SpringerLink
Log in

A popular reaction-diffusion model fractional Fitzhugh-Nagumo equation: analytical and numerical treatment

  • Published:
Applied Mathematics-A Journal of Chinese Universities Aims and scope Submit manuscript

Abstract

The main objective of this article is to obtain the new analytical and numerical solutions of fractional Fitzhugh-Nagumo equation which arises as a model of reaction-diffusion systems, transmission of nerve impulses, circuit theory, biology and the area of population genetics. For this aim conformable derivative with fractional order which is a well behaved, understandable and applicable definition is used as a tool. The analytical solutions were got by utilizing the fact that the conformable fractional derivative provided the chain rule. By the help of this feature which is not provided by other popular fractional derivatives, nonlinear fractional partial differential equation is turned into an integer order differential equation. The numerical solutions which is obtained with the aid of residual power series method are compared with the analytical results that obtained by performing sub equation method. This comparison is made both with the help of three-dimensional graphical representations and tables for different values of the γ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R Khalil, M Al Horani, A Yousef, M Sababheh. A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 2014, 264: 65–70.

    Article  MathSciNet  Google Scholar 

  2. A Atangana, Derivative with a new parameter: Theory, methods and applications, Academic Press, 2015.

  3. A Kurt, O Tasbozan, Y Cenesiz. Homotopy analysis method for conformable Burgers-Korteweg-de Vries equation, Bull Math Sci Appl, 2016, 17: 17–23.

    Google Scholar 

  4. Y Cenesiz, O Tasbozan, A Kurt. Functional Variable Method for conformable fractional modified KdV-ZK equation and Maccari system, Tbilisi Mathematical Journal, 2017, 10: 118–126.

    Article  MathSciNet  Google Scholar 

  5. Y Cenesiz, A Kurt, O Tasbozan. On the New Solutions of the Conformable Time Fractional Generalized Hirota-Satsuma Coupled KdV System, Annals of West University of Timisoara-Mathematics and Computer Science, 2017, 55(1): 37–50.

    Article  MathSciNet  Google Scholar 

  6. O Tasbozan, M Şenol, A Kurt, O Ozkan. New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves, Ocean Engineering, 2018, 161: 62–68.

    Article  Google Scholar 

  7. O Tasbozan, Y Cenesiz, A Kurt, D Baleanu. New analytical solutions for conformable fractional PDEs arising in mathematical physics by exp-function method, Open Physics, 2017, 15: 647–651.

    Article  Google Scholar 

  8. T Abdeljawad. On conformable fractional calulus, Journal of Computational and Applied Mathematics, 2015, 279: 57–66.

    Article  MathSciNet  Google Scholar 

  9. R Fitzhugh. Impulse and physiological states in models of nerve membrane, Biophys Journal, 1961, 1: 445–466.

    Article  Google Scholar 

  10. J S Nagumo, S Arimoto, S Yoshizawa. An active pulse transmission line simulating nerve axon, Proc IRE, 1962, 50: 2061070.

    Article  Google Scholar 

  11. D G Aronson, H F Weinberger. Multidimensional nonlinear diffusion arising in population genetics, Adv Math, 1978, 30: 336.

    Article  MathSciNet  Google Scholar 

  12. M Alquran. Analytical solutions of fractional foam drainage equation by residual power series method, Mathematical sciences, 2015, 8: 153.

    Article  MathSciNet  Google Scholar 

  13. O A Arqub. Series solution of fuzzy differential equations under strongly generalized differentiability, Journal of Advanced Research in Applied Mathematics, 2013, 5: 31–52.

    Article  MathSciNet  Google Scholar 

  14. O A Arqub, A El-Ajou, A Bataineh, I Hashim, A representation of the exact solution of generalized Lane Emden equations using a new analytical method, Abstract and Applied Analysis, 2013, Article ID: 378593, 10 pages.

  15. Y Cenesiz, A Kurt. New fractional complex transform for conformable fractional partial differential equations, Journal of Applied Mathematics, Statistics and Informatics, 2016, 12: 41–47.

    Article  MathSciNet  Google Scholar 

  16. H M Jaradat, S Al-Shara, Q J Khan, M Alquran, K Al-Khaled. Analytical solution of time-fractional Drinfeld-Sokolov-Wilson system using residual power series method, IAENG International Journal of Applied Mathematics, 2016, 46: 64–70.

    MathSciNet  Google Scholar 

  17. A Kumar, S Kumar, M Singh. Residual power series method for fractional Sharma-Tasso-Olever equation, Commun Numer Anal, 2016, 1: 1–10.

    MathSciNet  Google Scholar 

  18. W Malfliet. Solitary wave solutions of nonlinear wave equations, American Journal of Physics, 1992, 60: 650–654.

    Article  MathSciNet  Google Scholar 

  19. S Zhang, H Q Zhang. Fractional sub-equation method and its applications to nonlinear fractional PDEs, Physics Letters A, 2011, 375: 1069–1073.

    Article  MathSciNet  Google Scholar 

  20. M Alquran, I Jaradat, D Baleanu, R Abdel-MuhsenAn. A new two-mode coupled Burgers equation: Analytical Study of (2 + 1)-Dimensional Physical Models 11 Embedded Entirely in Fractal Space, Romanian Journal of Physics, 2019, 64: 103.

    Google Scholar 

  21. I Jaradat, M Alquran, R Abdel-Muhsen. An analytical framework of 2D diffusion, wavelike, telegraph, and Burgers’ models with twofold Caputo derivatives ordering, Nonlinear Dynamics, 2018, 93(4): 1911–1922.

    Article  Google Scholar 

  22. I Jaradat, M Alquran, K Al-Khaled. An analytical study of physical 10 models with inherited temporal and spatial memory, The European Physical Journal Plus, 2018, 133(4): 162.

    Article  Google Scholar 

  23. M Alquran, I Jaradat, A novel scheme for solving Caputo time-fractional nonlinear equations: theory and application, Nonlinear Dynamics, 2018, 91(4): 2389–2395.

    Google Scholar 

  24. M Alquran, H M Jaradat, M I Syam. Analytical solution of the time-fractional Phi-4 equation by using modified residual power series method, Nonlinear Dynamics, 2017, 90(4): 2525–2529.

    Article  MathSciNet  Google Scholar 

  25. E F D Goufo, S Kumar, S B Mugisha. Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos, Solitons and Fractals, 2020, 130: 109467.

    Article  MathSciNet  Google Scholar 

  26. Z Odibat, S Kumar. A robust computational algorithm of homotopy asymptotic method for solving systems of fractional differential equations, Journal of Computational and Nonlinear Dynamics, 2019, 14(8): 081004.

    Article  Google Scholar 

  27. A El-Ajou, M A N Oqielat, Z Al-Zhour, S Kumar, S Momani. Solitary solutions for time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative, Chaos: An Interdisciplinary Journal of Nonlinear Science, 2019, 29(9): 093102.

    Article  MathSciNet  Google Scholar 

  28. S Kumar, A Kumar, S Momani, M Aldhaifallah, K S Nisar. Numerical solutions of nonlinear fractional model arising in the appearance of the stripe patterns in two-dimensional systems. Advances in Difference Equations, 2019, 2019(1): 413.

    Article  MathSciNet  Google Scholar 

  29. B Sharma, Sunil Kumar, C Cattani, D Baleanu. Nonlinear dynamics of Cattaneo-Christov heat flux model for third-grade power-law fluid, J Comput Nonlinear Dynam, 2020, 15(1): 011009.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Hatay Mustafa Kemal University, Hatay, 31000, Turkey

    Orkun Tasbozan

Authors
  1. Orkun Tasbozan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Orkun Tasbozan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tasbozan, O. A popular reaction-diffusion model fractional Fitzhugh-Nagumo equation: analytical and numerical treatment. Appl. Math. J. Chin. Univ. 36, 218–228 (2021). https://doi.org/10.1007/s11766-021-3810-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11766-021-3810-x

Keywords

MR Subject Classification

Search

Navigation