Log in

A different approach for conformable fractional biochemical reaction—diffusion models

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

Abstract

This paper attempts to shed light on three biochemical reaction-diffusion models: conformable fractional Brusselator, conformable fractional Schnakenberg, and conformable fractional Gray-Scott. This is done using conformable residual power series (hence-form, CRPS) technique which has indeed, proved to be a useful tool for generating the solution. Interestingly, CRPS is an effective method of solving nonlinear fractional differential equations with greater accuracy and ease.

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

Access this article

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 includes VAT (Germany)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. G Adomian. The diffusion-Brusselator equation, Comput Math Appl, 1995, 29: 1–3.

    Article  MathSciNet  Google Scholar 

  2. Marius Ghergu. Non-constant steady-state solutions for Brusselator type system, London Mathematical Society, 2008, 21: 2331–2345.

    MathSciNet  MATH  Google Scholar 

  3. S Islam, A Ali, S Haq. A computational modeling of the behavior of the two-dimensional reaction-diffusion Brusselator system, Appl Math Model, 2010, 34(12): 3896–3909.

    Article  MathSciNet  Google Scholar 

  4. R C Mittal, R Jiwari. Numerical solution of two dimensional reaction-diffusion Brusselator system, Appl Math Comput, 2011, 217(12): 5404–5415.

    MathSciNet  MATH  Google Scholar 

  5. R Jiwari, J Yuan. A computational modeling of two dimensional reaction-diffusion Brusselator system arising in chemical processes, J Math Chem, 2014, 52: 1535–1551.

    Article  MathSciNet  Google Scholar 

  6. W R Holmes. An efficient, nonlinear stability analysis for detecting pattern formation in reaction-diffusion systems, Bull Math Biol, 2014, 76(1): 157–83.

    Article  MathSciNet  Google Scholar 

  7. J D Murray. Mathematical biology II: Spatial models and biomedical applications, Berlin, 2003.

  8. K M Owolabi, K C Patidar. Higher-order time-step** methods for time-dependent reaction-diffusion equations arising in biology, Appl Math Comput, 2014, 240: 30–50.

    MathSciNet  MATH  Google Scholar 

  9. P Gray, S K Scott. Autocatalytic reactions in the isothermal, continuous stirred tank reactor: isolas and other forms of multistability, Chem Eng Sci, 1983, 38(1): 29–43.

    Article  Google Scholar 

  10. P Gray, S K Scott. Autocatalytic reactions in the isothermal, continuous stirred tank reactor: oscillations and the instabilities in the system A + 2B → 3B, B → C, Chem Eng Sci, 1984, 39(6): 1087–1097.

    Article  Google Scholar 

  11. A M A El-Sayed, S Z Rida, A A M Arafa. On the Solutions of the generalized reaction diffusion model for bacteria growth, Acta Appl Math, 2010, 110: 1501–1511.

    Article  MathSciNet  Google Scholar 

  12. S Rida, A Arafa, A Abedl-Rady, H Abdl-Rahaim. Fractional physical differential equations via natural transform, Chin J Phys, 2017 55: 1569–1575.

    Article  Google Scholar 

  13. A Arafa, G Elmahdy. Application of residual power series method to fractional coupled physical equations arising in fluids flow, Int J of Diff Eq, 2018, 2018, 10 pages, doi: https://doi.org/10.1155/2018/7692849.

    MathSciNet  MATH  Google Scholar 

  14. A A M Arafa, A M S H Hagag. Q -homotopy analysis transform method applied to fractional Kundu-Eckhaus equation and fractional massive Thirring model arising in quantum field theory, Asian-Europ J Math, 2019, 12: 11 pages, doi: https://doi.org/10.1142/S1793557119500451.

    Article  MathSciNet  Google Scholar 

  15. S Z Rida, A M A El-Sayed, A A M Arafa. Effect of bacterial memory dependent growth by using fractional derivatives reaction-diffusion chemotactic model, J Stat Phys, 2010, 140: 797–811.

    Article  MathSciNet  Google Scholar 

  16. V Gafiychuk, B Datsko. Stability analysis and oscillatory structures in time fractional reaction-diffusion systems, Phys Rev E, 2007, 75: 055201, doi: https://doi.org/10.1103/PhysRevE.75.055201.

    Article  Google Scholar 

  17. B I Henry, T A M Langlands, S L Wearne. Turing pattern formation in fractional activator-inhibitor systems, Phys Rev E, 2005, 72: 026101, doi: https://doi.org/10.1103/PhysRevE.72.026101.

    Article  MathSciNet  Google Scholar 

  18. J Singh, M M Rashidi, D Kumar, R Swroop. A fractional model of a dynamical Brusselator reaction-diffusion system arising in triple collision and enzymatic reactions, Nonlinear Eng, 2016, 5: 277–285.

    Google Scholar 

  19. A Tahavi, A Babaei, A Mohammadpour. Analytical approximation solution of a mathematical modeling of reaction-diffusion brusselator system by reduced differential transform method, J Hyper, 2014 3: 116–125.

    MathSciNet  MATH  Google Scholar 

  20. M Y Ongun, D Arslan, R Garrappa. Nonstandard finite difference schemes for a fractional-order Brusselator system, Adv Difference Equ, 2013, 2013, 102, doi: https://doi.org/10.1186/1687-1847-2013-102.

    Article  MathSciNet  Google Scholar 

  21. H Jafari, Abdelouahab Kadem, D Baleanu. Variational Iteration Method for a Fractional-Order Brusselator System, Abst Appl Anal, 2014, vol 2014, 6 pages, doi: https://doi.org/10.1155/2014/496323.

    MathSciNet  MATH  Google Scholar 

  22. H Khan, H Jafari, R Ali Khan, H Tajadodi, S Jane Johnston. Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials, The Scient World J, 2014, 2014, 7 pages, doi: https://doi.org/10.1155/2014/257484.

    Google Scholar 

  23. E Pindz, K M Owolabi. Fourier spectral method for higher order space fractional reaction-diffusion equations, Commun Nonlinear Sci Numer Simul, 2016, 40: 112–128.

    Article  MathSciNet  Google Scholar 

  24. A A M Arafa, S Z Rida, H Mohamed. Approximate analytical solutions of Schnakenberg systems by homotopy analysis method, Appl Math Model, 2012, 36: 4789–4796.

    Article  MathSciNet  Google Scholar 

  25. A A M Arafa, A M S H Hagag. A new analytic solution of fractional coupled Ramani equation, Chin J Phys, 2019, 60: 388–406.

    Article  MathSciNet  Google Scholar 

  26. A A M Arafa, S Z Rida, M Khalil. The effect of anti-viral drug treatment of human immunodeficiency virus type 1(HIV 1) described by a fractional order mode, Appl Math Model, 2013, 37: 2189–2196.

    Article  MathSciNet  Google Scholar 

  27. A A M Arafa, S Z Rida, A A Mohammadein, H M Ali. Solving nonlinear fractional differential equation by generalized Mittag-Leffler function method, Commun in Theor Phys, 2013, 59: 661–663.

    Article  MathSciNet  Google Scholar 

  28. A A M Arafa, M Khalil, A Sayed. A Non-Integer Variable Order Mathematical Model of Human Immunodeficiency Virus and Malaria Coinfection with Time Delay, Complexity, 2019, 2019, 132 pages, doi: https://doi.org/10.1155/2019/4291017.

    Article  Google Scholar 

  29. M A Bayrak, A Demir. A new approach for space-time fractional partial differential equations by residual power series method, Appl Math Comput, 2018, 336: 215–230.

    MathSciNet  MATH  Google Scholar 

  30. M A Bayrak, E Ozbilge. A New Approach for the Approximate Analytical Solution of Space-Time Fractional Differential Equations by the Homotopy Analysis Method, Adv Math Phys, 2019, vol 2019, 12 pages, doi: https://doi.org/10.1155/2019/5602565.

    MathSciNet  MATH  Google Scholar 

  31. R Khalil, M Al Horani, A Yousef, M Sababheh. A new definition of fractional derivative, J Comput Appl Math, 2014, 264: 65–70.

    Article  MathSciNet  Google Scholar 

  32. O S Iyiola, O Tasbozan, A Kurt, Y Cnesiz. On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion, Chaos Solit Frac, 2017, 94:1–7.

    Article  MathSciNet  Google Scholar 

  33. H Thabet, S Kendr. Analytical solutions for conformable space-time fractional partial differential equations via fractional differential transform, Chaos Solit Frac, 2018, 109: 238–245.

    Article  MathSciNet  Google Scholar 

  34. Emrah Unal, Ahmet Gokdogan. Solution of conformable fractional ordinary differential equations via differential transform method, Optik, 2017, 128: 264–273.

    Article  Google Scholar 

  35. A Kurt, H Rezazadeh, M Senol, A Neirameh, O Tasbozan, M Eslami, M Mirzazade. Two effective approaches for solving fractional generalized Hirota-Satsuma coupled KdV system arising in interaction of long waves, J Ocean Eng Sci, 2019, 4: 24–32.

    Article  Google Scholar 

  36. O Tasbozan, M Şenol, A Kurt, D Balean. Analytical and numerical solutions for time-fractional new coupled mKDV equation arising in interaction of two long wave, Asia Pac J Math, 2019, 6:3, doi: https://doi.org/10.28924/APJM/6-13.

    Google Scholar 

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

    Article  Google Scholar 

  38. M Senol, O Tasbozan, A Kurt. Numerical Solutions of Fractional Burgers Type Equations with Conformable Derivative, Chin J Phys, 2019, 58: 75–84.

    Article  Google Scholar 

  39. A El-Ajou, O Abu Arqub, S Momani, D Baleanu, A Alsaedi. A novel expansion iterative method for solving linear partial differential equations of fractional order, Appl Math Comput, 2015, 257: 119–133.

    MathSciNet  MATH  Google Scholar 

  40. S Das. Functional fractional calculus, Springer Science Business Media, 2011.

  41. I Podlubny. Fractional Differential Equations, Academic Press, New York. 1999.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Anas Arafa.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arafa, A. A different approach for conformable fractional biochemical reaction—diffusion models. Appl. Math. J. Chin. Univ. 35, 452–467 (2020). https://doi.org/10.1007/s11766-020-3830-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11766-020-3830-5

Keywords

MR Subject Classification

Navigation