Skip to main content

Advertisement

SpringerLink
Log in

Fractional-Order Genocchi–Petrov–Galerkin Method for Solving Time–Space Fractional Fokker–Planck Equations Arising from the Physical Phenomenon

  • Original Paper
  • Published:
International Journal of Applied and Computational Mathematics Aims and scope Submit manuscript

Abstract

In the current study, we present the fractional-order Genocchi–Petrov–Galerkin method to investigate the approximate solution of time–space fractional Fokker–Planck equations (FFPEs). In the proposed approach, due to fractional Genocchi functions (FGFs) and their operational matrices transform these equations into a system of algebraic equations. The operational matrices of fractional-order integration and derivative are obtained by utilizing Riemann–Liouville fractional integral operator and Caputo fractional derivative, respectively. The convergence analysis of the proposed technique is rigorously discussed. To illustrate the applicability of the error formulas, we present the process of obtaining results for some examples. These examples are carried out to confirm the effectiveness of the proposed method.

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
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others

References

  1. Aminataei, A., Vanani, S.K.: Numerical solution of fractional Fokker–Planck equation using the operational collocation method. Appl. Comput. Math. 12(1), 33–43 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Araci, S., Acikgoz, M., Bagdasaryan, A., Sen, E.: The Legendre polynomials associated with Bernoulli, Euler, Hermite and Bernstein polynomials (2013). ArXiv preprint arXiv:1312.7838

  3. Bayad, A., Kim, T.: Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials. Adv. Stud. Contemp. Math. 20(2), 247–53 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Chen, S., Liu, F., Zhuang, P., Anh, V.: Finite difference approximations for the fractional Fokker–Planck equation. Appl. Math. Model. 33, 256–273 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Dehghan, M.: A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications. Numer. Methods Partial Differ. Equ. 22(1), 220–257 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Dehghan, M., Tatari, M.: The use of He’s variational iteration method for solving a Fokker–Planck equation. Phys. Scr. 74, 310–316 (2006)

    MATH  Google Scholar 

  7. Dehestani, H., Ordokhani, Y., Razzaghi, M.: Fractional-order Legendre–Laguerre functions and their applications in fractional partial differential equations. Appl. Math. Comput. 336, 433–453 (2018)

    MathSciNet  MATH  Google Scholar 

  8. Dehestani, H., Ordokhani, Y., Razzaghi, M.: Application of the modified operational matrices in multiterm variable-order time-fractional partial differential equations. Math. Methods Appl. Sci. 42, 7296–7313 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Dehestani, H., Ordokhani, Y., Razzaghi, M.: Hybrid functions for numerical solution of fractional Fredholm–Volterra functional integro-differential equations with proportional delays. Int. J. Numer. Model. 32, e2606 (2019)

    Google Scholar 

  10. Dehestani, H., Ordokhani, Y., Razzaghi, M.: Fractional-order Bessel functions with various applications. Appl. Math. 64(6), 637–662 (2019)

    MathSciNet  MATH  Google Scholar 

  11. Dehestani, H., Ordokhani, Y., Razzaghi, M.: Numerical technique for solving fractional generalized pantograph-delay differential equations by using fractional-order hybrid bessel functions. Int. J. Appl. Comput. Math. 69, 1–27 (2020)

    MathSciNet  Google Scholar 

  12. Dehestani, H., Ordokhani, Y., Razzaghi, M.: Pseudo-operational matrix method for the solution of variable-order fractional partial integro-differential equations. Eng. Comput. (2020). https://doi.org/10.1007/s00366-019-00912-z

    Article  MATH  Google Scholar 

  13. Deng, W.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47, 204–226 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Firoozjaee, M.A., Jafari, H., Lia, A., Baleanu, D.: Numerical approach of Fokker–Planck equation with Caputo–Fabrizio fractional derivative using Ritz approximation. J. Comput. Appl. Math. 339, 367–373 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Hashemi, M.S.: Group analysis and exact solutions of the time fractional Fokker–Planck equation. Phys. A 417, 141–149 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Heinsalu, E., Patriarca, M., Goychuk, I., Schmid, G., Hanggi, P.: Fractional Fokker–Planck dynamics: numerical algorithm and simulations. Phys. Rev. E 73, 1–9 (2006)

    Google Scholar 

  17. Heinsalu, E., Patriarca, M., Goychuk, I., Hanggi, P.: Fractional Fokker–Planck subdiffusion in alternating force fields. Phys. Rev. E 79(4), 041137 (2009)

    Google Scholar 

  18. Isah, A., Phang, C.: New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials. J. King. Saud Univ. Sci. 31(1), 1–7 (2019)

    Google Scholar 

  19. Jumarie, G.: A Fokker–Planck equation of fractional order with respect to time. J. Math. Phys. 33, 3536–3542 (1992)

    MathSciNet  MATH  Google Scholar 

  20. Khan, A., Abdeljawad, T., Gomez-Aguilar, J.F., Khan, H.: Dynamical study of fractional order mutualism parasitism food web module. Chaos Solitons Fractals 134, 109685 (2020)

    MathSciNet  Google Scholar 

  21. Khan, H., Gomez-Aguilar, J.F., Alkhazzan, A., Khan, A.: A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler Law. Math. Method Appl. Sci. 43(6), 3786–3806 (2020)

    Google Scholar 

  22. Khan, A., Gomez-Aguilar, J.F., Khan, T.S., Khan, H.: Stability analysis and numerical solutions of fractional order HIV/AIDS model. Chaos Solitons Fractals 122, 119–128 (2019)

    MathSciNet  Google Scholar 

  23. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)

    MATH  Google Scholar 

  24. Lakestani, M., Dehghan, M.: Numerical solution of Fokker–Planck equation using the cubic B-spline scaling functions. Numer. Methods Partial Differ. Equ. 25(2), 418–429 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Liang, J., Ren, F., Qiu, W., Xiao, J.: Exact solutions for nonlinear fractional anomalous diffusion equations. Phys. A 385, 80–94 (2007)

    MathSciNet  Google Scholar 

  26. Loh, J.R., Phang, C., Isah, A.: New operational matrix via genocchi polynomials for solving Fredholm-Volterra fractional integro-differential equations. Adv. in Math. Phy. 2017, 12 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Metzler, R., Jeon, J.H., Cherstvy, A.G., Barkai, E.: Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014)

    Google Scholar 

  28. Odibat, Z., Momani, S.: Numerical solution of Fokker–Planck equation with space- and time-fractional derivatives. Phys. Lett. A 369(5), 349–358 (2007)

    MATH  Google Scholar 

  29. Pinto, L., Sousa, E.: Numerical solution of a time–space fractional Fokker Planck equation with variable force field and diffusion. Commun. Nonlinear Sci. Numer. Simulat. 50, 211–228 (2017)

    MathSciNet  Google Scholar 

  30. Prakash, A., Kaur, H.: Numerical solution for fractional model of Fokker–Planck equation by using q-HATM. Chaos Solitons Fractals 105, 99–110 (2017)

    MathSciNet  MATH  Google Scholar 

  31. Ray, S.S., Gupta, A.K.: A two-dimensional Haar wavelet approach for the numerical simulations of time and space fractional Fokker–Planck equations in modelling of anomalous diffusion systems. J. Math. Chem. 52(8), 2277–2293 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Risken, H.: The Fokker–Planck Equation. Springer, Berlin (1989)

    MATH  Google Scholar 

  33. Rivlin, T.J.: An Introduction to the Approximation of Functions. Dover Publications, New York (1981)

    MATH  Google Scholar 

  34. Shkilev, V.P.: Subdiffusion in a time-dependent force field. J. Exp. Theor. Phys. 114(5), 830–835 (2012)

    Google Scholar 

  35. Saravanan, A., Magesh, N.: An efficient computational technique for solving the Fokker–Planck equation with space and time fractional derivatives. J. King. Saud Univ. Sci. 28, 160–166 (2016)

    Google Scholar 

  36. Sepehrian, B., Radpoor, M.K.: Numerical solution of non-linear Fokker–Planck equation using finite differences method and the cubic spline functions. Appl. Math. Comput. 262, 187–190 (2015)

    MathSciNet  MATH  Google Scholar 

  37. Sokolov, I.M., Klafter, J.: Field-induced dispersion in subdiffusion. Phys. Rev. Lett. 97(14), 140602 (2006)

    Google Scholar 

  38. Tatari, M., Dehghan, M., Razzaghi, M.: Application of the Adomian decomposition method for the Fokker–Planck equation. Math. Comput. Model. 45, 639–650 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Xie, J., Yao, Z., Gui, H., Zhao, F., Li, D.: A two-dimensional Chebyshev wavelets approach for solving the Fokker–Planck equations of time and space fractional derivatives type with variable coefficients. Appl. Math. Comput. 332, 197–208 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Yang, Y., Huang, Y., Zhou, Y.: Numerical solutions for solving time fractional Fokker–Planck equations based on spectral collocation methods. J. Comput. Appl. Math. 339, 389–404 (2018)

    MathSciNet  MATH  Google Scholar 

  41. Yang, Q., Liu, F., Turner, I.: Computationally efficient numerical methods for time and space fractional Fokker–Planck equations. Phys. Scr. T136, 014026 (2009)

    Google Scholar 

  42. Yildirim, A.: Analytical approach to Fokker–Planck equation with space- and time-fractional derivatives by means of the homotopy perturbation method. J. King. Saud Univ. 22(4), 257–264 (2010)

    Google Scholar 

  43. Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical treatment for the fractional Fokker–Planck equation. ANZIAM J. 48, 759–774 (2007)

    MathSciNet  MATH  Google Scholar 

  44. Zhang, H., Jiang, X., Yang, X.: A time–space spectral method for the time–space fractional Fokker–Planck equation and its inverse problem. Appl. Math. Comput. 320, 302–318 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We express our sincere thanks to the anonymous referees for valuable suggestions that improved the final manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Mathematical Sciences, Alzahra university, Tehran, Iran

    Haniye Dehestani & Yadollah Ordokhani

  2. Department of Mathematics and Statistics, Mississippi State University, Starkville, MS, 39762, USA

    Mohsen Razzaghi

Authors
  1. Haniye Dehestani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yadollah Ordokhani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohsen Razzaghi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yadollah Ordokhani.

Ethics declarations

Conflict of interest

The authors declare that they have 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

Dehestani, H., Ordokhani, Y. & Razzaghi, M. Fractional-Order Genocchi–Petrov–Galerkin Method for Solving Time–Space Fractional Fokker–Planck Equations Arising from the Physical Phenomenon. Int. J. Appl. Comput. Math 6, 100 (2020). https://doi.org/10.1007/s40819-020-00859-6

Download citation

  • Published:

  • DOI: https://doi.org/10.1007/s40819-020-00859-6

Keywords

Mathematics Subject Classification

Search

Navigation