H.: Exact solutions of time-fractional heat conduction equation by the fractional complex transform. Therm. Sci. 16(2), 335–338 (2012)

    Article  Google Scholar 

  15. Li, Z.B., He, J.H.: Fractional complex transform for fractional differential equations. Math. Comput. Appl. 15(5), 970–973 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Chen, J., Liu, F., Anh, V.: Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338, 1364–1377 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain. Comput. Math. Appl. 64, 3377–3388 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Luchko, Y.: Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59, 1766–1772 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jumarie, G.: Fractional partial differential equations and modified Riemann–Liouville derivative new methods for solution. J. Appl. Math. Comput. 24(1–2), 31–48 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jumarie, G.: Modified Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results. Comput. Math. Appl. 51(9–10), 1367–1376 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jumarie, G.: Cauchy’s integral formula via the modified Riemann–Liouville derivative for analytic functions of fractional order. Appl. Math. Lett. 23(12), 1444–1450 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. He, J.H.: Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus. Physica Lett. A 376, 257–259 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Tarasov, V.E.: On chain rule for fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 30(1), 1–4 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rui, W.: Applications of homogenous balanced principle on investigating exact solutions to a series of time fractional nonlinear PDEs. Commun. Nonlinear Sci. Numer. Simul. 47, 253–266 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rui, W.: Applications of integral bifurcation method together with homogeneous balanced principle on investigating exact solutions of time fractional nonlinear PDEs. Nonlinear Dyn. 91(1), 697–712 (2018)

    Article  MathSciNet  Google Scholar 

  26. Wu, C., Rui, W.: Method of separation variables combined with homogenous balanced principle for searching exact solutions of nonlinear time-fractional biological population model. Commun. Nonlinear Sci. Numer. Simul. 63, 88–100 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rui, W.: Idea of invariant subspace combined with elementary integral method for investigating exact solutions of time-fractional NPDEs. Appl. Math. Comput. 339, 158–171 (2018)

    MathSciNet  MATH  Google Scholar 

  28. Rui, W.: Dynamical system method for investigating existence and dynamical property of solution of nonlinear time-fractional PDEs. Nonlinear Dyn. 99, 2421–2440 (2020)

    Article  Google Scholar 

  29. Li, J., Liu, Z.: Smooth and non-smooth traveling waves in a nonlinearly dispersive equation. Appl. Math. Model. 25(1), 41–56 (2000)

    Article  MATH  Google Scholar 

  30. Li, J., Zhang, L.: Bifurcations of travelling wave solutions in generalized Pochhammer–Chree equation. Chaos Solitons Fractals 14(4), 581–593 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  31. Li, J., Li, H., Li, S.: Bifurcations of travelling wave solutions for the generalized Kadomtsev–Petviashili equation. Chaos Solitons Fractals 20, 725–734 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, J., Chen, G.: Bifurcations of traveling wave solutions for four classes of nonlinear wave equations. Int. J. Bifurc. Chaos 15(2), 3973–3998 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  33. Malaguti, L., Marcelli, C.: Sharp profiles in degenerate and doubly degenerate Fisher-KPP equations. J. Differ. Equ. 195(2), 471–496 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  34. Pablo, A.D., Vázquez, J.L.: Travelling waves and finite propagation in a reaction-diffusion equation. J. Differ. Equ. 93(1), 19–61 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  35. Aronson, D.G.: Density-dependent interaction systems. In: Stewart, W.E., Ray, W.H., Cobley, C.C. (eds.) Dynamics and Modelling of Reactive Systems. Academic Press, New York (1980)

    Google Scholar 

  36. Byrd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Scientists. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11361023), the Science and Technology Commission in Chongqing City of China (Grant No. cstc2018jcyjAX0766) and the Research Project of Chongqing Education Commission (Grant No. CXQT21014).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Weiguo Rui

Authors
  1. Weiguo Rui
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weiguo Rui.

Ethics declarations

Conflict of Interest

The author declares that he has no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Rui, W. Separation method of semi-fixed variables together with dynamical system method for solving nonlinear time-fractional PDEs with higher-order terms. Nonlinear Dyn 109, 943–961 (2022). https://doi.org/10.1007/s11071-022-07463-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07463-x

Keywords

Search

Navigation